What is the difference between skewness and kurtosis?

Skewness measures how lopsided a distribution is — which way it leans. Kurtosis measures how heavy its tails are — how prone it is to extreme values. Skew is about asymmetry; kurtosis is about tail weight. Both describe shape but answer different questions.

What does positive skewness mean?

Positive skewness means the distribution has a long tail on the right, made of a few large values. The bulk of the data sits on the left and the mean is pulled above the median. Incomes and house prices are typically right-skewed.

What is excess kurtosis?

Excess kurtosis is ordinary kurtosis minus 3, so that a normal distribution scores 0. Positive excess kurtosis (leptokurtic) means heavier tails and more outliers than a normal curve; negative excess kurtosis (platykurtic) means lighter tails.

Skewness & Kurtosis: Definition, Formula & How to Read It

The shape numbers. Skewness tells you which way a distribution leans; kurtosis tells you how heavy its tails are. Together they describe what a mean and standard deviation can't see.

What it is

The mean tells you where a distribution sits and the standard deviation tells you how wide it is — but neither says anything about its shape. Two columns can share a mean and an SD and still look completely different: one symmetric and bell-shaped, the other lopsided with a long tail. Skewness and kurtosis are the two numbers that capture that difference.

Skewness measures asymmetry — whether the distribution leans left or right. Kurtosis measures tail weight — how often the data stray far from the center into extreme values. Skew is about which side; kurtosis is about how heavy the tails.

The formula

Both are standardized moments: you measure each value's distance from the mean in units of standard deviations, then average a power of that distance. Skewness is the standardized third moment:

g₁ = (1/n) · Σ ( (xᵢ − x̄) / s )³

Cubing keeps the sign, so values far to the right (positive) and far to the left (negative) push the total in opposite directions. Kurtosis is the standardized fourth moment:

kurtosis = (1/n) · Σ ( (xᵢ − x̄) / s )⁴

The fourth power makes far-out values dominate, so kurtosis is really a measure of the tails. A normal distribution has a raw kurtosis of exactly 3, which is why the more useful quantity is excess kurtosis:

excess kurtosis = kurtosis − 3

Subtracting 3 re-centers the scale so that a normal distribution has excess kurtosis 0. That makes the number easy to read: positive means heavier-tailed than normal, negative means lighter.

How to read it

Value	Shape	What it means
Skewness > 0	Long right tail	Few large values; mean > median (right-skewed)
Skewness ≈ 0	Symmetric	Balanced around the center
Skewness < 0	Long left tail	Few small values; mean < median (left-skewed)
Excess kurtosis > 0	Heavy tails (leptokurtic)	Outlier-prone; sharper peak, fatter tails than normal
Excess kurtosis ≈ 0	Normal-like tails (mesokurtic)	Tails match the bell curve
Excess kurtosis < 0	Light tails (platykurtic)	Few extremes; flatter than normal

A rough rule of thumb: skewness beyond roughly ±0.5 starts to look visibly lopsided, and beyond ±1 it's strongly skewed. Both numbers connect back to other statistics — positive skew is exactly the case where the mean exceeds the median, and high kurtosis is a warning that you have more outliers than a normal model expects.

A worked example. Take a column of household incomes. Most households cluster in a band, but a handful of very high earners stretch far to the right. Computing the third moment gives a skewness of, say, +1.4 — strongly positive, confirming the long right tail you'd see in a histogram. The mean income comes out well above the median, the signature of right skew. The fourth moment gives an excess kurtosis of +3.0 — heavy tails, meaning extreme incomes appear far more often than a normal curve would predict. Neither number is visible from the mean and SD alone; together they tell you a normal-distribution assumption would badly underestimate the high end.

Why it matters

Model assumptions. Many tests (t-tests, regression) assume roughly normal data. Large skewness or excess kurtosis is a flag that those assumptions may not hold.
Outliers. High excess kurtosis means the data are outlier-prone — a few extreme values do a lot of the work.
Choosing a center. Strong skew is the case where the median, not the mean, is the honest summary.
Always pair with a picture. The numbers are a summary; a histogram shows the shape directly.

How Stratum computes it. Stratum reports both skewness and excess kurtosis in its descriptive statistics, using the excess convention so that a normal distribution scores 0 — no mental subtraction of 3 required. Plot the same column as a histogram and the shape becomes visual: the lean is the skew, the tails are the kurtosis. See the conventions page for how missing and non-finite values are excluded before the moments are computed.

Skewness & kurtosis

What it is

The formula

How to read it

Why it matters

See also