The Face of the Data: Herman Chernoff, SYSTAT, and the Art of Seeing Statistics

I have always had a particular affection for Chernoff Faces.

Decades ago, when I was using SYSTAT for statistical analysis, I would occasionally generate these peculiar little icons. Some smiled. Others frowned. A few looked surprised, suspicious, or vaguely irritated. They resembled cartoon characters, but not in the ordinary sense. They were data.

Every feature could represent a numerical measurement: the width of a face, the size of the eyes, the angle of the eyebrows, or the curvature of a mouth. A table of multivariate observations became a small crowd of expressive characters. I thought the idea was brilliant then. I still do.

What if one of our best instruments for recognizing patterns was something the human brain already knew how to read: a face?

A statistician gives the data a face

In 1973, Herman Chernoff published “The Use of Faces to Represent Points in k-Dimensional Space Graphically” in the Journal of the American Statistical Association. His proposal appears both practical and wonderfully eccentric. Human beings are skilled at distinguishing faces, so perhaps facial features could carry several dimensions of numerical information at once.

Chernoff’s original design allowed as many as eighteen numerical features to be encoded. That does not mean viewers can reliably recover eighteen numbers from a glance. It means a single observation can be rendered as an object whose shape varies systematically with a collection of measurements.

The distinction is critical. Representation is not the same as interpretation. An image can contain information without making all of that information equally accessible.

From a baseball season to a face

To investigate the idea, I returned to familiar territory: baseball. I used an existing Lahman-derived offensive dataset covering 1954 through 2025 and selected twenty notable player-seasons, one season per player, using a five-variable offensive score. This is a deliberately selected comparison set, not a random sample of all MLB seasons or a definitive ranking of the greatest twenty seasons.

Five measurements went into the experiment: on-base percentage (OBP), isolated power (ISO), walks per plate appearance (BB/PA), strikeouts per plate appearance (SO/PA), and net stolen bases per plate appearance (NetSB/PA). Strikeout rate was reverse-coded so that the corresponding standardized measure rewarded fewer strikeouts. The first four measures emphasize reaching base, power, and discipline; the fifth features a base-running dimension.

My broader Model C uses seven offensive variables. This experiment uses five because those are the measures available in the comparison dataset. Runs per plate appearance and RBI per plate appearance are not included in this particular set of faces.

We can begin by describing each season as a 5-dimensional vector:

Because the variables have different ranges, we standardize them before building a composite:

The resulting score is the sum of the five appropriately oriented standardized measures:

Standardization was carried out in the historical source analysis; the Chernoff experiment uses the resulting player-season values. The purpose of the score is to select a diverse set of outstanding performances, not to settle which player was best.

Same numbers, different faces

The first experiment is simple. Draw each of the twenty seasons as a face, assigning each offensive variable to one visual feature. Then draw the same twenty seasons again, changing the assignments while leaving every numerical observation untouched.

In one mapping, OBP might control mouth curvature while another variable changes eye size. In the second mapping, the same variables are redirected to different features. Facial shapes should therefore change, although the players’ numbers and their underlying multivariate relationships do not.

Figure 1. The same 20 player-seasons under two assignments of numerical variables to facial features. The data do not change, but the expressions do.

Barry Bonds’s 2004 season is especially memorable. In the first mapping, his face smiles. In the second, it frowns. Nothing happened to his on-base percentage or his power. Only the drawing rules changed.

This is both the charm and the danger of Chernoff Faces. A smiling face appears positive, even when the statistical variable driving the smile may have no positive emotional meaning at all. A face with an ominous expression may represent a remarkable performance.

Why a face can mislead

We are not equally sensitive to every feature of a face. Mouth curvature and eye size can dominate a first impression, while a comparably large change in nose length may be barely noticed. Therefore, a variable mapped to the mouth can have more perceptual influence than an equally important variable mapped to a subtler feature.

Chernoff and M. Haseeb Rizvi studied a version of this problem in 1975. Their work showed that reassigning variables to facial characteristics could affect classification errors. The issue is not simply philosophical. Choice of visual encoding can change how easily observers classify the same observations.

Figure 2 concentrates on selected players so that this transformation is easier to see.

Figure 2. Selected player-seasons displayed with alternative facial mappings. These are illustrative Chernoff-style glyphs, not measurements of player personality or emotion.

There is an important asymmetry here. The underlying distance between any two observations stays fixed. But the viewer’s impression of similarity may move with the mapping. A graphical convention has introduced a psychological variable into the interpretation of a mathematical relationship.

Every statistical graphic involves choices. A histogram depends on bin width, a heat map on its palette, and a scatterplot on its axes. Chernoff Faces make that dependence unusually visible, because our response to faces is immediate and difficult to suppress.

Chernoff meets Tukey

John Tukey emphasized that exploratory data analysis starts with discovering structure, not simply confirming a predetermined hypothesis. On that view, Chernoff Faces make sense. They are not a statistical test and do not supply a confidence interval. They ask us to notice something unexpected and then check it by more rigorous methods.

Imagine two player-seasons with nearly identical composite scores. One may derive its advantage from power and walks, the other from contact and base running. Their total scores may conceal substantially different offensive profiles. A set of faces could make that contrast apparent before we examine the component measurements.

The key word is “might.” A face that attracts our attention is not evidence of a genuine cluster or outlier. It is a reason to look for one.

What if the faces lived inside a PCA?

I also tried a hybrid: using a principal component analysis to position the player-seasons while drawing Chernoff Faces instead of ordinary dots. PCA finds directions through the numerical data that account for as much variation as possible. The faces retain the separate measurements in their features.

