The Basics: Defining Impulses on the Turning Ball

To start, some bookkeeping. Below is the trajectory of a spinning delivery (Ashwin to Pooran in the 2024 IPL). The ball leaves the hand at some angle across the pitch. This angle changes due to drift in the air before the ball pitches. Upon pitching, interaction with the pitch and the revs on the ball make it turn away from the line before bouncing. In terms of physics, the ball has received an “impulse” from the act of bouncing on the pitch, which is technical terminology for getting a change in speed due to some force. This impulse changes the line of the ball, as can be seen from the top-view of the pitch here. It also changes the vertical speed of the ball, which is not discernible from this view.

We can write this “impulse” as three components that are calculated by a change in velocity due to pitching, computed along three orthogonal dimensions:

1. Along the pre-pitching line of the ball (blue arrow): a result of the motion of the ball in the air, and the revs imparted on it (the direction of the seam and how hard the ball is spinning). We call this dv_par or J_par.

2. Perpendicular to the pre-pitching line (orange arrow): this is entirely due to the revs on the ball, the impulse in the direction normal (at 90 degrees) to the direction in which the ball was travelling before bouncing. We call this dv_perp or J_perp.

3. In the vertical direction: the change in vertical speed. We call this dv_z or J_z.

Here is how J_par and J_perp are related to turn in degrees:

We see here that the change in velocity along the line of pitching is always negative (the ball loses speed), and is correlated with the turn.

In the second plot, we have J_perp, the change in speed perpendicular to the pitching line, which can go in either direction (two directions of spin). This is very tightly correlated with the turn. Now, if we plot both against J_z, we see that both are proportional to J_z:

This tells us that the force responsible for “spin” is friction, which is always proportional to the normal force. This normal force is represented by J_z in our setup. The more the change in vertical velocity, the higher the friction in general, giving higher grip off the pitch.

To nullify the effect of J_z on the other Js, we divide J_par and J_perp by J_z to get the “grip”:

• line_grip = |J_par| / J_z: grip in the direction of pre-bounce travel of the ball.

• turn_grip = |J_perp|/J_z: grip in the direction perpendicular to the travel of the ball.

We see that both these are proportional to each other. That makes sense, they are both resulting from the same underlying mechanism, broadly: friction. We also notice that the turn grip is roughly 0.2-0.3x the line grip. Despite this lower magnitude, the turn_grip, or the change in speed perpendicular to the ball’s travel, is mainly responsible for turn.

Turn is the change in the angle of the ball upon pitching. This is influenced by both the parallel change in speed and the perpendicular change, but the former has a much weaker effect. This can be shown by expressing the formula for turn in terms of J_par and J_perp, which finally tells you that the relative sensitivity of turn to J_perp vs J_par is proportional to the turn itself! At 5 degrees of turn, the turn is 11x more sensitive to J_perp than J_par! We shall see this in our results soon!

What Influences Turn?

Here, I will show you four plots, plotting mean turn vs different kinematical quantities of the delivery before bounce. This helps us explore the physical relationships between the trajectory of the ball and the resulting turn.

Top Left: Vertical Acceleration

As you go rightward on the x-axis, the vertical acceleration decreases in magnitude. The “dip” due to the overspin and the Magnus Force decreases, the ball falls “flatter” into the pitch. We see that in general, less dip = less turn. We have to be careful here: there is a confounding variable. Bowlers usually slow it down when they bowl with more dip, so this high turn and high dip is a result of both high dip and lower speeds on average.

Top Right: Sideways Acceleration

The sideways acceleration measures the force on the ball that causes drift in the air. My initial thought was that drift should be proportional to turn, which is what comes out of this “averaged” plot. My friend Aaron Briggs, cricket ball dynamicist and ECB analyst, maintains that the “drift” we see on spinning balls is mostly due to swing and not the Magnus effect. When you check the dependence of this lateral acceleration on the forward release speed, a neat proportionality emerges, lending credence to his idea. Nevertheless, we see here that the turn holds a nice linear relationship with the drift, indicating that the sidespin revs that (possibly) cause drift also have a role to play in turn.

Bottom Left: Forward Speed

This one is pretty obvious: the turn decreases as the ball speed in the forward direction increases. Turn is, after all, a change in direction of the ball. The direction is the ratio of the forward to sideways speed. A higher forward speed means you need a higher sdieways speed change to change the angle upon pitching. This leads to lower turn.

Bottom Right: Downward Speed

This one is cognate with the top-left: a higher downward speed before bouncing leads to more turn as the ball bounces into the pitch at a steeper angle, hitting it harder. This makes for a higher J_z (normal force) and higher grip.

Three Other Factors

The above four plots show the effect of the primary properties of the pre-bounce trajectory on turn. In addition to these, we have three other factors:

1. Bowler identity: this captures the skill of an individual bowler that is not captured in the pre-bounce drift / speed numbers in a simple manner.

2. Innings: the condition of the pitch changes across the Test match, with the turn increasing monotonically in most countries as the match progresses.

3. Ball condition: the state of the ball (new or old). We do not have any data for this.

4. Ground identity: the most important, the contribution of the specific ground.

The Turn Model

Model

Our primary objective in this work is to establish a method to rank individual grounds on the basis of the “turn” they offer. The naivest approach to this would be calculating the average turn on a ground, but this leads to spurious results. Grounds in SENA nations where a few, very competent spinners have bowled only on days 4 and 5 get a very high ranking by this method. For instance, this method tells you that Australian pitches have the most turn, which clearly goes against cricket logic.

A smarter way to do this pitch rating is to take “known turners”, spinners who have bowled a lot on different grounds. By comparing a spinner’s baseline turn vs the turn at a given ground and systematically summing these ratios up, we can arrive at a +/- turn rating for every ground.

A more systematic way of doing this while also accounting for the effects of the pre-bounce trajectory and the bowler is using a mixed effects linear model to predict the turn. In our model, we have four continuous trajectory variables and one categorical variable, which make up the fixed effects:

1. magnus_z: pre-bounce vertical acceleration with gravity subtracted, in m/s^2. +ve is dip.

2. vz0: magnitude of pre-bounce vertical velocity in m/s

3. vx0: magnitude of pre-bounce forward velocity (along the pitch towards the batter) in m/s

4. abs_ay0: magnitude of pre-bounce sideways acceleration in m/s^2.

5. Inns: 1,2,3,4

In addition, we have two categorical variables encoding random effects:

6. Bowler

7. Ground

The former 5 variables give us one number for each variable: a global, average effect encoding the relations between the variables and turn. The latter two variables give us one number for each bowler and ground, treating both as random variables.

Result Summary

A few points from this fitting summary table:

• The innings effect: The inns coefs show how much turn changes by the inns, with inns=1 being the baseline. The turn systematically increases as the match goes deeper. The second inns shows 0.23 degrees more turn than the 1st. The 3rd shows 0.28 degrees more than the second, and so on.

• Magnus_z: a small positive coef, which matches our earlier visual inpection: more dip = more turn. Keep in mind that this model already subtracts the effects of slower speeds and higher vz0 at the linear level, so this coef encodes a relatively uncontaminated effect of dip on turn.

• abs_ay0: This too confirms our visual inspection. The turn degrees change by 0.515 per unit of sideways acceleration. A deeper debate with Aaron pending, we shall for now take this as the effect of sidespin on turn.

• vx0: a negative coef, implying that a higher forward speed leads to lower turn, as we saw earlier.

• vz0: positive coef, showing that a ball banging harder into the pitch gets more turn. This coef mostly absorbs the effect of magnus_z, because the vz0 is the downward speed of the ball right before bouncing, which is influenced by vertical acceleration.

• Bowler variation: the spread (variance) of the turn^2 resulting from the variability in individual bowlers.

• Ground variation: the same as above for individual grounds.

• Share of varation: The ‘scale’ = 2.2545 is the variation in turn outside of the variation caused by the bowler and the ground. The total ball-by-ball variance in turn is 2.25 + 0.51 + 0.38 = 3.14.

• Bowlers explain 0.51 / 3.14 = 16.2% of the variance.

• Grounds explain 0.38 / 3.14 = 12.1% of the variance.

Test

The above results were from the model fit on the training set based on a 75:25 train-test split. Here is a calibration plot on the test set. We see the prediction works well on average, with a slope close to 1 and intercept close to 0.

Ranking Grounds

With this. we can now extract the coeffs for each ground, having accounted for the pre-bounce trajectory and the bowler’s individual characteristics. With the coeffs, we can compute the “average” turn on a “typical” ball on each ground. We take the means of the 4 trajectory variables and take a weighted mean of the inns coeffs from our model to predict the turn (setting bowler effects to zero). The following plot results. It shows the average expected turn by ground:

The placement of most grounds on this ranked list makes sense. Optus, Nottingham, Perth are the lowest ranked. Aus and NZ grounds populate the lower half. “Flat” or “slow turn” pitches in Asia are also seen in the lower half of the table. At the top, you have Dominica, Wankhede, PSC, SSC, Galle, Dubai, Chennai, Chittagong, Rajkot, Dhaka etc. The ordering is far from a perfect match with our colloquial, anecdote-based rankings, but it broadly does match up. There are a few SA grounds in the top half, which I can only put down to selection effects. Delhi (Kotla) is the other disquieting entry, placed firmly in the lower half. These rankings are correlated with the bare rankings based on average turn (Delhi sits in the bottom 20 there too), but nevertheless come from a more rigorous method that attempts to account for bowler identity and ball trajectory.

Extending the Model

Next, let’s fit a similar model to predict both line_grip and turn_grip. We will extricate the ground coeffs from both the line_grip and the turn_grip fitting and rank the grounds based on these, along with ranking them on the average turn.

The plot below shows the correlations of the line_grip and turn_grip ranks with the turn ranks:

We first see that both the ranks are well-correlated with the turn (deviation) rank, but the sideways grip rank has a much tighter relation with turn rank. This goes back to our point that sideways impulse impacts turn a lot more than line impulse. The correlations of the ranks are:

line-grip vs deviation    : 0.867
sideways-grip vs deviation: 0.967

We can take the line_grip and turn_grip coefficients from these latest fits and plot them on a plane for each ground:

As is apparent, these two coeffs are correlated (correlation coeff of 0.78), which points us to our very first plots showing the high correlation between sideways and line grip. What we need to do is rotate these points to get a new scatter plot. This rotated scatter plot will show us two numbers: one along the common, shared, trendline (dashed line), and one along the direction perpendicular to it. Thus, a rotation will “decorrelate” the two dimensions. Here is the rotated plot:

The x-axis shows you the total grip: the distance of the point along the dashed line in the first plot. The higher this number is, the more “grippy” a surface is for spin. The y-axis shows you something uncorrelated with this total grip: it shows you the propensity for sideways impulse off the pitch. In terms of the first plot, it shows you how far away the point is from the dashed line. The colour shows you the “turn effect” on the ground: how much the ball turns compared to the average ground in the dataset.

