Shape: skewness, kurtosis and the box plot

The first four lessons gave you two summaries of a data set: where it is centred and how far it spreads. Neither says anything about its shape. Picture the incomes of the sixty Ashanagar households. The mean is ₹28,783 and the median ₹25,000. Most households earn around the median, and then a few earn a good deal more. Draw the histogram and the right-hand side trails off into a long tail; there is no matching tail on the left, because no household earns less than ₹10,500.

That lopsidedness is called skewness. A distribution is symmetrical if its right half is the mirror image of its left (Hyderabad 2018 words it exactly that way). It is positively skewed if the long tail runs to the right, and negatively skewed if it runs to the left. The word describes the tail, not where the bulk of the data sit.

A second feature matters too. Two symmetrical distributions with the same mean and standard deviation can differ in how much of their weight sits in the centre and how much out in the tails. That is kurtosis, and it gets its own section below. Both ideas are about the same question: what does the picture look like when the mean and the standard deviation have already been taken out?

The mean, median and mode respond to a tail in different ways. The mode sits under the highest point of the curve and ignores the tail. The median is the middle value and moves only a little. The mean adds up every value, so it is pulled furthest toward the tail. That gives an order you can read straight off:

The order holds for the usual single-peaked shapes. Oddly shaped distributions can break it, but exam questions treat it as the rule.

Try it below. The two sliders set how much weight the curve puts on the left and on the right; the three averages are drawn on it, and three coefficients of skewness (explained in the next section) are worked out beside it. Start with the right-skewed preset, then flip to left-skewed and watch the mean and the mode swap places with the median always between them.

The examiners ask the same thing in many forms:

• Mean 23, median 24, mode 25.5: the mean is the smallest, so the distribution is negatively skewed (CUET 2020). The same order, mean < median < mode, is the answer in CUET 2019 and Hyderabad 2019.
• Mode 5, median 15, mean 20: the mean is largest, so the data are skewed to the right (CUET 2021). In a positively skewed distribution the mean lies to the right of the mode (Hyderabad 2011), and the order is mean > median > mode (CUET 2021, JAM 2025).
• The right tail is the mirror image of the left: the distribution is symmetrical (Hyderabad 2018). For a symmetrical distribution the coefficient of skewness is 0 (CUET 2022).
• For a symmetric distribution it is the skewness that is zero (JNU 2021). The mean need not be zero, the variance is certainly not zero, and the kurtosis of a symmetrical curve can be anything.

The order of the averages tells you the direction. To say how much, and to compare two distributions measured in different units, you need a coefficient with no units. There are three families, and exams name all of them.

Karl Pearson’s coefficients

The gap between the mean and the mode grows with the skew. Dividing it by the standard deviation removes the units:

A mode is awkward to find in real data, and it can be several values or none. For a moderately skewed distribution, one that is not too lopsided, there is an empirical relation that lets you drop it: the median sits roughly one third of the way from the mean toward the mode, which gives

With it, the mode can be replaced and Pearson’s second coefficient needs only the mean, median and standard deviation: $S{k}_P=3(\bar{x}-\text{median})/\sigma$. For the Ashanagar incomes this gives 3 × (28,783 − 25,000) ÷ 13,366 = 0.85, a clear right skew.

The relation also lets you recover a missing average. If the mean is 28 and the mode 25, the median is (2 × 28 + 25) ÷ 3 = 27 (CUET 2021). A longer chain: a distribution with coefficient of variation 5%, variance 4 and skewness 0.5 has SD = 2, so the mean is 2 ÷ 0.05 = 40, and then 0.5 = (40 − mode) ÷ 2 gives a mode of 39 (JNU 2021).

Bowley’s coefficient

The quartiles give a measure that ignores the extreme tails altogether. In a symmetrical distribution the median sits exactly halfway between Q₁ and Q₃. In a right-skewed one it sits closer to Q₁. Bowley’s coefficient measures how far off-centre it is:

For the Ashanagar incomes Q₁ = ₹20,000, the median ₹25,000 and Q₃ = ₹38,375. The median is only 5,000 above Q₁ and 13,375 below Q₃, so Bowley’s coefficient is (38,375 + 20,000 − 50,000) ÷ 18,375 = 0.46.