For this selected sample, the first two principal components explain approximately 51.7% and 28.0% of the standardized variation, respectively, or about 79.7% together. Those percentages describe this selected set of twenty seasons, not all baseball players.

Figure 3. Chernoff-style faces over a two-component PCA display. Small markers show exact PCA positions; some faces are offset to preserve readability.

The result is intriguing. The coordinates provide a reproducible mathematical geography of offensive profiles. The faces add a visual indication of how those profiles differ. At a glance, we can ask not only which observations are near one another, but what combination of traits makes them distinctive.

We should resist reading too much into the appearance of a particular face, especially in a small, hand-selected sample. Still, as a device for generating questions, the hybrid has real appeal.

Distance is mathematical, resemblance is perceptual

Suppose we want to compare two seasons in standardized five-dimensional space. Their straight-line distance is:

Changing which variable controls the mouth or the eyes does not change this distance. Yet if the visual features attract different amounts of attention, the apparent resemblance between two faces can change dramatically.

We can even construct an illustrative model in which facial features receive unequal weights. In a preliminary exercise with assumed weights, the rank correlation between numerical distance and visually weighted distance was about 0.70 under one mapping and 0.59 under another. These are properties of the chosen weighting model, not results from a study of human observers. They should not be interpreted as measured perception.

The proper experiment would ask actual viewers to sort faces into groups or identify outliers, with facial assignments randomized across conditions. We could compare those responses with a distance matrix or clustering solution computed from the original set of variables. That should test the gap between statistical structure and perceived similarity rather than simply illustrate it.

Would I use Chernoff Faces today?

Certainly (I just did). Not as a replacement for regression, PCA, clustering, or conventional graphics, but as a companion to them. I would use them to provoke questions, compare profiles, and remind readers that any plot is a translation of numbers into something our senses can grasp.

I might even use them in a larger study of great offensive seasons, expanding the historical dataset to include Babe Ruth and restoring the full seven-variable Model C. But that would require rebuilding the dataset and rechecking historical comparability; our present experiment begins in 1954 and cannot legitimately include Ruth.

I remember the pleasure of producing these images in SYSTAT, back when statistical graphics were much less elaborate than they are now. Their simplicity was part of their appeal. A face could tell you where to look without pretending to tell you everything you needed to know.

More than fifty years after Chernoff’s paper, I still think this is a marvelous idea. It sits at the intersection of mathematics and perception, where many of the most interesting questions within data analysis begin.

The numbers haven’t changed; only the way we look at them has.

References

Chernoff, H. (1973). The use of faces to represent points in k-dimensional space graphically. Journal of the American Statistical Association, 68(342), 361–368.

Chernoff, H., & Rizvi, M. H. (1975). Effect on classification error of random permutations of features in representing multivariate data by faces. Journal of the American Statistical Association, 70(351), 548–554.

Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley.

The DJ Who Shuffled Every Song and Still Got Track Seven

About halfway through a twelve-track album, the DJ hears a familiar transition. The sixth song has just ended, the opening notes of the next one come through the speakers, and somebody at the other end of the room says, “I thought you put this on shuffle.” The DJ checks the display. Sure enough, the seventh song on the album is playing seventh.

I sympathize with the listener. If I bothered to shuffle an album and then heard a song exactly where I expected to hear it, I might check the display myself. But before we send the equipment out for repair, let me offer a wager. Which is more likely on a fair shuffle: that every song moves to a different position, or that at least one winds up right where it started?

I’ll take the second bet. On a twelve-track album, it wins about 63.2 percent of the time.

A three-song experiment

Let’s give the DJ a three-song EP and call the tracks A, B, and C. They appear on the original recording in that order. For our experiment, the player uses each song once, and each of the six possible running orders has the same chance of appearing. You can check the wager yourself by covering the right-hand column and looking for songs that kept their seats.

Only BCA and CAB move every song. Even playing the album backward leaves B sitting comfortably in the middle. Four of the six orders give our suspicious listener something to point to, so the bet on at least one match wins two times out of three.

The two orders in which everything moves have a name. Mathematicians call each one a derangement, a word I would have reserved for the fellow keeping written records of what the DJ played.

When did you make your prediction?

Back at the twelve-track album, suppose you write “track seven will play seventh” on a slip of paper and hand it to the DJ before he presses shuffle. Only one of the twelve slots will satisfy that prediction. Your chance of being right is one in twelve, or about 8.3 percent.

Now put the pencil away and listen to the whole album. If track two plays second, you can point that out. If track nine plays ninth, you can point that out just as readily. By the end of the evening, you may remember one striking coincidence, but the shuffle had twelve opportunities to supply it.

There is a wrinkle in working out those odds. The twelve opportunities depend on one another because every song uses up a slot. Imagine that eleven songs have somehow kept their original positions. Where, exactly, do you propose to put the twelfth?

One song, on average

We can answer one question without untangling those dependencies. If the DJ shuffles the album night after night, how many songs per night should he expect to find in their original positions?

Consider the three-song experiment again. Read down the right-hand column and count the matches: three, one, one, zero, zero, and one. That makes six matches distributed across six equally likely playlists, for an average of exactly one. The original order, ABC, contributes three matches all by itself, enough to balance both evenings when nothing matched.

The same reasoning works for twelve songs. Each song has a one-in-twelve chance of keeping its position, so adding the twelve individual chances gives an expected count of exactly one. Put a hundred songs on the playlist, and the arithmetic becomes a hundred times one hundredth. I rather like that trade: every individual prediction gets worse, yet the expected number you get right stays precisely where it was.