This plot confirms that turn is mostly a result of the overall grip of a surface (x-axis): the colour gradient goes from left to right in the plot. The y-axis presents an interesting decomposition, telling us the relative importance of the two mechanisms causing the turn. A higher-than-zero turn bias implies that sideways grip plays a significant role in the generation of turn on a pitch. This half of the plane has mostly WI and Asian pitches. The lower half, with negative turn bias, has pitches that rely on the robbing of forward pace for turn instead of sideways impulse. That does not mean that sideways impulse is the less dominant factor. Sideways pace is still dominant over line-impulse. However, the lower-half grounds see line impulse act significantly.

Perspectives

This was an unstructured wandering into the realm of pitches and how they spin. The idea was to connect the anecdotally known and discussed nature of pitches with respect to spin with some hard numbers. This is part of a broader research movement that seeks to codify the character of pitches using ball-tracking data to make rigorous connections between the two.

This methodology is by no means complete, but my hope is it will serve as a launchpad for more detailed studies into pitch character. One of the possible applications of this is to build a pitch rating system that runs live based on the fits done above, furnishing a “turn rank” as the match goes on, akin to PitchViz.

The question of the pace and bounce of a cricket pitch has gnawed away at me (and a few other cricket analysts) for a very long time. The traditional appellation of “slow” pitches evokes ideas of a surface on which the pace of the ball towards the batter is lost upon pitching. How much does the data verify this? What really is the “bounce” of a pitch? I hope to jot down my preliminary explorations of these questions here.

At the outset, let me state that the “trueness” of a pitch is a related but different matter: it depends on the variance in the pace, turn, and bounce, especially within one game or innings. Here, I will talk mostly about mean values of “pace” and “bounce” and how the perception of a pitch relates to these.

First, a few definitions. Below is a picture of the cricket pitch seen from front-on with three axes. The x-axis goes down the pitch, the y-axis goes across the pitch (y increases from left to right from the umpire’s persepctive), and the z-axis is height.

We have information about the ball’s velocity right before and right after bouncing. I will be using the terms vx0, vy0, vz0 to denote the three components of pre-bounce velocity, and vx1, vy1, vz1 to denote the corresponding post-bounce values. First, we have the coefficient of restitution (COR), the ratio of the post-to-pre-bounce z-velocity:

This just tells us how much vertical velocity a ball retains upon pitching. For the velocity loss in the x- and y-directions, we have the corresponding quantity which I name COX: the ratio of post-to-pre velocities along the pre-bounce line of the ball. For simplicity, assume that the ball has no change in y-velocity (no seam movement or turn). Then,

Our actual definition of COX will involve both x and y velocities, and keep the fraction of speed along the ball’s pitching line. COX tells you how much speed is retained in the direction along the pitch.

So we have the pair of COR and COX, which tell us about the fraction of velocity retained along the vertical direction and in the x-y plane (forward and sideways). Another useful quantity is the ratio of angles post-and-pre-bounce:

This just tells us the steepness of the bounce relative to the angle at which the ball drops to the ground. In terms of the velocities (again assuming v_y = 0 for simplicity), we have

The above fraction is approximately equal to (vz1/vz0) / (vz1/vx0), which of course is COR/COX. This shows that the steepness of bounce on a pitch is a combination of both forward pace and vertical pace.

You can get steep bounce by either high COR or low COX. Both cases will lead to high bounce, and this makes sense. Imagine a pitch A on which the ball loses more forward pace than on pitch B. The ball takes a longer time to reach the crease on A, rising more than on pitch B by the time the batter makes contact. Although pitch A has lost more pace, the batter perceives pitch A to have more “bounce”, even if the COR (vertical speed) on both pitches is the same. The height at which the ball reaches the stumps depends on the post-bounce angle and not the forward or vertical speeds alone.

Now let’s have a look at pitches in different countries, considering only pace balls faster than 128 kmph. Our analysis will be restricted to these, since both slower balls and spinning balls live in an entirely different regime, needing slightly different surface contact physics.

Below, I plot the average angle ratio (x axis) and the average fraction of speed lost upon pitching (y axis), for 5-10m length pace balls. Both these quantities are related to the length, but I confirmed that restricting to different length bands within the 5-10m region also gives similar results.

We can see three (excluding England) distinct clusters here. At the top right we have the “pace friendly” countries: NZ, SA, WI. They have a high angle ratios: the ball bounces steeply here. In contrast, the 5 Asian venues have low angle ratios — low bounce. The surprising revelation is on the y-axis. It is seen that the high-bounce countries lose the most pace off the pitch. This is highly counterintuitive to the colloquial labeling of these pitches. They are supposed to be “fast” and “quick”, but the data points us in the opposite direction. The “harder” surfaces outside Asia and England absorb more energy from the ball and bounce higher. In fact these two phenomena are related (more on that later). Finally, we see UAE and Pak live in their own little cluster: they lose a lot of pace and have low bounce angles — truly “low and slow”.

Let’s now break the angle ratio apart into COR and COX. First, here are the distributions of both:

You’ll notice that COR has a much larger range than COX — it varies from 0.5 to 0.8, whereas COX varies from 0.875 to 0.975. We can reason that the variation of COR is a bigger driver in the variation of angle ratio.

COR and COX are not exactly independent of the incoming speed. After all, a ball that bangs harder into the pitch elicits a harder response. This is shown in the next two plots, where I have shown the mean COR and COX (normalised by the population COR and COX) in bins of vz0 (rightward is lower speed).

The direction of the curves makes sense: banging the ball harder into the surface leads to a lower relative retention of speed. However, the variation is not very steep for either. They both move about 0.4 σ for a wide range of vz0. Nevertheless, let’s restrict ourselves to the 6-8 m length band and plot the mean COR and COX in different countries:

Again, we see the same pattern and the same clustering. NZ, SA, WI are the “fast” countries: they show high vertical speed retention (high COR) but also show low COX (high amount of forward speed lost). Ind, SL, Ban lie in their own cluster close to UK, with lower COR (lower than the full set mean) and high pace retention. UAE and Pak lie at very low COR and low COX — again truly low and slow.

The remarkable thing in the above plot is the loose negative correlation: high COR correlates with low COX. Pitches that absorb more vertical velocity absorb less forward velocity. From a surface physics perspective, this makes sense. On harder pitches, the ball rebounds more vertically, the change in vertical velocity is higher. A ball that falls at 8 m/s and rebounds at 6 m/s changes speed by 14 m/s, with a COR of 0.75. If the same ball rebounds at 5 m/s, the change is lower (13 m/s) with a COR of 0.625. A higher COR means a greater change in vertical speed, which means longer contact with the pitch. In physics terms, a higher COR implies a larger normal impulse. This larger normal impulse means that the frictional impulse is higher, robbing the ball of more of its forward speed (remember that Friction Force is proportional to Normal Force; higher normal force = more friction = more loss of forward speed).

So, a harder pitch leads to higher bounce (the ball reaching the batter higher) in three correlated ways:

1. The COR is higher — the ball retains more of its vertical speed

2. The COX is lower (weakly correlated) — the ball loses more of its forward speed

3. And, in combination, the post-bounce angle (COR/COX) increases due to both of the above, leading to a greater height at the crease.

So, contrary to perception, pitches in NZ/WI/SA lose more pace in the forward direction than Asian pitches. They also have higher vertical speed retention. Both these come together to make the ball reach the batter higher. It is this perception that is called the “bounciness” of a pitch.

We can quantify this with solid data. The average pace-on ball landing close to the 7m mark reaches the pitching point at a forward speed of 34.28 m/s (vx0) and a downward speed of 7.26 m/s (vz0). For a fixed length, we can assume that COR and COX are independent, COR is independent of vz0, and COX is independent of vx0. With the trajectory equations (below), we can plot the trajectory of the post-bounce path for any COX and COR values.

where all v_i and a_i are post-bounce values. a_x is set to the median value in the set, a_y = 0 (no swing after bounce). We obtain v_x1 = v_x = v_x0 COX and v_z1 = v_z = v_z0 COR for computing the trajectory using the equations.

We plot the projected height at the stumps and the time to stumps from pitching, vs the normalised COX ([COX - Mean COX] / Std(COX)).

Two things to see here:

1. A 2σ variation in COX changes the projected height by less than 3 cm.

2. The same 2σ variation in COX changes the time to stumps by a mere 0.02s.

Let’s see the projected height plot for COR (the time to stumps is independent of COR):

A 2σ variation in COR produces a whopping 20cm difference in the height at stumps.

COR (vertical speed) is the predominant driver of where the batter sees the ball approach them. Fast pitches (NZ/SA/WI) make the ball lose a lot of its forward speed. They are not “fast” in the usual sense. But the same mechanism drives these pitches to kick the ball higher vertically. Crucially, both these effects work to increase the height of the ball.

My sense is that this “perception of bounce” is called the “quickness” of a pitch. A “fast” pitch is a bouncy pitch, a “slow” pitch is a low pitch.

There’s much else to be done in this regard, but I cannot go into deeper detail in a public post. I will end this piece with a plot of the projected height at the stumps and the time taken to reach the stumps on the "average pitch” in different countries:

We have three distinct clusters, with England sitting somewhere in the middle of everything.

1. The “bouncy” countries: pace-friendly (SA/NZ/WI), high height at stumps, longest time to get there;

2. The “low” countries: Ind/Ban/SL, the ball arrives at a ~5cm lower height than in the “bouncy” cluster. The ball arrives about 0.004 s faster.

3. The “slow-low” countries: Pak/UAE, where the ball arrives 15 cm lower than in the bouncy countries, but also takes a bit of time to arrive.

Conclusion

A “slow” pitch is mostly a low pitch. The final plot above demonstrates that between pitches which lose a lot of forward pace and those that do not, the reaction time-difference for a batter is a measly 0.004 s. Now, the actual reaction times will have a spread, but this small a variation is not perceptible or actionable for humans. This leads me to conclude that it is not the time, but the height at which the ball arrives, which determines how “fast” a pitch is to a batter. A few points:

• Pitches show a much greater variation in stumps height than in the time-to-stumps. A 5cm difference in height is the difference between a thick edge and a nick that goes to the keeper. A 0.01s difference in reaction time is mostly imperceptible.

• The bat comes in a natural downward motion, which means adjusting to height variation is extremely difficult. My conjecture is that a “lower” pitch means the ball is met at a closer trajectory to the downward swing, making it appear “easier” or “not quick”. High bounce pitches demand adjustment, or quicker reactions, giving them the “quick”moniker.

• Another conjecture is that pitches with higher stumps height make the ball hit the bat higher, jarring it more, making it feel “heavier”, giving the impression that the ball loses less pace.

There is much confusion among cricket parlance about the character of a pitch. This is a preliminary investigation into a thorny and deep question: what makes a “fast” pitch? This question will need more research in the coming years if people are willing to do it. But for now, I believe that a low-bounce pitch is often called “slow”, and the simple definition of a pitch losing lower pace in the forward direction does not hold for traditionally labelled “fast” pitches.

To paraphrase my friend Kartikeya Date: in the longer formats batter plays the ball, while in T20 they play the field. As the common sense of the player pool in the shortest format rapidly imbibes fast-scoring, batters are looking for any and all advantages in their pursuit for boundaries. In the last 14 overs of an innings, this pursuit can take two forms: six-hitting or gap-finding. The former needs raw power and the ability to hit across lengths, whereas the latter takes finding the one undefended boundary with 5 fielders outside the inner ring.

