The Shape of a Distribution
A histogram or boxplot reveals whether a distribution is symmetric, skewed, or multi-peaked, and that shape decides which measure of center you can trust. · 12 min
A column of numbers hides its own shape. Two datasets can share a mean, a median, and a spread, and still be built completely differently — one balanced, one lopsided, one with two separate peaks. To see a distribution you group the values into bins and draw a histogram: bars whose heights show how many values fall in each range. The picture answers questions a summary cannot, and one of them is decisive: which measure of center you are entitled to trust.
Guess before you learn
A very easy exam is graded, and most students score near the 100-point ceiling, with a few trailing down to low scores. What shape will the histogram of scores have?
Left-skewed. The bulk of scores pile up against the high ceiling, and the tail stretches down toward the low scores. The tail — not the peak — names the skew, so a tail on the left means a left skew. If you picked right-skewed, keep the mark: naming a skew by its crowd instead of its tail is the most common mix-up, and this folio fixes it.
9–12
3–5
Turn numbers into a picture by stacking a block for each one. A symmetric shape looks the same on the left and the right, like a hill with matching sides. Most heights of people make a shape like that.
A skewed shape leans: the blocks pile up on one end and a thin tail trails off the other. And a two-peak shape has two separate hills — usually a sign that two different groups got mixed together.
6–8
A histogram can take a few named shapes. Symmetric: the left and right halves mirror each other, with the peak in the middle. Skewed: one tail is longer — a right skew trails toward large values, a left skew toward small ones. Uniform: every bar about the same height. Bimodal: two distinct peaks, usually two groups combined.
Name the shape by its tail, not its peak. The peak is where the crowd is; the tail is the thin part that stretches out. A right-skewed distribution has its crowd on the left and its tail on the right — and that tail is what pulls the mean away from the median.
9–12
The shape is described along three axes. Symmetry: is the distribution a mirror image about its center, or does one tail run longer? A right (positive) skew has a long right tail, a left (negative) skew a long left tail. Modality: one peak (unimodal), two (bimodal), or more — extra peaks usually betray mixed subpopulations. Tails and outliers: are extreme values present, and how heavy are they?
Shape decides which center to trust. For a symmetric, single-peaked distribution the mean, median, and mode coincide, and the mean is the natural report. Under skew they separate — the mean chases the long tail — so the median becomes the honest center and the IQR the honest spread. Bimodality is a warning that no single center is typical at all: the right response is often to split the data into the two groups it is hiding.
K–2
Stack blocks to show how many. If the pile is even on both sides, the shape is balanced — the same left and right.
If the blocks bunch up on one side and trail off on the other, the shape is lopsided. The long thin tail points the way it leans.
Undergrad
Shape is quantified by the standardised moments. Skewness, E[(X − μ)³]/σ³, signs the asymmetry: positive for a right tail, zero for symmetry. Kurtosis, E[(X − μ)⁴]/σ⁴, indexes tail weight relative to the normal's value of 3; excess kurtosis above zero means heavier tails and more outliers. These are summaries, not substitutes for the plot — sample skewness and kurtosis are themselves badly outlier-sensitive, since they average third and fourth powers of deviations.
Modality is a subtler, non-moment feature. A histogram's apparent peaks depend on bin width, so a kernel density estimate with a defensible bandwidth is the sounder read; formal tests (the dip test, Silverman's bandwidth test) exist for claiming multimodality. The practical discipline is unchanged from the school rule: plot before you summarise, because a mean and a standard deviation reported for a bimodal or heavy-tailed sample describe a distribution that may not exist anywhere in the data.
Postgrad
Beyond low moments, shape is the province of the full distribution function and its functionals. L-moments — linear combinations of order statistics — give skewness and kurtosis analogues with bounded influence and existence under far weaker integrability than conventional moments, which is why hydrology and extreme-value work prefer them. Tail behaviour is classified by regular variation: a distribution is heavy-tailed when 1 − F(x) ~ x^(−α)L(x), and the tail index α governs which moments even exist.
This reframes the unit. Center and spread are the first two moments; skew and tail weight the next order of structure; and for heavy tails (α ≤ 2) the variance is infinite and the standard deviation estimates nothing. The choice of summary is downstream of the distribution's shape, and mistaking a mixture or a power-law tail for a well-behaved unimodal law is the error the histogram, read honestly, exists to prevent.
skew
A distribution is skewed when one tail is longer than the other. The direction is named by the tail: a right (positive) skew trails toward large values; a left (negative) skew trails toward small ones. The mean is pulled toward the longer tail.
A histogram is one view of shape; a boxplot is a second, built from the five-number summary — minimum, Q1, median, Q3, maximum. Its box spans the middle half (the IQR from folio 3), a line marks the median, and whiskers reach toward the extremes. A boxplot reads skew at a glance: if the median line sits nearer one end of the box and one whisker is far longer, the data lean the way the long whisker points.
Why is this true?
Why does a clear skew in the histogram mean you should report the median instead of the mean?
Because a skew is a long tail, and the mean is dragged toward that tail while the median stays with the crowd. Reporting the mean would describe a value few subjects are near; the median names where the bulk of the data actually sits, which is what typical is meant to mean.
You have now described a single variable completely: its kind, its center, its spread, and its shape — each choice constrained by the one before. That closes Unit I. Next the ground shifts: instead of describing data you already hold, you ask where it came from and whether a sample can speak for the population behind it.
Practice — new ink and old, interleaved
1.A boxplot of a skewed variable shows Q1 = 12 and Q3 = 39. Report the IQR, the spread figure that pairs with the median.
2.Five quiz scores are 80, 85, 90, 95, 100, with a mean of 90. A sixth student scores 0. What is the new mean of all six?
3.You roll a fair die 6000 times and make a histogram of the results 1 to 6. What shape do you expect?
4.A histogram is clearly left-skewed. Where is the mean relative to the median?
5.Why is it wrong to say 30°C is twice as hot as 15°C?
6.You want to draw a histogram of a variable. Which kind of variable can a histogram sensibly display?
7.Sketch a right-skewed distribution: a tall peak at low values and a long thin tail toward high values (values 0 to 50).
8.Reaction times in an experiment are right-skewed by a few very slow trials. Which center should the report headline?
9.For the sorted data 3, 5, 6, 8, 10, 11, 14, compute the IQR (Q3 − Q1).