Where the 63.2 percent comes from

The average count leaves one piece of the wager unsettled. A playlist with three matches earns the listener the same single win as a playlist with one, so we need to count playlists with a match. We can do that by correcting for the occasions we counted twice, then correcting for the occasions that correction went too far.

The three-song EP shows it in action. Adding the individual matches gave us six, but ABC appeared in that count three times. Subtract the three overlapping pairs of matches, all of which belong to ABC, and we have erased that playlist altogether. Add it back once for the three-way overlap, and the count comes out right: six minus three plus one gives four winning playlists.

For a twelve-track album, the same bookkeeping gives the following probability. The exclamation point means factorial, so, for example, 4! is the product of 4, 3, 2, and 1.

Those fractions get small in a hurry. The first four terms already give 62.5 percent, and carrying the calculation through the twelfth gives approximately 63.212 percent. As the playlist grows, the probability approaches a familiar mathematical quantity in an unexpected setting: one minus the reciprocal of Euler’s number. Rounded to three decimal places, that limit is 0.632.

The DJ makes a small adjustment

By the following weekend, the DJ has devised a cure for the complaints. He lets the player generate a playlist, checks it against the back of the CD case, and presses shuffle again if even one song has kept its slot. Eventually a playlist passes inspection, and the evening proceeds without incident.

I wonder how many listeners would judge the new arrangement an improvement. Every song now goes somewhere else, which was what they expected from shuffle in the first place. All the DJ had to do to make his twelve-track album sound properly random was throw away nearly two-thirds of the orders a fair shuffle would have given him.

A pseudo-review of Haruki Murakami’s What I Talk About When I Talk About Running

Author’s Note: Haruki Murakami is a perennial contender for the Nobel Prize in Literature; I am not. He has run over 25 marathons; I have 6 under my belt. He might run more; I certainly will not. My right hip is in bad shape, as are numerous other joints. He has sold over 150,000,000 books; I have not. I don’t think he’ll be awarded a Nobel Prize because he sells way too well. Historically, the prize goes to authors who are not popular. Even though I am not popular, I also will not be awarded any such prize. That said, he and I do have some things in common, and I greatly admire him. Let’s get to work.

I used to write pretty much all my blog posts from ideas that came to me while running. I wrote the pieces later, slowly at the keyboard, but the idea came to me during the run. A random thought would become interesting as the miles clicked off. Two or three things (almost invariably three) that I’d never thought of putting together would suddenly seem to be connected. By the time I’d sat down to write, I’d have a full outline.

I’m older now, and that particular creative process is over. I miss coming home from my runs with lots of ideas. Running had become an integral part of how I thought, and I’d gotten used to finding ideas in it. Now I get absolutely nothing. I suppose it’s just the normal thing with age (alongside the fact that I’m pretty much in pain in all the right joints as I hobble along at a sub-14:00 pace).

That’s the personal baggage I bring to Haruki Murakami’s What I Talk About When I Talk About Running. It’s a memoir about his relationship between running and writing, viewed through the lens of training, memory, and reflection on his own work. The NYC Marathon is the catalyst for the book, but the real interest lies in the routine habits that accompany the work. In some sense, we are kindred spirits.

Murakami writes about those little habits with a most engaging lack of fuss. The weather, the music, the mileage, and the sheer practicality of fitting one thing into another’s schedule all are addressed. I quite like the amount of attention he pays to the mundane. A life spent writing still involves many hours of deciding when to get up, what to do next, and whether you’ve got the energy to do it. And let me tell you, the energy part comes into clear focus as I age.

There’s a very interesting difference between his experience and mine. Murakami describes running as a way to calm the mind. My runs used to generate thoughts that I wanted to keep. What I see in his description, though, is the relief of being able to step away from the need to make every thought useful. In my case, the useful thought seemed to like being left to itself; it was happy to be a passing thought.

I can’t actually explain why it worked. Maybe it was just the repetition. Maybe the idea needed time to grow up in its own right rather than me just sitting on it and demanding an answer. I’m not in any position to give a scientific explanation of something I know essentially through experience (imagine that?!). All I know is that I went running and came back with stuff to write about.

The setup was absolutely ideal for a writer. You didn’t need to make a separate appointment to sit down and think of a subject. Thinking was part of the activity you were already engaged in, and the writing simply grew out of that. Only now do I realize how much of the work was done before I even got to the desk.

Muraki’s view is that you have to leave yourself enough energy for the next run. You have to be able to stop a day’s writing and still have something to say. His point is about continuity, really, about being able to go back the next day. I think that’s a more useful observation than any sweeping statement about inspiration. If you’re writing regularly, you’re going to wake up the morning after a good idea and still have a lot of work to do.

Still, I have issues with the running and writing comparison. You can take it too far. Writing involves false starts, wrong turns, and there are days when the only thing you can do is delete a paragraph rather than add a page. Tenacity matters, but so does recognizing when what you have been doing needs to change.

That’s also where the book’s small size is both a real strength and a real limitation. The observations are very accessible, and you do get a sense of things, but they don’t always go on to really explore the creative process. If you’re looking for a detailed insight into how a particular novel was written, you may be disappointed. I’m one of those people who sometimes does want more. For me, the book is at its best as the company of someone who has spent a lifetime thinking about this.