The batter who can most successfully find that one gap in the boundary is one who possesses multiple shot options for the same ball. In my view, this is the future of T20 hitting and batter analyses. There have already been efforts to this end with the availability of wagon wheel and shot zone data. Most prominently, Arnav Jain’s all-encompassing app attempts to do this via an array of metrics.

The app has an intelligent wagon wheel that weights a batter’s wagon wheel by the rarity and relative SR of the zones accessed, a sector importance map, a nifty “360 score” that uses the angular variance of a batter’s wagon wheel spokes, and a simple but useful “field coverage”. The last metric asks how much of a batter’s runs can be covered using “n” outfielders — essentially telling you how many runs you leave undefended if you cover the batter’s “n” best zones. The app also paints an ideal field setup for a given batter. All these are extremely useful tools for both batters and fielding teams planning to thwart them.

Let’s look at this formally using the distribution of shot zones after weighting by line and length. Arnav’s aforementioned app already does this, but I want to explore a breakdown.

We want to compute the zone distribution of the “average” batter given the lines and lengths faced by our chosen batter. For now, let us consider only IPL 2025-26 data, counting only controlled scoring shots vs pace. We are considering 8 equal field zones. For each line (lin) and length (len) combination, we first construct the distribution (q) of wagon zones (z) for the pool of all batters:

Now, let’s take a batter and get the distribution of lines and lengths he has faced over the time period we are covering, and call this w(lin,len). Using this and q(z | lin,len), we construct a “counterfactual” distribution of zones:

This is the distribution of zones if an average batter from our pool had faced the same lines and lengths. In a similar vein, we can compute the counterfactual runs-per-ball in each zone from the average batter facing the same lines and lengths:

where R(z | lin,len) is the average runs per ball in zone “z” from balls of line “lin” and length “len”. R_cf(z) weighs this average run distribution by the proportion of balls faced by our chosen batter and the zones the ball would have gone to, if the average batter were facing them. Mathematically,

• the w(lin,len) part tells you about the distribution of lines and lengths faced by the chosen batter;

• the q(z | lin,len) part translates that to the wagon zone distribution that the average batter would have sent these balls to;

• finally, the R(z | lin,len) tells you the average runs scored in each zone from each line/length combo.

We now have two “counterfactual” arrays: 1) the zone distribution of the average batter, and 2) the runs per ball in each zone from the average batter, both if they had faced the same balls as our chosen batter. These can be translated into a field plot like the one below, which shows a comparison on both these metrics for Abhishek Sharma.

The left plot shows the % distribution of zones accessed, with blue showing Abhishek’s distribution, and green showing the average batter’s. All batters have been converted to RHB for this. You can see that Abhishek accesses the offside (two sectors on the left of the plot) more than the other batters given the ball areas he faces. The dashed circle in the left panel shows the 12.5% level: this would be the distribution if all zones were uniformly accessed. On the right you can see a similar comparison of his SR in different zones, with the white circle showing 200 SR just as a guide.

Below is the same plot for Vaibhav Sooryavanshi vs pace. As I discussed in my recent piece on ESPNcricinfo, he accesses the straight legside V much more often than other batters due to his unique backlift, bend of the back, and downswing.

These figures can be a useful starting point for not just looking at strong and weak zones of batters, but also beginning to quantify the “diversity” of a batter’s shot options. For instance, in the two plots above, we can see that the bulk of both Abhishek Sharma’s and Sooryavanshi’s scoring is in 2-3 zones. Their SRs exceed the field’s SR by far, but their target areas are 3 zones. Compare this to Prabhsimran Singh’s zone breakdown, which seems to be more “evenly” distributed.

We are already getting a sense of different “styles” of batters: Abhishek and Sooryavanshi target fewer areas compared to Prabhsimran but they gun for maximising those areas. How can we formalise this notion?

Enter entropy. Entropy is commonly and colloquially called “disorder”, borrowing from its meaning in the thermodynamics / statistical physics sense. However, the same entropy plays a vital role in information theory, where it encodes the “ability to surprise” of a given distribution. A sharply lopsided distribution has low entropy. Let’s take a hypothetical batter who plays just one shot. When they play a shot, no opposing captain is surprised. Compare this to another hypothetical batter who plays 7 shots with equal probability: the fielding captain is unsure about which shot will be played, and is “surprised” more than they were facing the one-shot batter. The idea is that low-probability events possess more “information”. Entropy codifies this concept.

The formula for the entropy H(p) of a distribution p is

The entropy is is the expected “information” in a distribution, where the information (surprisal) is log(p).

(thanks to Omkar Walunj for pointing me in this direction as he was trying to replicate Cricviz’s “innovation” metrics)

Now, what does H(p) tell us about a batter? Let’s pivot to the distribution of shots for the batter. If the batter has just 2 shots, the entropy is low, since there are only two options, both with high probability. In contrast, if he has 8 shots, the entropy is high (the -log(p) term is high since the probability of each kind of shot is a smaller number). The 8-shot batter, one with more options, has a higher entropy. His shots can “surprise” the opposing team a lot more than the batter with just two shots.

So, the entropy of a batter’s shot distribution tells us the inherent unpredictability of a batter’s options.

A related number is the cross-entropy. The entropy H(p) told us about the average surprise of a batter’s shots, considering his own shot distribution. The cross-entropy, on the other hand, is

where p_i is the batter’s shot distribution and q_i is the average batter’s shot distribution. This quantity is weighing the batter’s shot options by the general rarity of shots. It can be increased by either of the two components of the product: if a batter plays a shot very often (high p_i), or/and if the shot is rare (low q_i).

Cross-entropy tells us the average unpredictability in a batter’s shots given we only know the average batter’s shot distribution. If I as a captain had studied only the average IPL batter, how much would this batter’s shots surprise me?

This kind of categorical cross-entropy is often used in deep learning applications when building multi-class classifiers.

A third, related quantity is the Kullback-Leibler Divergence (KL Divergence). This is simply the difference between the cross-entropy and self-entropy:

What does this tell us about a batter? A high D_KL means that the opposing captain stands to gain specific information about a particular batter by studying the batter’s patterns. In other words, the batter is unusual compared to the other batters in very specific ways. Doing your “homework” can give you substantial advantages in setting a field for a high D_KL batter. A low D_KL batter does not have any individual patterns in his shotmaking.

So, in summary:

• High entropy H(p): batter’s own shotmaking is inherently unpredictable

• High cross-entropy H(p,q): batter’s shotmaking is unusual compared to the average batter / the batter is difficult to set a field for if you have not studied him individually

• High D_KL: specific homework can close the gap.

Let’s calculate this for IPL batters. We will do a more nuanced computation, since the shot hit is a function of line and length. For each batter, we will compute the entropy, cross-entropy and D_KL for each line and length combo, comparing to the average batter’s shot distribution for that line/length bin. We will then find the weighted mean of these three quantities for the batter, weighted by the number of balls faced by the batter.

Here is a plot of the D_KL (y axis) vs Entropy (x axis):

We have four quadrants here, divided using the medians of the entropy and D_KL. It’s worth exploring the cricketing meaning of each of these.

1. Top-right: high-entropy, high D_KL. These batters have a wide range of shots, and their shots are unusual compared to the other batters given the lines and lengths they face. Suryakumar Yadav and Rishabh Pant are typical of this kind of batter.

2. Top-left: low entropy, high D_KL. These batters have a small oeuvre of shots, but their shots are highly unusual compared to the average batter given the lines and lengths they face. They don’t have a lot of shots, but they need specific homework for field settings. Sunil Narine, Tim David and Ayush Mhatre are typical of this set: they have only a few shots (which they play well), but they can hit balls into their favourite areas from a variety of locations.

3. Bottom-left: low entropy, low D_KL. These batters have a small set of shots, like those in the top-left, but their shots are also “typically” distributed — close to the overall shot distribution given the balls they face. They do not need funky fields.

4. Bottom-right: high entropy, low D_KL. These batters have multiple options to the same ball, but their shot distribution is “usual”. These are classical batters and good pace hitters, as evidenced by the presence of Gill, Kohli and Rahul in there.

Here is a companion plot showing Entropy vs “score”. “Score” in this context is a weighted sum of the efficacy of each shot played by the batter for a line-length combination.

I would like to point out Vaibhav Sooryavanshi and Sunil Narine have low entropy (they have a small number of shots) but they execute them very well and have a high “score”. There are multiple ways to skin the cat when it comes to T20 hitting. Some, like Kohli and Gill, play a wide range of classical shots, sticking to conventional choices for the lines and lengths they face. Some, like Suryakumar play many shots but unconventionally. And some, like Sooryavanshi, play only a few shots, play them in zones like the average batter would, but don’t care about the field too much, executing them perfectly.

This was just a little idle pondering and Sunday doodling, and I don’t know how useful this can be, but I liked this as an academic tool to explore the “types” of batters by their field access. Batter profiling is mostly done by line/length outcomes, but field access is extremely important in T20s, and I hope this kind of analysis can enter the broader discussion. We do not possess fieldset data, which I think is a crucial component of T20 analysis going forward. While we wait for that, I think cricket analytics should start building the machinery to profile batters via their shots. After all, batter intent is the dominant determinant of a ball’s outcome in T20s (~65%).

The IPL does not exist to enrich the Indian cricket team, but rather as a private money-making entity. Despite this, it remains the best arena for Indian players to hone and showcase their skills at the shortest format, in addition to being the best audition for a spot on the India T20 team. How is it faring on the counts of preparing Indian players and showing their mettle for international cricket?

I first want to specifically talk about the recent turn of the IPL, with the Impact Player and increasingly flatter pitches.

It is clear that playing with 8-9 batters on account of essentially having 12-vs-12 cricket has freed up batters to play more aggressively. Strike rates have gone up across phases and across lengths over the last three years. Even the good length strike rate in the Powerplay in 2025 was around 150, up from about 120 three years ago. There are two aspects, linked in series, to hitting in T20: intent and execution. These two exist in a complex feedback loop: higher intent comes from better execution, but better execution is the result of more intent and more practice. The modification of the game state with the extra batter first led to higher intent, which led batters to try more attacking shots. This then gradually improved their execution as they played more and more attacking shots.

Most importantly, this feedback loop of trying to hit more and succeeding at it (due to flatter pitches) shifted the baseline of intent: batters now just know they can hit. In the early years of the IPL, conventional thinking held them back. Now, the range of possibilities has increased — batters have seen clearly that higher strike rates are possible, which enables them to have more intent in general. Like a shifting of a “batting Overton window”.

The Indian team under Rohit Sharma and Rahul Dravid had already started moving away from staid batting in T20s in 2023. The Indian side in 2025 is a natural progression of that, with a steady feed of hitters with high intent supplied by the new avatar of the IPL.

But does this translate well to international cricket? Shot-making in T20 is a high-speed activity. As such, it is highly sensitive to minute variations in the pace and more importantly, bounce of a pitch. Playing on flatter pitches shifts the intent of Indian batters higher, but if the pitch deviates from the standard conditions seen in the IPL, high-intent shot execution takes a hit. Thus, the pitches in the IPL might be contributing a false sense of achievement when it comes to high-octane T20 hitting.