Moments: the general method

Both of those are shortcuts. The statistician’s own measure uses central moments, the average of the deviations from the mean raised to a power:

Older books give β₁ = m₃² ÷ m₂³ as the measure of skewness. It is just γ₁ squared, so it loses the sign, and the sign has to be taken from m₃ separately. Here is a worked example small enough to do by hand.

The three coefficients live on different scales, so only the sign and the rough size are comparable between them. For the Ashanagar incomes Pearson gives 0.85, Bowley 0.46 and the moment coefficient 0.88: all positive, all agreeing on a right skew. Bowley looks smaller because it uses only the middle half of the data and ignores the tail where the skew lives.

Take the fourth power of the deviations and a value far from the mean counts enormously: twice as far out counts sixteen times as much. The fourth moment is therefore a measure of how much of the spread comes from the extreme values in the tails. Divided by the standard deviation to the fourth, it gives the coefficient of kurtosis, β₂. For the normal curve it equals 3, and that value is the yardstick:

The test is the same however it is written: β₂ < 3 or γ₂ < 0 means platykurtic (UGC NET 2025), and kurtosis defined as m₄ ÷ σ⁴ − 3 is positive for a leptokurtic distribution (JAM 2026). If you are given the moments directly, just divide: with a fourth moment of 16 and a variance of 2, β₂ = 16 ÷ 2² = 4 (CUET 2023).

The picture below shows three symmetric curves that have exactly the same mean and standard deviation. Everything that differs between them is kurtosis. Select one to highlight it and compare the share of the distribution lying more than two standard deviations from the mean.

Two points to hold on to. First, kurtosis is mostly about the tails. “Peakedness” is the usual one-word summary, but the number is driven by how far the rare extreme values go. Second, skewness and kurtosis are separate properties: a distribution can be skewed without unusual tails, or have heavy tails and still be symmetrical, so the two can exist individually (Hyderabad 2016).

The quickest picture of a distribution’s shape is the box plot, built from the five-number summary: the minimum, Q₁, the median, Q₃ and the maximum. The box runs from Q₁ to Q₃ with a line at the median, so it holds the middle half of the data. Whiskers run out from it toward the extremes.

Whiskers stop short of values that are unusually far out. The standard rule is Tukey’s fences: any value more than 1.5 × IQR beyond a quartile is flagged as an outlier and drawn as a separate point, and the whisker ends at the most extreme value still inside the fences.

For Ashanagar, Q₁ = 20,000, Q₃ = 38,375 and the IQR is 18,375, so the upper fence is 38,375 + 27,562.5 = ₹65,937.50. Exactly one household, the one on ₹72,000, lies beyond it; the next highest, ₹58,500, is inside. The lower fence is negative, so nothing can fall below it.

The slider below changes only that richest household’s income. Watch which summaries respond. The median and the interquartile range cannot, because they depend on position, not size. The mean and the standard deviation follow the household wherever it goes.

The box also shows skew. In a symmetrical distribution the median is in the middle of the box and the two whiskers are about equal. If the median is close to Q₁ and the upper half of the box and the upper whisker are much longer, the distribution is right-skewed. A boxplot with the median near the first quartile and a much wider upper half is exactly that picture, and the answer in DSE 2016 is right-skewed. Bowley’s coefficient is this same comparison, written as a number.

One caution on quartiles: this lesson uses the (n + 1)p rule, which puts Q₁ at the (n + 1) ÷ 4-th value. Spreadsheets and statistics packages offer several other rules and can differ in the last digit on a small sample. For the sixty incomes this rule gives Q₃ = ₹38,375, while the spreadsheet function QUARTILE.INC gives ₹38,125. If an exam names a rule, use that one.