It’s all the more pertinent in the context of aging. Murakami talks about continuing to train as his marathon times get worse and the difficulty of accepting what those times tell him. The relationship between effort and improvement which has always held true, starts to fall apart. That’s a very unsettling experience, particularly when that same relationship has worked for you for so many years.

Over four decades ago, “The Beav” and I used to go for 10-mile training runs. We would finish in under an hour with no difficulty, talking as we cruised through a local, well-known park. Now I cannot run 6 miles in an hour. Honestly, I can not get close. Is there a relationship between inspiration and speed? I’ll leave that for a later post. As of now, I have no idea. All I know is that I am slow and uninspired.

My disappointment is harder to quantify. There’s no record of when running stopped supplying me with ideas for my posts. All I have is the memory of how easily it used to do so, contrasted with the reality that it doesn’t anymore.

I link that change to aging, though I can’t prove age is the explanation. The experience is different; the explanation is less clear. I’d rather own up to that uncertainty than turn a personal bereavement into some sort of universal rule about what happens to us all.

What I miss is the ease of it. A very significant part of my life used to dovetail with another very significant part without any negotiation at all. Running and writing went hand in hand. Now I have to think about a relationship I once took for granted, which is, of course, something familiar to anyone who’s been around long enough to see an old routine change.

Murakami is the perfect example of that, as he absolutely allows for dissatisfaction. His love of running makes room for fatigue and disappointment. That’s what gives the book its weight. You can love a pursuit and still admit it’s gotten harder, that it doesn’t always pay off as much as it used to.

I’d particularly recommend What I Talk About When I Talk About Running to people whose lives are organized around a single activity. The book’s appeal lies less in the mileage Murakami covers than in how a repetitive act can become part of your sense of self. It’s very restrained in its prose, very personal in its scope, and the insights build up slowly. Some people will find that a bit thin, I’m sure; I think much of its value is in the space it leaves for my own experience.

For years, running was integral to my creative process. I had no idea how long that would last, and I certainly never thought to ask. Murakami’s book has helped me appreciate the arrangement far more now. However, appreciation alone does nothing to lessen the fact that I usually come home from a run too tired to do anything but sit on the floor.

I’d still like to come back from a run with the notion that I ought to sit down and write. Hope springs, but I am not counting on a return to my glory days,

 Postscript

Well, it appears I have lied throughout this essay, both on purpose and by omission. I know exactly why I am no longer inspired when I go for runs. I also refused to tell you why I read the book in the first place. Both situations involve much younger women who caught my attention. I guess I didn’t mention either of them because every author has a serious decision to make: how much of themselves are they willing to reveal to a total stranger, someone they do not know and will never meet. In this instance, I decided to do the opposite of what I have done in previous posts; I invoked my Fifth Amendment privilege and moved on.

The Thief Who Stole Only the Peel: How a seemingly modest agreement can cost you almost an entire orange

A thief approaches an orange merchant (what?) with an unusual proposal. He wants to take an orange, but promises to leave its interior untouched. Is he drunk or otherwise impaired? Is he, perhaps, some sort of citrus provocateur?

He wants only the peel. More precisely, he wants the outer layer extending inward through one-tenth of the orange’s radius.

The merchant considers the offer. If the orange has a radius of ten centimeters, the thief will remove a one-centimeter-thick layer, leaving a ball with a radius of nine centimeters.

That sounds like a substantial peel, but an acceptable bargain. The merchant agrees, assuming that most of the orange will remain on the counter.

For an ordinary orange, he is right. Unfortunately, the thief has selected an orange with one hundred dimensions.

The first orange

Before considering the merchant’s unfortunate predicament, let us examine an ordinary three-dimensional orange. We will treat it as a perfect sphere with uniform density, and “peel” will mean the specified outer layer rather than its actual skin.

The volume of a sphere is proportional to the cube of its radius. If we reduce its radius to 90 percent of the original, we retain 90 percent × 90 percent × 90 percent of its volume.

The fraction remaining is therefore: (0.9)³ = 0.729

The merchant keeps 72.9 percent of the orange, and the thief receives 27.1 percent. A layer occupying only one-tenth of the radius has taken more than a quarter of the volume.

That result already deserves a moment’s attention. A percentage of the radius is not the same thing as a percentage of the volume, because volume depends on three dimensions working together.

The merchant may have underestimated the thief’s breakfast. Nevertheless, he still has most of his orange.

An orange we cannot picture

A hundred-dimensional orange is difficult to imagine, which is a polite way of saying that I cannot picture one. Fortunately, you don’t need to picture an object to define it mathematically.

On a flat surface, a disk consists of all points within a specified distance of a center. In three dimensions, the corresponding solid object is a ball, with each point located by three coordinates.

We can extend that definition to one hundred coordinates. A hundred-dimensional ball consists of all points whose Euclidean distance from its center is no greater than its radius.

These dimensions do not have to represent additional directions we could physically walk. In data analysis, coordinates can represent measured variables, although the choice of units and scaling then matters.

We can study this ball’s volume even when our visual imagination has stopped cooperating. The key is that its volume scales with its radius raised to the number of dimensions.

In three dimensions, the power is three. In one hundred dimensions, the power is one hundred.

The same agreement, a different breakfast

The thief’s agreement remains unchanged. He takes the outermost tenth of the radius, leaving an inner ball with 90 percent of the original radius.

For a ball with dimensions, the fraction of the volume left inside is: (0.9)ⁿ

The fraction taken by the thief is whatever remains after subtracting that quantity from one. We can therefore write: 1-(0.9)ⁿ

There is a constant in the full formula for the volume of a ball, and that constant changes with the number of dimensions. Here it cancels, because we are comparing two balls in the same number of dimensions.