In this flawed translation from the IPL to the tougher pitches in T20Is too, there is a spectrum. This is purely conjecture, but I think that the stability of execution across conditions is a function of a batter’s baseline hitting ability and shot profile. To elaborate, I think the more you tend towards classical batting, the more your strike rates are inflated artificially by Impact Player / IPL pitches, making your execution less likely to translate to tougher conditions. As a more “classical” batter, you tend to rely more on conventional shots, which can be thwarted by movement and bounce coupled with accurate bowling. In the IPL, the former doesn’t exist for the most part, and the latter too is not very frequent. Thus, the differences between the elite and the average hitters are flattened.

My theory is that the only batters who can really translate hitting to tricky conditions are either extremely solid hitters (think Kohli vs pace) or those who have crossed the limits of high-intent batting (think Abhishek Sharma). In Abhishek’s case, the Hyderabad pitch has unlocked an intent so high that his shot profile and execution have improved to transcend tricky conditions. This is prompted by his knock at the MGC — which also involved some luck in terms of matchups — but his general performances in T20Is across slow and fast conditions lend some support to this very nebulous theory. The bottom line is that the advantage of high intent translating across conditions is like a step function: it increases massively after a high intent barrier. A large mass of batters in the middle of the distribution have simply upped their strike rates without really building a robust hitting game.

This brings me to the question of the IPL as a selection signal for India. The IPL is a very capricious tournament, 14 matches at different grounds in a highly volatile format, with seasons separated by 10 months. It can be almost impossible to predict a batter’s season based on the previous ones, and form can be extremely fickle. Take Tilak and Abhishek, who did not have the most stellar IPL seasons in 2025, but are setting the world on fire in T20Is. With the inflation of SRs in the IPL, the divergent team structures with 8-9 batters, and the flat pitches, how much does the IPL tell us about a batter’s skill in international cricket? It can serve as the perfect test with its high quality of cricket and the best international players participating, but as it steadily diverges from the format and dynamics of T20Is, its signal is going to get exceedingly unreliable.

As it stands, advanced ball-tracking data is not used for selection. Vanilla strike rates and averages are not enough to profile players. The chief selector has very ably relied on his judgement of quality to build the Indian side, which has worked well to date. However, it remains to be seen if selecting the T20 side will become more and more divorced from the short-term signals provided by the IPL, which yields a high volume of granular data, pregnant with the potential for advanced tracking of players.

As bad as the situation is for batters, it is worse for bowlers. Run rates and first-order statistics give next to no signal about bowling in the modern IPL. Performance is becoming increasingly de-correlated with conventional attributes like the percentage of “good” balls. Batters have successfully de-linked their shot choice from the line and length of the delivery, which means pretty much anything can go the distance. IPL pitches are seeing unprecedentedly low amounts of turn, prompting spinners to go shorter and quicker. Pacers are bowling more wides and short balls to evade the ever-increasing hitting arcs of batters. While this is improving bowler skill at multiple levels, it makes the IPL essentially useless for gathering signals about bowlers for selection. What works in T20Is in terms of lengths and variations is starting to differ from what works in the IPL. The selectors and team management need to back long-term quality, and more importantly, select on attributes rather than IPL performances, which might need advanced data analysis.

The first plot in our journey shows the False Shot Proportion on the x-axis vs the Wicket lost per false shot on the y-axis. Each point represents one batting innings, which is labelled. The colors signify Ind and Eng.

The farther an inns is along the x-axis, the more the chances taken by that batting team, or the better they bowling they faced. We can easily see that most of England’s inns in this series have had noticeably higher false shot proportion than India’s. This results from the combination of:

1. Indian bowlers have bowled better and created more chances

2. English batters have taken more risks, leading to more false shots

The y-axis shows how many of those false shots have yielded wickets. Both India inns in Vizag and one in Ranchi was extremely unlucky for them: the wicket-to-false shot ratio was higher than 0.1 - more than 10% of false shots lost them wickets.

On the contrary, England were extremely lucky in Rajkot (their 1st inns) and Hyderabad. Particularly in the latter, owing to Pope’s knock, their innings had extremely frequent false shots. Very few of these resulted in outs.

The second plot tells us how much run-scoring output was generated in each batting inns: the y-axis shows run rate (runs / ball) and the x-axis shows the false shot proportion. This plot explains the previous one a bit. There are two clear clusters separated by team. England have played a lot more false shots, but they have also got a lot more runs per ball in most cases. In Ranchi and in the 4th innings at Rajkot, they had a low scoring rate, out of which the 1st inns in Ranchi was where they played “normal” cricket - low risk, low run rate.

Except for the 3rd innings blitz in Rajkot, India scored at rates more in keeping with traditional Test cricket.

The proof of the pudding is ultimately, though, in the runs / wicket, or the batting average. The probability of winning in Test cricket depends on the difference between the averages of the two competing sides.

The plot below shows the batting average (y-axis) vs the false shot proportion (x-axis) for both teams. England averaged under 35 in all innings except Hyderabad Inns 3, which contained the Pope knock. In 4/7 innings, India averaged nearly the same or higher than the the cluster of England points on the plot. This despite India falling to unforced errors in the first two games.

The fundamental theorem of cricket is that risk and reward are tied together. If you attempt more shots, you will get more reward in terms of higher scoring rates, but also run the risk of losing wickets. For every format, there is an agreed-upon balance between risk and reward that sets the equilibrium of scoring rate vs wicket rate in the format.

Each format and each situation has an "ideal" attack function: basically the decision-making apparatus of attack vs rotate vs defend depending on the ball. As the format goes shorter, this moves towards rotate/attack from defend. In Test cricket, the "conventional" rules are set, and have been set. You attack only rank "bad" balls.

In Bazball, England are changing that, upending the convention. You take the initiative to attack/rotate on "good" balls if the ball is doing nothing. They succeed because they have the shotmakers and because teams have not formulated a consistent response to this attacking batting yet.

There is a payoff between risk and reward. They have discovered that the risk is overestimated by the conventions of Test cricket, and you can hit more while maintaining the balls per wicket, raising averages. There are also second-order effects of this, like pushing the field back and gathering risk-free singles (which is an important paradigm in ODI batting).

In summary, the “judiciousness” or “selectiveness” of attacking becomes looser as the format goes shorter, because the value of one wicket goes lower. Bazball has disrupted the age-old conventional, comfortable level of “judiciousness” in Test cricket: they attack even the “good” balls - those at good lines and lengths - and they have managed to double the averages in these zones vs pace, without reducing their balls-per-wicket. This has succeeded in part due to brilliant shots by the England players, in part due to the Dukes balls not being as naughty as they once were, and in part due to the pitches not doing as much as they were in England for a majority of Ben Stokes’s captaincy period.

In this series, the last plot and the resulting averages (39:26) have demonstrated that Bazball, or whatever this new approach is, has not worked. Yes, England have run India close at many points in this series, but a chunk of that can also be attributed to India’s inexperience and trials with injuries to major players, in addition to indifferent form on-and-off for their bowlers. As astounding as it has been to witness England’s disruption of Test cricket’s conventional risk-reward curve to some attractive success, the final frontier remains.

A slower ball basically is an attempt to bowl at a lower pace given the same arm action. It is well-known that batters pick cues from a bowler’s action before and after delivery to set themselves up. Therefore, maintaining the same arm speed becomes paramount in deceiving the batter. The action cues signal to the batter that a fast ball is coming, but the resulting delivery is slower. This is accomplished by somehow altering the contact of the ball with the bowling hand. In some way, the bowler breaks the transfer of energy from their arm to the ball - either cut the ball, or hold it with the knuckles, or hold it in the webbing. The slightly slower pace will ideally surprise the batter - who is too late to adjust their bat speed, and will mistime the ball.

This beautiful and skillful variation employs not only change in speed, but also dip and cut off the pitch to fox the batter. An illustration of this symphony lies in studying Harshal Patel and why he’s been so effective. This thread illustrates how all these work in symphony to induce a mishit.

Egged on by an exploration of Harshal’s deliveries, and a conversation on Twitter where the ever so insightful Dweplea pointed out that Bumrah’s great record against lefties at the death is probably down to his brilliant off-cutter, I decided to check out a few basic elements of slower balls at the death. In the following analysis, I have taken recorded slower balls from pacers in the death overs (17-20) in all of the IPL, where I define slower ball as having a speed less than 125 km/h. I’ve chosen the over slot so that we can look at run and wicket rates without having to control for innings phase.

Good Areas Are Important

To begin, where are slower balls bowled. In the following graphic, all batters are treated as RHB, and the boxes show the percentage of slower balls bowled at that line and length. The most popular line is between the 4th and 5th stumps, and the most popular lengths are between 6-8 metres from the stumps. There is also that extremely wide and slightly short region where 5.2% of slower balls are bowled, presumably to keep the ball away from the batter’s hitting arc. In makes sense that the most frequent lines are wide - away from the bat swing, and the frequent lengths are between good and full, attempting to draw out the big shot in the slot in hopes of a mishit.

Now let’s look at where slower balls are most effective compared to fast balls, both in terms of run concession and wicket taking ability. The graph below shows the difference in RPO between slow and fast balls, grouped by line and length. A negative value means slower balls are better. The data shows that the ideal slower ball is wide in the 4-6 m zone, which makes it the toughest for the batter to hit. This makes sense: the farther away the hitting point is from the batter, the more the chances of a mishit if they have to adjust their bat speed. The 4-6 m length ensures that batter is drawn onto the front foot and is loaded up for a big swing, making changes difficult.

Also, going short and wide is generally a very bad idea. The batter has time to sight the ball and adjust, and the ball often sits up for them to free their arms.

If you’re going for the slow yorker, aim for the stumps or go really wide.

What about wicket taking? The plot below shows the difference between the wicket rates (wickets / 100 balls) on slow and fast balls. A positive value means slower balls are better. Surprisingly, almost all zones show an advantage for slower balls. Here too, we see that zoning in on that 4-6 m length is the most beneficial. Since the slower ball aims to get mishits, it’s paramount to entice the batter into hitting the ball hard. That is what that 4-6 m length does. Slower balls are good, but you need to be skillful enough to put them in the right areas while also doing whatever jiggery-pokery you’re doing with your hands/fingers to release the ball.

Loss of Pace

Of course, the change in expected speed is not the only challenging aspect of facing a slower ball. Often, you might know a slower one is coming, but the fact that bowlers usually impart some spin on their slower deliveries makes them lose pace off the surface. And this is the hardest thing about hitting a slower ball well. As a batter, you calibrate your hitting based on the pace of the pitch. A slower ball slowing down massively upon pitching messes with those calculations that your brain has already done. This makes your bat speed too fast for some balls, making you mishit. This is why pitches with inconsistent pace, like the Chennai surface, are so tough to bat on.

A slower ball loses more pace off the pitch on two accounts. It has less pace to begin with, which means it stays in contact with the ground a little longer, letting friction impede it for a little longer. It also has revolutions on it like a spinner’s ball, increasing friction and the loss of energy off the pitch. The graph below shows a probability distribution of the loss in speed after pitching for slow and fast balls.

