Skewness and kurtosis

What are the four moments of a distribution?

Memorize

Four numbers describe most of a distribution’s shape. They are called its moments of a distribution .

Moment	Measure	Tells you about
First	Mean	Center
Second	Variance	Spread
Third	Skewness	Symmetry
Fourth	Kurtosis	Tail thickness

Each has a population version (μ, σ, N) and a sample version (X̄, s, n). The skewness and kurtosis formulas divide by the standard deviation, so, like correlation, they have no units.

Shape of a distributionfour numbers describe most of it

• Mean Memorize first moment: center
• Variance Memorize second: spread
• Skewness Memorize third: symmetry
• Kurtosis Memorize fourth: tail thickness

What does skewness tell you?

Mean and standard deviation do not say whether the big moves are gains or losses. Squaring hides the sign. Skewness restores it: it measures how lopsided a distribution is around its mean. A distribution that is not symmetric is skewed .

Skewnessasymmetry of returns around the mean

• Symmetric (skew 0) Memorize mean = median = mode
• Positive skew Memorize long right tail: more high outliers, a few large gains
  • Order mode < median < mean
  • Examples lottery-like payoffs, bought options
• Negative skew Memorize long left tail: frequent small gains, a few large losses
  • Order mean < median < mode
  • Examples selling insurance, equity index returns

For a unimodal skewed distribution, the order of the averages follows the tail: the mean is dragged toward the tail, the mode stays at the peak and the median falls between them.

Positive skew means more unusually high values: a higher chance of large gains than of equally large losses. For returns that reads as frequent small losses and a few large gains. Negative skew is the mirror image: frequent small gains and a few extreme losses. Investors hope for positive skew: for a given mean and standard deviation, it means limited but frequent downside and rarer, unlimited-looking upside. Equity index returns are usually negatively skewed.

The measure itself is built from cubed deviations. Cubing, unlike squaring, keeps the sign, so a long right tail gives a positive number. The sample skewness averages the cubed standardized deviations:

$$\text{Skew} = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{X_i - \bar{X}}{s}\right)^{3}$$

You will not need to compute it by hand from raw data; read its sign.

What does kurtosis tell you?

Kurtosis measures the combined weight of both tails relative to the rest of the distribution: how often extreme returns, up or down, occur. The normal distribution has a kurtosis of 3. Because 3 is the benchmark, analysts usually work with excess kurtosis .

$$K_E = K - 3$$

Kurtosisweight of the tails relative to the normal (kurtosis 3)

• Mesokurtic Memorize kurtosis = 3, excess = 0, like the normal
• Leptokurtic Memorize kurtosis > 3, excess > 0: fat- or long-tailed
  • Consequence extreme returns more often than the normal predicts
• Platykurtic Memorize kurtosis < 3, excess < 0: thin- or short-tailed

A leptokurtic distribution is fat-tailed (also called long-tailed). A platykurtic one is thin-tailed (short-tailed). A mesokurtic distribution has tails like the normal’s.

Compared with a normal distribution with the same mean, standard deviation and skewness, a fat-tailed one puts more of its observations far out (beyond about 2.5 standard deviations) and more in the middle (within one standard deviation), and fewer in between. Positive excess kurtosis means more extremes in both tails; which tail is longer is skewness’s job. Most financial return series have positive excess kurtosis, so models that ignore fat tails understate the chance of extreme outcomes.

For a large sample (100 observations or more), the sample excess kurtosis is:

$$\text{Kurt}_E = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{X_i - \bar{X}}{s}\right)^{4} - 3$$

Kurtosis or excess kurtosis? Check the software

Use it

Function	What it returns
SKEW (Excel and Google Sheets)	Sample skewness
SKEW.P (Excel)	Population skewness
KURT (Excel and Google Sheets)	Excess kurtosis

Some programs report kurtosis and others excess kurtosis, sometimes under the same label. If KURT shows 2.3, the kurtosis is 5.3; do not subtract 3 again.

Slide the skew and watch the mean, median and mode reorder. Raise the excess kurtosis and watch the tails above and below the normal curve.

Why do shape measures matter for risk?

Negative skew plus fat tails is the combination that risk managers fear. Returns look steady for long stretches and then lose a lot at once, and a standard deviation computed from the whole series understates the chance. A model that assumes a normal distribution therefore underestimates the probability of a large loss, which is why value at risk (VaR) from a normal curve can be too optimistic (Learning Module 7).

Exam tip Skew = which way the tail points. Kurtosis = how fat the tails are. A question that gives kurtosis may want excess kurtosis: subtract 3 first.

Go deeper Why the log-normal distribution is positively skewed

Stock prices modeled with compounding returns follow a log-normal distribution. It cannot fall below zero, but it has no ceiling, and compounding makes gains grow faster than losses shrink. The result is a long right tail and positive skew. It is covered in Learning Module 6, Statistical Distributions.

Go deeper How the formulas work

Skewness averages the cubed deviations and divides by s³ so the answer has no units. Kurtosis averages fourth powers and divides by s⁴. Fourth powers magnify large deviations enormously: a deviation twice as large counts sixteen times as much, which is why kurtosis is driven by the tails and barely notices what happens near the mean. Statistical software often reports excess kurtosis under the plain label “kurtosis”.