The results make the thief’s interest in unusual produce easier to understand. Each row applies to the same agreement.

In ten dimensions, the merchant has already lost nearly two-thirds of the orange. In fifty dimensions, he keeps approximately half of one percent.

At one hundred dimensions, the calculation becomes: (0.9)¹⁰⁰ ≈ 0.00002656

That is a fraction, so we multiply by one hundred to express it as a percentage. The merchant retains approximately 0.002656 percent of the original volume.

The thief receives approximately 99.997344 percent. He has taken almost the entire orange while honoring every word of the agreement.

Where did the interior go?

The natural temptation is to imagine that something has pushed the orange’s contents outward. Perhaps the center has become hollow, or the fruit has accumulated against its skin.

Neither has happened in our mathematical model. The ball is uniformly filled, and the amount of material per unit of volume is the same throughout.

The central region still exists. It simply accounts for an extraordinarily small fraction of the total volume.

The arithmetic explains why. Every increase in dimension adds another factor of 0.9 to the fraction the merchant keeps, and repeated multiplication by a number smaller than one steadily reduces the result.

Our everyday experience gives us little reason to expect such a dramatic reduction. We live in three spatial dimensions, where keeping nine-tenths of the radius still leaves nearly three-quarters of the volume.

In one hundred dimensions, the same proportional reduction leaves almost nothing by comparison. Our familiar intuition carries over into a setting where it no longer provides a reliable estimate.

Choose a point anywhere

Another way to understand the result is to imagine selecting points uniformly at random throughout the hundred-dimensional orange. “Uniformly” means that regions with equal volume have equal chances of receiving a selected point.

Approximately 99.997 percent of those points will fall in the outermost tenth of the radius. If we select one hundred thousand points, the expected number in the inner ball is only about 2.7.

This does not mean that every sample will leave exactly two or three points inside. It means that, over repeated samples, the average count there would be approximately 2.7.

The points have no preference for the edge. The outer layer receives almost all of them because it contains almost all the available volume.

There is also a distinction between being near the surface and being on it. The mathematical surface has zero volume, so a uniformly selected point has probability zero of landing exactly there, even though landing nearby is overwhelmingly likely.

Why this matters beyond imaginary fruit

A dataset with many measured variables can be represented as a collection of points in a space with many coordinates. A baseball player, for example, might be described by on-base percentage, isolated power, strikeout rate, walk rate, sprint speed, and numerous other measurements.

Add enough variables, and the number of dimensions quickly exceeds anything we can visualize. We may still describe observations as “close,” “central,” or “far away,” but those descriptions now depend on geometry that can behave quite differently from the geometry of a baseball field.

Real datasets do not automatically behave like points distributed uniformly inside a ball. Variables may be correlated, observations may cluster, and changing a measurement’s scale can change the distances between points.

The orange therefore teaches us about our expectations, rather than providing a universal description of data. It shows how a familiar geometric judgment can become unreliable when the number of dimensions changes.

Methods such as principal component analysis help us investigate structure in data with many variables. They can also give us a lower-dimensional view, although that view necessarily leaves out some information unless the data already lie within the retained dimensions.

There is a useful discipline here. Before trusting a picture formed in three-dimensional experience, we should ask whether the mathematics supports carrying that picture into a space with fifty or one hundred dimensions.

Reading the agreement

The merchant understood that the thief would remove a layer one-tenth of the radius thick. His mistake was assuming that this measurement told him, intuitively, how much orange he would surrender.

In three dimensions, that assumption was imperfect but survivable. In one hundred dimensions, it cost him almost everything.

In this world, the thief leaves with his peel, and the merchant is left studying a very small remainder. Word on the street is that next time, he plans to negotiate in percentages of volume.

The Distribution Was There All Along: Why Scale Matters in Stem-and-Leaf Plots

Author’s Note: Are there any aspiring scientists reading this post? If so, pay attention to what you are about to read. I always start any serious study with stem-and-leaf plots, and you should, too. Read on and find out why. And yes, archaeologists work in meters, but most people who read my blog are from the USA, so I will use the imperial system. That said, I want to recognize the growing number of Chinese readers clicking on my site; I see you, and I deeply appreciate that you are here. Let’s begin…

When I taught archaeology in an anthropology department at a Midwestern university about a million years ago, I sometimes started class with a simple exercise. I asked the students to write down their heights.

No archaeological theory. No artifacts. No radiocarbon dates. Just height. Then we put the observations together and constructed a stem-and-leaf plot.

The exercise had a larger purpose. Human height data can contain more than one concentration, especially when a classroom contains individuals drawn from populations with different average heights. If those concentrations are sufficiently separated, the resulting distribution can be bimodal. But that exercise hides another lesson, and it is an important one. Whether we see bimodality depends on how we construct the stem-and-leaf plot.

A Synthetic Classroom

To demonstrate the idea clearly, I created a synthetic dataset containing 200 student heights. I generated the observations from two overlapping height distributions, with 100 observations in each group. The first is concentrated around approximately 64 inches, and the second around approximately 71 inches.

The combined dataset has an overall mean of:

67.44 inches

and a median of:

67.00 inches

Those numbers are mathematically correct. But they do not tell us the most interesting thing about the data.

There are two concentrations. That distinction is critical because a measure of central tendency answers a particular question: where is the center of the combined data?

It does not necessarily answer another question: what does the distribution actually look like?

Those are not always the same problem.