All deliveries lose speed on bouncing, but slower balls lose noticeably more. This slight discrepancy makes them tough to smash out of the park. How can this loss in speed be employed well by an aspiring slow-ball exponent? The answer lies in the fundamentals: bowling the right length. As I said before, inducing a big shot is important to the process of deception on slower balls. The graph below shows that the average (runs/outs) decreases drastically as the ball loses more speed off the pitch for balls pitching in the 4-8 m length zone. The data shows that the loss of pace is indeed an important factor in slower ball success.

On the other hand, if you bowl short, you’ll get punished harder the more the pace lost by the ball. This is because the ball sits up, giving the batter time and space to hit it.

Lateral Deviation

Because bowlers usually implement slower balls by some sort of spin-like hand action, either off-cutting or leg-cutting the ball, these deliveries also turn off the pitch. The plot below shows the distribution of deviation (turn) from the pitch in degrees for fast and slow balls. It shows how slower balls have more extreme deviations. It also shows that more slower balls are bowled leaving the batter, which should be the ideal strategy.

What effect does deviation have? The next plots show the variation of RPO and wicket rate with the deviation off the pitch. A negative value of deviation means the ball leaves the batter. In terms of run scoring, extreme deviation both inward and out lead to curtailment, as is expected. But for wicket-taking, the outgoing delivery is most effective, as it leads to higher chances of mishits as batters hit “against the turn”.

Dip

Finally, because there are forward revolutions on most slower balls, these balls also have dip. Dip is the effect that pulls the ball downward in addition to gravity. It results from the motion of the rotating ball in the air.

The cricketing effect of dip is to make the ball land slightly earlier than it would had there been only gravity.

In the illustration below, this ball from Ashwin to Roston Chase has a dip of 0.5 m/s^2, pulling it downward with an extra force. This makes the ball land earlier and shorter: by 1.1 metres. This deception of length is instrumental in making the batter mishit the ball, like we saw in the Harshal example above, because it changes the distance between pitching and bat contact, making the ball deviate more between the two points.

Anyway, so how much do slower balls dip? Below is shown a distribution of the vertical acceleration of balls, with gravity subtracted out. This enables us to more clearly see the presence of dip (and its alter-ego lift).

Most fast balls from pacers get lift, because pacers backspin fast balls. The orange curve shows this: most fast balls have a positive value of vertical acceleration, which means they are being pushed slightly upward in the air, in addition to being pulled down by gravity. On the other hand, slower balls (blue curve) have dip (negative acceleration) on almost 40% of deliveries.

Can the benefits of this dip be seen in the data? Yes! As the figures below show, the higher the dip (more negative the acceleration), the lower the RPO, and the higher the wicket rate.

So there we have it. A slower ball is not just about the change in pace, but a concert of various factors that flummox the batter and wrong-foot them in terms of timing and contact. The first key to good slower balls is the right areas, which draws the batter out to hit a high-energy, high-risk stroke, increasing chances of a mistiming if the bowler does execute the other aspects well. These other aspects are break off the surface, dip in the air, and a good loss of pace.

In all of the IPL, I counted bowlers who have bowled 60 or more slower balls at the death on record. Here are their RPO and wicket rates:

And here is their deviation off the pitch and their speed loss (more negative = better).

Using ball-tracking data enables us to follow the full trajectory of a delivery, including turn, drift, dip, speed and the various angles at which the ball is released and then bounces off the pitch. Such data is available for four of Axar’s five Tests, and we will try to use it to confirm certain ideas about his bowling and uncover new ones. All ball tracking data mentioned in this piece is updated till the 1st  India vs New Zealand Test at Kanpur.

Release

We start at the beginning: the release of the ball. Axar’s point of release is unusual compared to other left arm spinners, both in terms of height and the distance from the stumps. The first plot below shows the probability distribution of the height at which the ball is released, comparing the data for Axar (blue) with all data for slow left arm bowlers around the wicket (black). The two curves seen together show the relative odds of the ball being released from a given height by Axar and the average slow left-armer. Most of Axar’s deliveries are released from between 2.15 to 2.30 metres above the ground, which is very high compared to the usual release height for other slow left arm bowlers; bowlers of his kind seldom release the ball from that high. Such an oddly tall release compels the batter to track the ball from a height he normally doesn’t. Batting is based on familiarity, and Axar’s anomalous height makes his bowling slightly tougher to follow.

More than the height of release, it is the width of release that poses a greater risk to a batter facing Axar. His height and long arm-span together mean that he bowls from a height and from an abnormally wide angle. The plot below shows the probability distribution of the width of release – the sideways distance from the plane of the middle stump at the point when the ball is released. Here too, Axar stands a league apart from other slow left-armers. Most of his balls are delivered from 1 to 1.4 metres from the middle stump, whereas most slow left-armers around the wicket bowl theirs from 0.5 to 1 metres away from the middle stump.

This sharp inward angle creates doubt in the mind of the batter, who is always looking to play to protect his stumps. In addition, the angle also has implications for judging how to play for the ball turning away, as we shall see next.

Turn (Or The Lack Of It)

The most frequently talked about aspect of Axar’s bowling is turn, or rather, the lack of it. Ball tracking data shows that Axar’s deliveries indeed turn less than the average left-arm finger spinner in India. But this is not for the lack of putting revolutions on the ball. Spinners try to impart sidespin on their deliveries, which generates both drift in the air and turn off the pitch. Drift can be measured by the force on the ball in the air, which is directly related to the number of revolutions on the ball and how square the seam is rotating in the air.

The plot below shows the median turn obtained for a given range of drift force in the air before the ball pitches.  For instance, the leftmost point on the curves shows the median turn off the pitch for all balls that have a certain amount of drift force on them (in this case, 0.2 to 0.4 m/s2). To be clear, this is not the amount the ball drifts in the air before bouncing, but a measure of the actual drift force on the ball that results in its sideways motion in the air. For us, this is a proxy for the amount of sidespin on the ball.   

The black curve on the plot below shows that the amount of turn increases with the drift force on average, confirming that sidespin is crucial to both phenomena. Looking at Axar’s average turn, shown by the blue curve, for the same amount of drift, Axar gets a lower amount of turn than the average slow left-armer in India. Axar gets noticeably less purchase off the surface, regardless of the revs he puts on the ball.

The reasons for this are varied. As Karthik Krishnaswamy explains in this piece, Axar utilises “undercut” on the ball, which refers to releasing the ball with the palm facing upward, almost throwing the ball like a frisbee. Axar does this by rotating the ball into his hand before releasing it, which makes the spinning ball come out of his hand with the leather of the ball facing the pitch. As a result, sometimes the ball lands on the leather, gripping less, and turning less. Even when Axar releases the ball normally, with sidespin, the seam sometimes wobbles in the air, making the ball randomly land on the leather, which also makes it turn less.

This lack of turn contributes to Axar’s lethality. Among slow left-arm bowlers who have taken ten or more wickets in India, Axar has the lowest median turn on wicket deliveries, as the table below shows. It’s not that his wicket taking balls are arm balls. He does turn the ball away from the right-hander but gets very little turn on average.

The lack of big turn alone isn’t responsible for why Axar has been such a menace to right handers. Remember the extremely wide release points creating the sharp inward angles from around the wicket? Because of these inward angles, Axar’s deliveries continue moving into the right-hand batter after pitching, even if they turn out. The plot below shows Axar’s deliveries that have ball tracking data. On the x-axis, we have the turn off the pitch, in degrees. Almost all the balls have positive values for turn, which means they are turning away from the RHB. On the y-axis, we have the deviation of the ball from the point of pitching to the plane of the stumps, as calculated by the same ball-tracking prediction that is used for LBW reviews. Most of Axar’s deliveries have positive values on the y-axis, which implies that they continue moving into the RHB after pitching.

Batters play spin by trying to read the direction of turn from the hand and in the air. A batter facing Axar reads an outgoing ball from the motion of his fingers at release and confirms this by seeing the seam rotate towards the offside. The batter’s training kicks in, setting him up to play outside the pitching line of the ball because he expects the ball to move away after pitching. Here is where the deviousness of the sharp angle defeats him: the ball still goes in from the line of pitching, past the defense and into the pad or stumps. A great example is Zak Crawley’s LBW in the 1st innings of the third Test in Ahmedabad. The ball is delivered like an away-spinner, but it starts wobbling mid-air, reducing the purchase off the surface. This one actually turns – 1.75 degrees away, but the delivery angle means that it is still going 0.05 degrees into Crawley after bouncing.

Ball tracking data allows us to reconstruct the path of the delivery, making a visual representation of this ball possible. The figure below shows a bird’s eye view of the ball. The first panel on the left shows the actual track of the ball from release to bouncing as observed from above the pitch. Notice how wide the release is. The second panel then shows the angle of the ball when it was released as a blue line. The third panel shows the angle of the ball just before it pitches, as a grey line. Notice that the release angle and the angle before pitching are different – the ball has drifted in the air into Crawley, making the inward angle even sharper. Finally, the fourth and rightmost panel shows the path of the ball after pitching in red. Notice how the ball has actually turned outward. The red line makes an angle with the grey line, which was the path of the ball before bouncing: this is the actual turn obtained off the pitch. Although the ball has turned away, it continues moving into Crawley after pitching, beating his inside edge.

This illustration of the three different angles that a batter must play against while facing a competent spinner sums up the threat posed by Axar. Very few of Axar’s wicket taking balls are actual arm balls, but it is the extreme release angle coupled with the inward drift and slight turn, sometimes by design, sometimes via wobbly imperfect release, that beats the batter who conventionally expects the ball to move away or straighten on pitching. The table below shows the median deviation of a wicket-taking ball between the pitching point and the stumps for all slow left-arm bowlers who have taken 10 or more wickets in India (till the 1st India vs New Zealand Test at Nagpur). A positive value means the ball moves in, a negative one means it moves away. Axar’s values are extreme: his wicket balls move 18 centimeters inward on average.

Darting The Ball, Literally

While the anomalous release and the lack of outward movement is often talked about while discussing Axar, his trajectory is given less attention. Given the tall release, one would assume that batters are more likely to be on the front foot playing him, but ball tracking data reveals otherwise. Spinners usually flight the ball, throwing it slightly upward to get the loop they need. In contrast, Axar almost never flights his deliveries. Shown below are the distributions of the vertical speed at release, for Axar and all slow left-arm bowlers whose deliveries are available in the ball tracking data. A positive value means that the ball is thrown upwards, and a negative value implies it is thrown downwards. Almost all Axar’s balls have a negative value: he literally darts all his deliveries in.

Batters subconsciously use the trajectory of the ball to decide if they want to go forward or back. Axar’s flat dart-like trajectory compels batters to play back more often than they would facing other spinners. According to Hitting Against The Spin, a new book on cricket data by Nathan Leamon and Ben Jones, 83% of balls delivered by spinners in the 4 to 5 metre length are played on the front foot. The corresponding figure for facing Axar is 76%1. About 70% of balls in the 5 to 5.5 metre length are played on the front foot, while facing Axar, only 51% balls are played on the front foot.

Lift: The Opposite of Dip

To create wicket-taking threats, this aversion to front-foot play combines with another anomaly of Axar’s bowling: the lift he gets. In addition to the sidespin that we covered earlier, spinners also put some amount of overspin on the ball by spinning it in the forward direction. As a result of this overspin, the ball experiences a downward thrust before pitching. This makes it land shorter than the batter initially predicts. Spinners can also get “lift”, which is an extra upward force on the ball, the opposite of dip. This can happen when the delivery is angled sharply across the pitch. Lift puts an extra upwards thrust on the ball, making it hang for slightly longer in the air, and therefore makes the ball pitch fuller than initially anticipated.

The plot below shows the distribution of dip or lift forces on the ball. A positive value of vertical acceleration means lift, a negative one means dip. As we can see, Axar gets lift much more often compared to the average spinner. This, again, is a direct result of the sharp angle of his delivery – the sharper the angle into the batter, the more the upward lift force on the ball.

This lift along with flat trajectory ensures a two-step trap for the batter. The flatness makes him move back, the lift makes the ball pitch fuller than predicted. The batter is now playing a full ball on the back foot – a recipe for disaster, especially because Axar bowls so quick – his median speed is 90 km/h while the median speed for the average slow-left armer is 84 km/h.

There are many examples of this mechanism bringing Axar wickets, like Dom Sibley’s leg before in the second Test in Chennai. That ball had a lift of 1 m/s2. This means that the ball had an extra upward force on it, which was 10% as strong as the pull of gravity downwards. As a result, the ball pitched a little fuller than it would have without the effect of lift. Sibley was stuck on the backfoot and unable to bring his bat down in time to a ball that ultimately ended up being fuller than he had initially calculated. Similarly, Jofra Archer in the 3rd Test in Ahmedabad tried to play an incoming ball on the backfoot. The lift on that was a humongous 2.2 m/s2, making it pitch much fuller than Archer had gauged from the flat trajectory. The speed meant that Archer, stuck on the backfoot, could not prevent it from hitting the stumps.

This lift also has a subtle role to play in the common front foot LBWs that Axar gets. Lift makes the ball stay longer in the air, which means it carries on along that sharp inward angle, giving it a better chance of beating the inside edge of the RHB. The dismissal of Ben Foakes in the 3rd Test in Ahmedabad involved a lift of 1.2 m/s2 on the ball, making it pitch fuller and crash into the pad going into the batter.

The Basics: Accuracy of Length

Finally, it is often said that Axar’s extreme accuracy contributes to his success. Screenshots of his pitch maps with tight bunches of deliveries abound on the internet. Ball tracking data allows us to quantify this and put an exact number on Axar’s precision. Since length is the primary determiner of a spinner’s success, I will look at the standard deviation of the length for different bowlers, which measures how much variance there is in the lengths they bowl. The lower this number, the more accurate the bowler is. Out of all the bowlers who have tracking data for at least 1000 balls and have taken 20 or more wickets, Axar is the most accurate, as the table below shows. His standard deviation in length is 0.85 metres.

Summary

In summary, freakish angles are central to Axar’s success. His extreme width of release and a lack of turn through natural variation make the ball come in even when it spins, beating the conventional way of playing spin outside the line if you think it will turn away. A flat dart-like trajectory hoodwinks batters into going on to the back foot. The exceptional lift he gets on the ball because of his sharp angles changes the length of his deliveries. To go with this, his extreme accuracy in length ensures full control and all this happens at high speeds, giving batters very little time to adjust. Although far from a conventional spinner, Axar succeeds through simple anomalies in a game based on familiarity.

And thus, a measure of chances created (for bowlers) or defence/shots attempted (for batters) is a more accurate descriptor of the game, which actually attempts to describe what the players did on the field, rather than what the players’ actions resulted in. The latter is a clearer indicator of ability over the kind of sample sizes cricket has. Unfortunately, cricket is conditioned to talk in terms of result statistics, and barring Cricinfo’s control percentage, and the random drip of nuggets from the CricViz handles, such stats are almost never in the public domain.

I often ask myself why cricket “analytics” has suddenly skyrocketed in popularity in the recent years. The easier availability of data in the public domain through Cricmetric and Cricsheet has had a major role to play, but taking a broader view, the demand for analytics beyond archaic cricket numbers comes from the blossoming of franchise T20, which demands that players be quantified better; there is value attached to their performance, and real consequences of how well franchises can purchase personnel. Also, conventional metrics are inept at describing T20. Neither of these factors was relevant in an age when ODI and Test cricket between countries (that are not teams made in “free” markets) dominated the landscape.

I think “impact” style numbers in T20 are pretty well-established now. This kind of metric compares a player’s output with the average player’s output in the same “situation”. The quality of your metric depends on how well you can define your “situation”, by taking various levels of context into account. This is the basis of Cricketingview’s Misbah/Jogi measures, my own runs-above-average metric, and also of Cricviz’s Impact calculation. We now need to move ahead of these and look to numbers like “boundaries attempted” and “outcomes when the ball is attacked”, which is in line with the philosophy of measuring the ability to execute a skill, rather than capturing the final outcome of a delivery.

The day is not far when T20 will need further specialisation. A “middle-overs enforcer” might be needed to bowl hard lengths at high speeds, a death specialist might be needed to bowl wide yorkers. It is suboptimal to judge them on their numbers, even after accounting for context. Instead, analysts should measure the ability to bowl that hard length, or bowl that wide yorker, and judge players on the efficiency of execution. This view analyses cricketers on the basis of purely how well they can ply their (limited) set of skills, isolating it from the recorded result. Such metrics are then directly relevant to designing strategy on and off the field. The question asked is: “How well is the player executing this skill, regardless of result?”

Small sample sizes, as ever, thwart the inferential power of cricket stats, but barring that, the two open questions in cricket, in my mind, are:

1. Quantifying second-order effects: How does the quality of other batters in the team affect how a player at the crease modifies their game? How does the non-striker being an “anchor” facilitate better hitting at the other end, if at all? How does conserving two overs of your best bowler for the end of the innings make the chasing side go harder in the middle, thereby losing wickets? These are all indirect effects of team composition and strategy, and it proves devilishly hard to isolate their effects from current data.

2. Matchups: Can there really be a good prediction for what happens when a certain batter faces a bowler? The sample sizes are small, and variances are high. It’s easier to summarise a batter’s performance against a bowling type, but it changes against each individual. In my recent article for Cricinfo, I’d constructed relative matchup indices that quantified how well a player did compared to the average player in that matchup. This didn’t make it to the final piece, but I had tried using products of individual indices to create a prediction for how a particular batter-bowler matchup would fare. Needless to say, the prediction and the actual data showed a fair bit of disparity. I’m trying a few other ideas (none of which show very high accuracy, but that’s an issue with cricket itself). This remains the Holy Grail of T20 analytics.

Currently recorded cricket stats are the highest level of record, in that they give descriptions most removed from the actual game. At a lower level, closer to the game, is detailed ball-by-ball tracking data, which enables us to measure attempts like I’ve mentioned before. At an even lower level, we have a fundamental description of the physical dynamics of cricket: biomechanical data. Recorded data tells you what happened, tracking data tells you how it happened on the pitch, but biomechanical data goes a step further to describe what the player did to make it happen on the pitch. Analysing and using biomechanical data is your best bet to increase the efficacy of players at executing specific skills. And that’s why I feel it’s the next step in cricket data.

The current times have seen an uptick in hobbyists doing great video analyses online. In addition, cricket Twitter has been abuzz with talk of biomechanics, of both fast bowling and hitting. While the footage available to common fans has been used brilliantly by amateur analysts, teams should be looking to extract a wealth of data from nets and practice sessions. Imagine installing markers on a batter and making them hit a certain shot 200 times against a certain kind of delivery. Then, combining this with biomechanical research on optimising the body’s dynamics for the best power hitting to make precise modifications to the batter’s movements. Some will say this robs the game of romance, but I say it will make for better cricket in the long run. In the next decade, high-res video cameras, tracking technology and increased research will act in concert to make biomechanical / detailed video analysis the centre of cricket analytics. This is of course going to be in addition to better models for measuring performance using just numbers, which should move in the direction of measuring attempts.

This gave me an idea: why not find similar clusters of good batting using this algorithm on a player’s career? After all, our casual definition of batting consistency is “does the batsman score high enough often enough”.

My latest article for ESPNcricinfo is a culmination of this idea. The aim is to check how much of a career is covered in such clusters. Players who are in these phases of scoring highly often enough are more “consistent”.

I hope you enjoy this article. It’s been one of my favourite projects, and I have mulled over it for such a long time. The graphics have taken some effort to make, but I’m happy with how they have turned out (thanks to Matplotlib, the python plotting library).

This is Tendulkar’s career with a “gap” of 3 innings, and a threshold of 50, which means a cluster is broken if he has a gap of 3 or more between consecutive 50s.

There are 6 important numbers here:

1. Proportion of innings covered in clusters.

2. Proportion of runs covered in clusters.

3. The ratio of 1 to 2: this tells you how evenly high scoring the player has been.

4. No of clusters.

5. Length per cluster on average: how long are the good streaks.

6. Length per void on average.

Other two numbers of importance are the runs per inns in clusters and voids, which tell you how good or bad a player is in his good or bad phases.

This contains the cluster plots for all batsmen with 5000 or more Test runs.

This file has the various numbers for all these players.

Since you’ve read 10,000 articles on Pujara, why not try the 10,001st?

Much was made of Pujara’s indifferent run of form in his 2014/15 overseas cycle. A closer analysis of his dismissal doesn’t reveal a pattern of being weak against pace or swing, except perhaps a susceptibility to the incoming ball. In the 2018 overseas cycle, Pujara plays a few solid innings in perhaps the most challenging conditions India have faced this millennium. The Dukes ball swings all day in the English summer of 2018. South Africa prepare green pitches as an act of retribution. If you look at his average, it isn’t very flattering. But he’s played 1068 balls for 13 dismissals, being run out thrice. He’s done his job: coming in early and blunting the ball. This pays dividends in Australia, of which we all know. In New Zealand, he is India’s second highest scorer, 2 runs behind Mayank Agarwal, having faced 350 balls in 2 matches.

To combat the incoming ball, his technique has changed slightly. He stays slightly leg side of the ball, keeps his pads away, and plays with almost a dead bat. The backlift is negligible, the ball almost goes backward hitting the bat. This is in stark contrast to someone like Shubman Gill or Rohit Sharma. High backlift, solid stride, energetic bat swing.

A Gill or a Kohli takes risk trying to make shots on even good balls. This brings them the runs, but also makes them more vulnerable to getting out. Pujara’s strategy is the diametrically opposite. He has zero interest in attacking the ball until and unless it’s absolutely required. People ask why Pujara seems to get out to jaffas. It’s because he simply doesn’t make unforced errors. A batsman gets out in a variety of ways, but you’ll never see Pujara get out flashing at a wide one, or lofting one into the hands of an outfielder. He deflates the chances of him getting out to a non-jaffa. And therefore, only jaffas seem to get him out.

Pujara is the examination in a graduate level Test bowling course. Test bowling is about bowling the same ball again, and again, and again. Pujara makes the bowler do that, trusting his technique. He asks the bowler to construct a wicket, to bowl 50 balls at the same place and hope for natural variation to get him out. He strips it to the elements of cricket: I’ll try to stay in, you try to get me out.