Start With a Coarse Stem

Suppose we begin with a conventional-looking stem-and-leaf plot in which the tens digit serves as the stem.

A height of 64 inches would therefore appear as:

6 | 4

A height of 71 inches would appear as:

7 | 1

That seems reasonable enough.

Figure 1. The 200 synthetic heights displayed using a coarse stem-and-leaf scale.

The problem becomes apparent almost immediately. There are only a few stems, and an enormous number of leaves are crowded onto them. The individual observations are technically still present, but the visual structure of the distribution has been compressed.

The plot contains the information; it just doesn’t display it effectively. This is an important distinction. A graph can be numerically correct while still being visually uninformative.

Change the Scale

Now consider exactly the same dataset with a different stem-and-leaf scale. Instead of using the tens digit as the stem, let each whole inch become a stem. Because our measurements were recorded to the nearest half-inch, we can use two possible leaves:

0 for a whole inch

and

5 for a half inch.

Thus,

64 | 0

represents 64.0 inches, while

64 | 5

represents 64.5 inches.

Nothing about the underlying dataset has changed; only the scale has changed.

Figure 2. The same 200 synthetic heights using whole inches as stems and half inches as leaves.

The distribution is much easier to interpret now. The observations are no longer compressed into only a handful of rows. Instead, the frequencies expand vertically across the range of heights.

The two concentrations become visible. One cluster occurs in the lower-to-mid 60s. Another occurs around the low 70s. Between them is a region with fewer observations.

That valley is the clue. We are looking at a bimodal distribution.

Same Data, Different Picture

This is what makes the example useful. Figure 1 and Figure 2 contain essentially the same information. The difference is not the sample. It is not the statistical method. It is not the population. The difference is scale.

A stem-and-leaf plot has a built-in parameter that we sometimes treat as mere formatting. It is not.

The choice of stem width determines how much of the distribution’s internal structure we can see.

Use stems that are too broad and separate features may be compressed together.

Use stems that are too narrow, and the opposite problem can occur. The distribution may become excessively sparse, with random sampling variation producing a jagged pattern that looks more meaningful than it really is.

Therefore, no stem width is universally correct.

Scale must fit the data.

Check It With a Histogram

We can verify what the finer stem-and-leaf plot is showing by constructing a histogram of the same 200 observations.

Figure 3. Histogram of the same synthetic height data.

The histogram confirms the pattern. A lower-frequency region separates two relatively high-frequency regions. The distribution doesn’t behave like one simple mound centered around 67 inches.

That makes the overall mean of 67.44 inches especially interesting. The mean is not wrong. It answers a question that may not be the one we actually care about.

If the underlying dataset represents a mixture of two populations, the combined mean can fall in a location that isn’t particularly representative of either population.

This is one reason exploratory data analysis should generally come before aggressive numerical summarization. Right? Do you see it?

The Average Can Hide the Population

Suppose someone gave us only these statistics:

Nothing looks obviously suspicious. We might reasonably imagine a single distribution centered somewhere around 67 inches. But that interpretation would be incomplete.

The graphical analysis tells us something the mean and median cannot: the observations appear to be concentrated around two different regions. This does not mean you can’t calculate measures of central tendency for bimodal data. They certainly can.

It means a single measure of central tendency may poorly describe a multimodal distribution. That is a much more important lesson.

Why Stem-and-Leaf Plots Are Still Relevant

Histograms are excellent for examining distributions, but stem-and-leaf plots have an easy-to-overlook advantage. They preserve the observations.

A histogram converts individual measurements into frequencies inside bins. Once we see a bar containing 18 observations, we know there are 18 observations in that interval, but we no longer see each value.

A stem-and-leaf plot provides both. We see the shape of the distribution while retaining the underlying measurements.

That makes the technique particularly useful for small and moderate-sized datasets, classroom demonstrations, exploratory analysis, and situations in which we want students to understand that a distribution is constructed from actual observations.

It also makes scale unusually important. Changing a histogram’s bin width can dramatically change its appearance. Changing the stem definition in a stem-and-leaf plot can do the same thing. The underlying principle is identical.

A General Rule for Exploratory Data Analysis

This broader lesson goes beyond stem-and-leaf plots. Whenever we visualize quantitative data, we make resolution decisions. Histogram bin width is a decision. The bandwidth of a density curve is a decision. The limits of an axis are decisions. The size of groups in an aggregated table is a decision. And the scale of a stem-and-leaf plot is a decision.

Those choices influence what patterns become visible. That does not mean we should manipulate scales until we find the pattern we want. Quite the opposite. It means we should examine a dataset at several reasonable resolutions and ask whether an apparent feature persists.

If a second mode appears only under one bizarre choice of scale, we should be skeptical. If the same structure appears across several sensible scales and is confirmed by another visualization, such as a histogram, then the evidence becomes considerably stronger.

The Archaeology Connection

This lesson is particularly relevant to archaeology. Imagine measuring projectile points, pottery thickness, vessel diameters, burial depths, artifact weights, settlement sizes, or some other quantitative attribute.

A single sample might actually contain objects from different chronological periods, manufacturing traditions, functional categories, or cultural populations. If those observations are combined, the mean may fall neatly between them.

The arithmetic may be impeccable; the interpretation may be terrible.

A bimodal distribution can therefore be more than an interesting statistical shape. It can be evidence that what we thought was one population may actually contain multiple processes or populations.

That is why the first question should often (always?) be:

What does the distribution look like? Only after that should we become overly interested in what its average happens to be.

The Lesson

The 200 synthetic heights did not become bimodal when we changed the stem-and-leaf scale. They were bimodal before we touched the graph. The finer scale simply allowed us to see what was already there.

That is the lesson I wanted students to encounter when they wrote their heights down in class. Statistics is not merely the calculation of means, medians, standard deviations, and other summary measures. It is the search for structure in data. Sometimes that structure sits right in front of us, waiting for us to choose the right scale.

The Distribution of Offensive Performance in Major League Baseball, 2026

A Statistical Examination of Baseball’s Best, Worst, and Most Typical Hitters

I’ve completed an analysis using FanGraphs data, and the results are quite interesting.

The most important initial discovery is that offensive performance among established MLB hitters is remarkably close to a normal distribution. However, that distribution changes substantially depending on the minimum number of plate appearances we recognize.

I created five figures to illustrate the findings.

Establishing our population

The expanded FanGraphs dataset contains 1,472 player records. Of those, 662 recorded at least one plate appearance during the 2026 regular season. For the primary analysis, I selected 250 plate appearances, giving us 318 players. However, I also examined several alternative thresholds.

Notice what happens as we increase the minimum playing-time requirement. The mean rises, while the standard deviation generally decreases. This is consistent with two processes operating together: small-sample extremes becoming less influential and better hitters tending to accumulate more plate appearances.

There is an important selection effect here. Players with poor offensive performances are less likely to receive 500 or more plate appearances. Consequently, the average performance of our qualified hitters is considerably better than the MLB-wide benchmark of 100 wRC+.

Is offensive performance normally distributed?

For our primary population of 318 players, the descriptive statistics are:

The mean and median are remarkably close. Furthermore, the skewness of 0.06 suggests that the distribution is approximately symmetrical. The kurtosis is also reasonably close to that expected under a normal distribution.

Figure 1. Distribution of wRC+ among players with at least 250 PA

The histogram shows a distribution that closely resembles the familiar bell curve. Kind of beautiful, isn’t it?

I also performed a Shapiro-Wilk normality test.

The result was:

At the conventional significance level of 0.05, we cannot reject the hypothesis that the observations came from a normally distributed population. This does not prove that offensive performance follows a normal distribution. Nevertheless, the descriptive statistics, histogram, and formal test all support approximate normality within this selected group. That gives us a useful starting point for identifying unusually good and unusually poor offensive seasons.

What happens when we change the qualification threshold?

This is where the analysis becomes particularly revealing.

Figure 2. Box plots by minimum plate appearances

These box plots have a story to tell. As the qualification threshold increases, the entire distribution shifts upward. The median moves from approximately 96 wRC+ among players with at least 100 plate appearances to approximately 111 among players with at least 502. At the same time, the lower tail becomes less extreme.

This suggests that playing time is strongly associated with offensive performance, although the relationship does not establish causation.

We must also remember that these groups overlap. The 502-PA sample is contained within the 400-PA sample, which is itself contained within the 250-PA sample.

Figure 3. Plate appearances versus offensive performance

This scatterplot makes the relationship clearer.

Among the 465 players with at least 100 plate appearances, the Pearson correlation between PA and wRC+ is approximately:

That is a moderate positive association. Players accumulating more plate appearances generally have better offensive statistics. However, this does not mean more plate appearances make a player a better hitter. Rather, teams are likely to give more opportunities to players who perform better. Injuries, defensive contributions, contracts, and roster decisions also influence playing time. This distinction will become especially important when we begin constructing regression models.

Identifying the best offensive performances

We can now begin identifying the extremes of the distribution.

Using our primary 250-PA threshold, the five highest wRC+ performances were:

Alvarez stands comfortably ahead of the rest. His wRC+ of 180.82 indicates offensive production per plate appearance approximately 81% above the league-average benchmark after the adjustments incorporated into wRC+. He is also an exceptionally unusual observation within our selected population.

Using the familiar box-plot definition of an outlier, we obtain an upper fence of approximately 156.74. Alvarez exceeds that threshold. Interestingly, Crow-Armstrong, with a wRC+ of 156.52, falls just below it.

The worst offensive performances

At the opposite extreme:

Narváez and Friedl fall below the conventional lower outlier fence of approximately 48.71. Thus, using the 1.5 × IQR criterion, we identify three statistical outliers in our 318-player sample: one exceptionally high and two exceptionally low.

Figure 4. The extremes of offensive performance

These rankings are based on offensive rate performance. They do not necessarily identify the players who contributed the most or fewest total offensive runs. That distinction is our next important finding.

Offensive excellence versus offensive contribution

Consider the following comparison.

Yordan Alvarez recorded a wRC+ of 180.82 over 692 plate appearances.

Pete Crow-Armstrong recorded a wRC+ of 156.52 over 726 plate appearances.

Alvarez was substantially better as a hitter on a rate basis. However, the difference between their total offensive contributions was much smaller.

Why?

Because FanGraphs’ Off statistic includes both batting and baserunning contributions. Alvarez’s extraordinary hitting performance was partially offset by negative baserunning value. Crow-Armstrong, meanwhile, accumulated substantial positive baserunning value.

This illustrates the importance of distinguishing among batting ability, offensive rate performance, and cumulative offensive contribution.

Figure 5. Offensive rate performance versus total offensive contribution

The relationship is extremely strong.

Among the 318 players with at least 250 plate appearances:

Players with higher wRC+ generally accumulate more total offensive runs. However, these measures are not independent. They share underlying batting information, and total offensive runs also depend on opportunities and baserunning. Consequently, the high correlation is expected. It should not be interpreted as evidence that wRC+ predicts Off independently.