This is the crux of Pujara’s batting: occupation of the crease. A plot of his runs scored as his innings progresses shows a clear strategy outside India, where conditions assist pace as the dominant form of bowling.

His scoring rate in Asia matches the average batsman, but the same outside Asia is significantly lower. This is not a bad thing. It’s proof of his method. He displays monk-like discipline outside the off stump, just leaving or blocking. Anything on the pads, he flicks and collect runs. But he shuts down any modes of dismissal outside the off stump. Does this work? Let’s have a look at the chances of crossing fifty balls.

Voila! Pujara has a median score of 15 runs at the 50-ball mark outside Asia (before the Brisbane Test). But… he crosses 50 balls more often than most batsmen, beaten only by AB de Villiers, Marnus Labuschagne and Shiv Chanderpaul. If his brief is to blunt the ball, his method works. It works better than most.

Since 2015, Pujara plays 103 balls per out outside Asia. This is the third highest among players with at least 30 innings, behind Smith (127.7) and Williamson (116.7). In a bowling-friendly era and coming on the back of unsure and inconsistent openers, Pujara does his job perfectly.

This series against Australia, the TV commentators have made repeated noise about Pujara’s scoring rate. This has been followed by endless chatter on social media. Theories fly around that somehow Pujara’s blockathons “put pressure on the batsman at the other end”. Unless a declaration is in the offing, I don’t see how playing time and occupying the crease are bad in any way in an era when most matches finish in four days.

Strike rate is not a knob on the batsman’s body that can be turned at will. It is a function of conditions and the bowling attack one faces. This series, the Australians have kept the pressure on with exceptionally consistent bowling. One can count on one’s fingers the number of rank bad balls Pujara has received from Hazlewood and Cummins. When he does get the bad ball, he puts it away. Like those three boundaries against Cummins.

But he's most focused on just staying there. As I type, he has already faced more than 800 balls this series. He is occupying, and he’s doing it well.

Yes, you’ll say that Cummins has had his number this series, perhaps. But that is a testament to the quality of Cummins (and Hazlewood). Pujara’s beside-the-line technique leaves him open to that magic ball that angles in and goes out. If a batsman were to pick one weakness, it would be this, because it’s a magnificent sort of delivery, unicorn-like in its frequency of appearance. Cummins bowls it once every 5 overs. Because he’s an all-time great. If that is the weakness my number three has, I’d take him with both hands, strike rate be damned!

Reverse swing is a commonly used but poorly understood term. It is said that the old ball swings towards the shiny side. But is that true reverse swing, or is it contrast swing?

Conventional Swing

The physics of conventional swing is fairly well understood. Some cricketers and enthusiasts speak about swing being influenced by the “wetness” or “heaviness” of one side of the ball compared to the other, but this in not true. The layer of air around a moving ball, much like the wind that hits you on a bike, is key to generating swing.

I’m going to be referring to Dr. Rabindra Mehta’s work throughout this post. Here is a comprehensive Cricinfo article on the physics of swing based on his work. It explains many kinds of swing in detail, and I’d encourage you to read it to understand the general physics behind the phenomenon.

The above figure, from Dr. Mehta's own paper on swing, shows that layer of air. The ball is going to the right of screen, the seam is pointing towards the leg slip for a right hander. The air that hits the ball in front is smooth (called laminar). The seam disturbs this smoothness and makes the air flow turbulent (as shown on the bottom side of the ball in the figure). Turbulent air clings to the ball more, which means it separates from the ball later than the smooth air on the top side. This difference in when these two kinds of flow separate from the ball leads to a sideways force on it. Which we call swing.

In the figure above, the turbulent layer separates later. And that is the direction in which the ball swings.

You can see that with the seam pointing into the right hander, this ball will be an inswinger.

So, conventional swing moves towards the seam.

Reverse: What Is It?

Reverse swing is a mysterious and somewhat maligned term even today in cricket. But what is it? How is it different from conventional swing?

Let’s get one thing clear: the scientific literature on this defines reverse swing clearly: if the ball swings away from the seam direction, it’s called reverse.

Notice that this definition is in contrast with “conventional” swing, where the ball swings towards the seam.

The key to achieving this reversal of direction is twofold: 1) the condition of the ball, 2) the speed.

At high speeds, even the new ball can be reverse swung: it can go opposite to the seam direction.

So, if I bowl with the seam tilted towards slip for a right hander, the ball will be an inswinger if I bowl it fast enough (145 km/h or faster).

The technical explanation for this is done well in the above linked paper and article, so I’m going to skip it. The point of this post is something else.

Moving on, can you achieve this reverse swing at lower speeds? Yes, by roughening up one side of the ball. A rough side can aid this process, and make the ball go against the seam direction at lower speeds.

Note: Hereafter, I use reverse for the reverse swing defined above, and “reverse” in quotes for reverse swing as we commonly define it in cricket.

What About The Shiny Side?

Now, in common cricketing discourse, “reverse” swing usually means something else. It is said that the old ball with one shiny and one rough side will always go towards the shiny side. Bowlers can usually bowl only one direction of swing, depending on their hand. Using a properly conditioned old ball, any bowler can now swing the ball both ways, just by flipping the shiny side.

This phenomenon is also covered well by Dr. Mehta. This is called contrast swing his work. While it is explained well in the article, I’ll point out a few relevant things.

• Contrast swing can be achieved with a straight seam. It is a hard skill to release the ball with a tilted seam, which swing bowlers perfect. But we see part-timers and other bowlers also get swing with the old ball sometimes, even though they release the ball with seam straight. This is contrast swing.

• The direction of contrast swing depends on the speed. At speeds lower than 80 km/h, the ball moves towards the rough side. At higher values, it moves towards the shiny side.

Now, we know that First Class level bowlers bowl at higher speeds than 80 km/h. This is why we see the ball go towards the shiny side when it’s old. You need to release the new ball with a slanted seam to get movement. You just need to release the old ball with an upright seam.

There are multiple examples of this. Mohammed Shami bowls with an upright seam with the old ball and gets “reverse”. In this interview, Waqar Younis, one of masters of “reverse”, says that the wrist isn’t that important for reverse swing, which I took to mean that the seam is to be released straight.

This explains the swing of the ball when the seam is upright. What about when it’s not?

Newer Research

This last question has troubled me for a long time. Dr. Mehta’s research left me with two questions. One, he mentions that reverse swing can be controlled by flipping the side on which the shine is, but the ball swings towards the rough side. Secondly, it’s never specified if the ball ever swings towards the shiny side when the seam is tilted.

Something wasn’t adding up. We’ve seen countless instances in cricket of a slanted-seam ball moving towards the shiny side. The whole crux of “reverse”, as we hear it from players and commentators, is the shiny side of the ball. Where’s that?

To look at this, I looked at an excellent new paper by Dr. Sanjay Mittal’s lab at IIT Kanpur. They did a comprehensive analysis of the motion of new and old SG balls tilted at different angles, measuring the swing force, modelling the flow of air around the ball, and computing the trajectory of a ball as it goes down the pitch.

The paper presents interesting results, which I’ll quote later. Let’s come to the “reverse” swing part first. My aim here is to check if the results of this paper match up with the following notion: the old ball swings towards the shiny side.

First up, let’s define the 4 kinds of balls they used, presented in the following figure. This is important to study the key graph which I present later.

Imagine the balls are going to the left of screen, to a right hander. The seam is tilted towards leg slip. Let’s call the side of the ball towards which the seam is tilted the seam side. In this case, the seam side is the upper part of the ball in the figure.

i) New ball: no sides rough

ii) S-R: The seam side is rough. The non-seam side is shiny.

iii) NS-R: The non-seam side is rough. The seam side is shiny.

iv) C-R: Both sides rough.

Cases ii) and iii) are relevant to our questions here.

Here comes the plot:

Understanding this graph

1. On the x-axis is shown the speed at which the ball is released (by the bowler).

2. On the y-axis is the amount the ball has moved sideways (swung) by the time it reaches the end of the pitch.

3. They did this analysis for 5 different balls. Four of them are the ones I detailed above, with the seam angle tilted at 20 degrees (which is the optimal angle for swing). The 5th one is an NS-R ball with a different angle (30 deg).

4. If the line is positive, it means the ball swung towards the tilt of the seam. Which means, if I bowled it like an outswinger, it swung out.

5. If negative, it means the ball swung away from the tilt. Which means, if I bowled it like an outswinger, it swung in.

Now, Understanding The Two Cases of “Reverse”

I keep putting “reverse” in quotes to refer to what we conventionally call reverse swing in cricket: swinging towards the shiny side. I’ll look at the plot above and study two cases. We’ll see if we can corroborate the colloquial definition.

The physical mechanisms of these two cases are well explained in the paper, and center around the effects of speed, seam and roughness on the surface layer of air around the ball, and when it gets tripped into turbulence. I do enough physics at my day job so we’ll leave those details be, and focus on interesting results.

Revise those S-R, NS-R definitions before we move on. Gave me a lot of grief.

Case 1: S-R - Green Line

Let’s define a cricket situation with an S-R ball. Imagine:

1. I’m angling the seam like I’m bowling an inswinger to a right hander.

2. The seam is thus tilted towards the leg side.

3. I’ve kept the shiny side towards off. This means the seam side is rough (S-R).

4. I’m bowling at a 135 km/h or higher.

This is a fairly typical situation. Now, conventional cricketing logic dictates that the ball will be an outswinger, because the shiny side is facing towards off.

Now, does the plot show that? Look at the green line (which shows the S-R ball) to the right side of the plot. It is in the negative region. Which means it swings away from the seam direction.

But “away from the seam” is the same direction as the shiny side!

This ball is moving towards the shiny side. Your best chance of getting this kind of swing is to bowl higher than around 130 km/h.

This is true reverse swing (as defined by going against the seam).

This wicket by Josh Hazlewood is a great example of this. In the replay from the batsman’s angle, you can see:

• The seam is facing towards the slips.

• The shiny side is facing towards the leg side.

• This means this is a case of S-R (seam side is rough).

• The ball swings in, i.e. towards the shine.

• Notice that this is also reverse swing as it is defined in literature: the ball is released like an outswinger, but swings in.

My postulate is that this must be the dominant mode of “reverse” swing for right armers. Their default release is for an outswinger, and their default “reverse” direction goes into the right-hander. So this is what happens when a right-armer “reverses” the ball with the seam tilted. This was Waqar Younis’s MO when he bowled “reverse”. His natural ball was the outswinger, but with the old ball he used to swing it in. As one can see from the plot (and the video), the magnitude of swing is huge in this case.

Here is a left-armer’s version of this exact mechanism by Zaheer Khan.

Case 2: NS-R - The Red Line

I haven’t been able to find video examples for this one. But let’s try to break it down like the S-R case.

1. I’m angling the seam like I’m bowling an inswinger to a right hander.

2. The seam is thus tilted towards the leg side.

3. I’ve kept the shiny side towards leg side. This means the non-seam side is rough (NS-R).

4. I’m bowling at a 135 km/h or higher.

Look at the red line in the plot above in the high speed region. Its value is greater than zero.

This means the ball swings towards the seam. Which means the ball is swinging in. This is also the same direction as the shiny side.

Notice that this is conventional swing as defined in the literature, as the ball swings towards the seam direction.

By covering these two cases, we’ve confirmed “reverse” swing as it is called in cricket:

1. In both cases, the ball swings towards the shiny side. Check.

2. This means that the bowler can change the direction of swing based on where he keeps the shiny side. Most importantly, he can do this without changing the seam direction. If I know how to bowl one kind of swing, I can still achieve it in both directions with the old ball. This is what bowlers do with the old ball, often hiding the direction of the shiny side to not let the batsmen know which way it will move.

Other Interesting Stuff

With this out of the way, there are some interesting conclusions to be drawn from the above plot and the rest of their paper.

• The magnitude of swing in the NS-R case is much lower than that in the S-R case. Maybe this is why bowlers use a “default” mode of reverse swing: Waqar brings it in, for instance. Getting it to go the other way is a good surprise, but maybe it doesn’t move as much.

• Looking at the blue line (new ball) in the plot, we confirm Dr. Mehta’s result: the new ball can indeed swing in the opposite direction to the seam if it’s bowled fast enough. This has to do with the surface layer becoming turbulent before it reaches the seam if the ball is fast enough (or rough enough).

• To get the maximum lateral movement with the new ball, bowl it at 90-100 km/h.

• For the NS-R case with a 30 degree tilt (orange), “reverse” swing is achievable at much lower speeds compared to the 20 degree case.

• The NS-R case with 20 degree tilt (red) shows that sometimes the old ball can go towards the rough side, if the ball is bowled at less than 130 km/h.

• Most strikingly, a ball which is roughed up on both sides (black) will get appreciable reverse swing (against the seam direction) at a wide range of low speeds (80-140 km/h). If you don’t have a leadership group taking care of the ball and it gets all scuffed up, you can use your Ganguly-Chris Harris clones to try and swing it. It will go the other way, though.

Two Final Plots

This, on the right shows a plot of swing vs release speed. The seam angle is kept the same (20 deg).

Here’s more confirmation that the new ball goes against the seam direction if it’s fast enough. Also, there’s an unfortunate speed range (125ish km/h) where you might not get any swing at all.

Now, a cousin of this plot: same release speed (146 km/h), different release angles.

This shows some wild results. Changing the angle of the seam has a massive impact on whether the ball will go with the seam direction or against it.

The most bonkers result is the case when the ball is released at a 30 degree-angle. The ball starts going one way, but as it speed reduces as it moves along, the direction of swing switches. This has to do with something called the critical Reynolds number, which is the magic threshold that dictates whether the ball will go this way or that relative to the seam angle.

God help the guy who has to face that 30-degree ball.

Conclusion

If you see the ball going towards the shiny side, and the seam is upright, it is contrast swing.

If you see it go towards the shiny side with the seam tilted, it is one of the two cases outlined above. The more commonly seen case (a la Waqar) is true reverse swing, as it goes away from the seam direction. The other case is actually conventional swing, as it swings in the same direction as the seam.

The Revolt of 1857 ended the rule of the Mughal Empire, and the reprisal of the British forces left their capital of Delhi ravaged in its wake. In the decades after, the British, now fully in control, decided to shift their capital to Delhi, which had long been a seat of temporal and spiritual power. They pushed a policy of urbanisation in their imperial capital, and as part of this, commissioned a cricket ground on the outskirts of the old city. Located next to the citadel of Feroz Shah, the 14th century ruler of Delhi who came from the Turkic Tughluq dynasty, this ground now takes his name, the Feroz Shah Kotla.

Despite the royal connections, the Kotla fails to evoke the romance associated with an Eden Gardens or a Chepauk. There are no iconic Pongal Tests held there, no mind-boggling crowd capacities to boast of. Instead, the modern North stand is often derided as a macabre cousin of a car park. Add to this the sorry image of the Delhi District Cricket Association as a hotbed of corruption and incompetence, and Delhi gets relegated to the backseats of the Indian fan’s perception.

***

The Kotla ground’s old Willingdon Pavilion, that sits behind the South Stand, was inaugurated in 1933 by the then Viceroy of India, The Earl of Willingdon. Since then, the ground has been witness to some iconic moments in Indian cricket history.

Independent India’s first Test match at home was played in 1948 in Delhi, on a turf pitch. A strong West Indies team made their highest ever Test total of 631, with Lala Amarnath’s home side holding on to a tense draw.

The same Hemu Adhikari who had held fort to a draw in 1948 combined with Ghulam Ahmed for a record 10th wicket stand of 109 runs in 1952 against Pakistan, giving India an innings win over the archrivals. This was the first Test between the two nations, and was inaugurated by India’s first president, Rajendra Prasad. The Pakistani team was led by Abdul Hafeez Kardar, one of three men to have played for both India and Pakistan.

Perhaps best known for Anil Kumble’s record 10-wicket haul against Pakistan in 1999, the Delhi ground brought the best out of India’s most successful spinner. He took 58 wickets in 7 Tests, averaging a measly 16.8. He also played his final Test match here, his last wicket being a running caught and bowled despite his hand carrying eleven stitches. He was carried off the ground on the shoulders of his teammates to a rousing reception.

The ground has been a happy hunting ground for India’s spinners before Kumble as well. In 1965, S Venkatraghavan’s 12 wickets on a dry surface, helped by tons by Dileep Sardesai and Tiger Pataudi, helped India beat the touring New Zealand team by seven wickets. In the Delhi Test of 1969 versus Australia, Bishan Singh Bedi and Erapalli Prasanna bowled 80 overs for 8 wickets on an “absolutely grassless pitch” (according to Wisden). In the second dig, Pataudi introduced the pair after just three overs from the “new ball bowlers”, and they took 5 scalps each, consigning Australia to a seven wicket loss.

Indian batting too has seen some milestone moments at the Kotla. In 1983, Sunil Gavaskar, the original Little Master, made his 29th Test hundred to overcome Don Bradman’s tally, stroking an attack boasting of Holding and Marshall for 121. 22 years later, Sachin Tendulkar, in many ways his successor as India’s legend from the Mumbai school of batting, overtook Gavaskar’s tally of 34 Test centuries on a hazy day here against Sri Lanka.

Delhi also was home to the relatively unknown coup-de-grace of the man who was called the “Bradman of India”, Vijay Merchant. Merchant returned to Test cricket after a gap of five years, making a sparkling 154 against England in Delhi, at the ripe old age of 40. At that time, this was the highest Test score for an Indian. In the same match, Vijay Hazare, often his competitor for the title of India’s best batsman, bested this with a 160. Merchant injured his shoulder in that match, and never played for India again.

India have been undefeated at the Feroz Shah Kotla ground since 1987, with their most recent victory being a 337-run walloping of South Africa that featured a fourth-innings blockathon for the ages. Hashim Amla and AB de Villiers put on a masterclass in defence, facing a whopping 541 balls for combined scores of 68. Victory was delivered by a fiery Umesh Yadav spell that lit up a grey final day.

***

That the Kotla has been a fortress for India is fitting, as the colloquial name itself means “fort”, taken from the structure bordering it.

Delhi is a city of cities, a landscape dotted with the settlements of successive kings, spread in a triangle between the ancient Aravali hills and the Yamuna river. Ferozabad, built by Feroz Shah Tughluq in 1354, is known as the fifth city of Delhi, which started from the northern ridge near the Delhi University campus, and went unbounded to the south.

The “Kotla” (fort) was its central citadel, and today houses one of two Ashokan pillars in Delhi (the other standing at the northern ridge). Erected in 240 BC, it features inscriptions in both Brahmi and Pali scripts. The structure that houses it sees devotees of the djinns, creatures from Islamic mythology that are said to have inhabited Delhi since medieval times. Every Thursday, the fort sees a procession of fervent followers in their honour. It is their love for the city that keeps the city thriving through the rise and fall of different empires. Devotees spend hours in prayer, pasting letters with their wishes and aspirations, lighting lamps and tying amulets all around.

The complex also has a baoli, a medieval step-well, a network of which ran throughout Delhi, supplying water. The Grand Mosque (Jami Masjid) that stands at the edge the fort was so magnificent that Timur stopped there for prayers in 1398.

The ground and the fort lie just outside the southern boundary of “Old Delhi”, the name given to emperor Shahjahan’s walled city, the seventh and final “old” version of Delhi. It is a world in itself, conserving the way of life of a bygone era. A drive along Delhi’s outer ring road, where the Yamuna used to flow, delivers views of the Red Fort, the Old Fort (the sixth city), and the Kotla, with the pillar tall, and the stadium lights in the background, juxtaposing the new and the old, giving an impression of moving through time while moving through space.

On Bahadur Shah Zafar Marg, where one enters the stadium, there lies the grand Kabuli Darwaza (gate), which served as an entrance to Sher Shah Suri’s city, built after Ferozabad. At the fall of the Mughals in 1857, the last heirs of the dynasty were shot dead and hung here for display for three days. Perhaps that is why it is now called the Khooni (bloody) Darwaza. A little further north is the Delhi Gate, the southern gate of emperor Shahjahan’s city. Sunday mornings used to see a mile-long bazaar ­of second-hand books, with everything from pulp fiction to niche collectibles available at a fraction of the market price, which was recently moved away.

Visits to the Kotla are never complete without gorging on the authentic Delhi food that neighbours the area. The stadium borders Ansari Road, which skirts the edge of Old Delhi. Before the morning session of a Test, this place offers a quintessential Delhi breakfast of aloo-poori (fried bread with potato curry), sold by migrants who came to Delhi after the Partition of India for more than 40 years. Further inside Old Delhi, one can find nihari, a rich, spicy mutton or buff stew, cooked overnight and served with thick flatbread.

After day-night games, one can go back to Old Delhi to visit the iconic Karim’s, an eatery serving authentic Mughlai food since the Delhi Durbar of 1911. It remains open until late in the night, and remains jam-packed after matches at the nearby Kotla. My fondest memory there was sharing a table with four other groups at 1 am after an IPL game, digging into red-hot chicken curries and kebabs. The frills and fancies of fine-dining are absent, the focus is on the grub.

The Feroz Shah Kotla ground courts history and culture, in more ways than one. It takes its name from a magnificent medieval fort, and borders an imperial walled city, replete with delectable food and unique heritage. And it has seen some of Indian cricket’s greatest moments. The next time you buy tickets to it, remember and explore the remarkable past and present of the place, both on and off the field.

The RAA is the difference of a batsman’s runs scored or bowler’s runs conceded and the average score in the situation. So, for each combination of (wickets down, innings phase, ground), I calculate the average runs scored per ball, and then subtract that from a player’s runs scored (or conceded). Adding this over all balls in a season gives the season RAA.

This can be converted into an impact on the player’s team’s win percentage. I report a Wins Above Average corresponding to an RAA value, which is defined as the wins earned above the average player in a 14-match season.

In the following tables, I have reported Season RAA, RAA per Ball, RAA per Inns, and WAA.

At the end, I’ve linked the full sortable tables.

Batting

Top 10

Bottom 10

Bowling

Top 10

Bottom 10

Full Table Links:

Batting

Bowling