The most interesting players may actually be those who deviate from the general relationship. Crow-Armstrong is an excellent example.

6. What about the worst cumulative offensive performances?

Here the rankings change.

Among players with at least 250 plate appearances, the lowest total offensive contributions were:

This illustrates why identifying the worst offensive season requires some methodological care. Carlos Narváez had the lowest wRC+ among players meeting our 250-PA threshold. However, Ezequiel Tovar accumulated the largest negative offensive contribution. Tovar was not the least effective hitter per plate appearance, but he accumulated substantially more playing time.

In other words, the worst offensive rate performance and the greatest cumulative offensive deficit are not necessarily produced by the same player. That is a distinction worth preserving throughout this project.

Some preliminary conclusions

Five important findings have emerged.

First, offensive performance among MLB players meeting moderate playing-time thresholds appears approximately normally distributed.

Second, the distribution changes considerably as we adjust the qualification threshold. This means that rankings, averages, and outlier classifications are sensitive to how we define the population.

Third, Yordan Alvarez clearly distinguished himself as the outstanding offensive-rate performer among players receiving substantial playing time. His performance is also a statistical high outlier using our primary sample.

Fourth, the distinction between hitting effectiveness and total offensive contribution matters. Exceptional baserunning can considerably change a player’s total offensive value.

Finally, players with more plate appearances tend to have better offensive statistics. This suggests that any attempt to model offensive performance needs to consider playing-time selection effects.

A methodological caution

We should keep one additional issue in mind. The closer agreement with a normal distribution at higher PA thresholds does not necessarily reveal the natural underlying distribution of hitting ability. We are observing a selected population. Its composition depends partly on decisions made by managers, front offices, and players themselves.

Thus, the distribution of observed offensive performance is not necessarily identical to the distribution of underlying offensive talent. That distinction could become important in subsequent studies.

Today’s date, written in the familiar American shorthand, is 10/24/26. At first glance, there is nothing particularly remarkable about it. But square those three numbers, and a familiar relationship emerges:

The arithmetic works out neatly. One hundred plus 576 equals 676. Today’s date forms a Pythagorean triple, a set of three positive integers that can serve as the side lengths of a right triangle.

That gives October 24 a small mathematical distinction. The calendar has handed us a geometry problem with the answer already attached.

A Familiar Triangle in Disguise

Most of us first encountered the Pythagorean theorem in school. For a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side, called the hypotenuse:

The theorem applies whether the side lengths are whole numbers, fractions, or irrational numbers. Pythagorean triples are the special cases in which all three lengths are positive integers. The most familiar example is probably 3, 4, 5. Another is 5, 12, 13.

That second triple explains today’s date. Divide each of our three numbers by two:

Our calendar triangle is a scaled version of the 5-12-13 triangle. Every side has doubled, while the angles remain unchanged. The two triangles are similar, although the larger one has four times the area.

Mathematicians call 5, 12, 13 a primitive Pythagorean triple because the three numbers have no common divisor greater than one. The triple 10, 24, 26 is not primitive. It belongs to the same family, enlarged by a common multiplier.

Where the Numbers Come From

Pythagorean triples can seem like numerical accidents when encountered one at a time. Yet there is a systematic way to generate them. For positive integers , the following expressions produce a Pythagorean triple:

Choose 3 for and 2 for . The first expression gives 9 minus 4, or 5. The second gives 12. The third gives 9 plus 4, or 13. Double those results, and we arrive at 10, 24, 26.

The formula works because of an algebraic identity:

Expanding the left side reveals why. The negative middle term from the first square is offset by part of the second term, leaving exactly the expansion of the expression on the right. What initially looks like a lucky match turns out to follow from a general relationship.

There are infinitely many Pythagorean triples. Dates, however, impose much tighter restrictions. A month cannot exceed 12, and a day cannot exceed 31. Only a small portion of that infinite mathematical landscape can fit onto a calendar.

The Calendar Supplies the Coincidence

There is a qualification here. The pattern depends on how we write the date. We use 26 for the year rather than 2026. In a day-first format, the date becomes 24/10/26, which preserves the relationship because exchanging the two shorter sides changes nothing.

Writing the full year removes this particular pattern:

That dependence on notation is part of the appeal. The calendar does not possess some hidden geometric force. We have chosen a way of recording dates that occasionally places meaningful numerical relationships in front of us.

Recognizing that distinction lets us enjoy the coincidence without asking it to explain more than it does. There is no reason to expect today’s events to follow a special course because its abbreviated date describes a right triangle. There is, however, a good reason to pause over the arithmetic.

A Small Invitation to Notice

I like these moments because they make familiar ideas briefly unfamiliar again. A theorem remembered from school appears in the corner of a computer screen. Three ordinary numbers become the sides of a triangle. A date becomes an invitation to investigate.

Following that invitation takes us from arithmetic to geometry, then to scaling, algebra, and the distinction between a pattern and the notation that reveals it. That is a respectable amount of mathematics to find in a date.

October 24, 2026, will pass like any other day. For a moment, though, we can picture it as a triangle: ten units across, twenty-four units up, and twenty-six units along the diagonal. The numbers fit, and noticing why is a pleasure.

Postscript

Check this out. We will have to wait a while for the next one.

So today is the last such date this century. The next is more than 78 years away, followed shortly by April 3, 2105, which reverses the two shorter sides. Not bad, in fact, that is pretty cool.