Skewness and Kurtosis

The mean tells you where a distribution is centered, and the standard deviation tells you how spread out it is. Neither tells you whether the distribution leans to one side, or whether it hides rare, violent surprises far from the center. Two distributions can share the exact same mean and standard deviation and behave completely differently — that gap is what skewness and kurtosis measure.

An analogy: two commutes with the same average

Commute A takes 30 minutes almost every day, occasionally 35. Commute B takes 25 minutes on a normal day, but once a month a signal failure turns it into 3 hours. Arrange the numbers so both commutes average 30 minutes with the same standard deviation. They still feel nothing alike: A is symmetric and predictable, B is usually short but occasionally catastrophic. Skewness captures "which side does the surprise usually fall on," and kurtosis captures "how extreme, and how often, are the surprises" — exactly the difference between these two commutes.

Skewness, one symbol at a time

Skewness measures asymmetry around the mean $\bar{x}$. The sample formula is

$$\text{skew} = \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^3}{s^3}$$

Read it piece by piece: for each observation $x_i$, take its distance from the mean, cube it — cubing keeps the sign, so distances above the mean count positive and distances below count negative — average those cubes, then divide by the standard deviation $s$ cubed, purely to make the number unitless. In plain English: if the big deviations tend to be positive (a long right tail), skewness is positive; if they tend to be negative (a long left tail), skewness is negative; symmetric data gives roughly zero.

Kurtosis, one symbol at a time

Kurtosis measures tail weight — how much of the variance comes from rare, extreme values versus everyday ones.

$$\text{kurt} = \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^4}{s^4}$$

Same structure, but raised to the fourth power instead of the third. The fourth power makes every deviation positive (so this measures magnitude of extremity, not direction) and disproportionately rewards the largest deviations, because $10^4$ is ten thousand times $1^4$ while $10^3$ is only a thousand times $1^3$. In plain English: kurtosis is dominated by the few most extreme points in the sample, and a normal distribution has kurtosis exactly 3 — so quants usually report "excess kurtosis," $\text{kurt} - 3$, where 0 means normal-tailed, and positive means fatter-tailed than normal.

A normal curve, like the one above, has zero skew and zero excess kurtosis by construction — it is the reference point both statistics are measured against.

Now switch to a Poisson shape with a small rate, like the one above — visibly lopsided, with a long right tail. That lopsidedness is positive skew made visible: most outcomes cluster low, with a thin trail of rare, large ones stretching to the right, exactly the pattern a positively-skewed return series shows.

Worked example 1: skewness of five trades

Five daily P&L outcomes, in $thousands: $1, 1, 1, 1, -6$. Mean: $\bar{x} = \frac{1+1+1+1-6}{5} = \frac{-2}{5} = -0.4$.

Deviations from the mean: $1.4, 1.4, 1.4, 1.4, -5.6$. Squared: $1.96, 1.96, 1.96, 1.96, 31.36$, averaging to $\frac{39.2}{5} = 7.84$, so $s = \sqrt{7.84} = 2.8$.

Cubed deviations: $2.744, 2.744, 2.744, 2.744, -175.616$, averaging to $\frac{-164.64}{5} = -32.93$.

$$\text{skew} = \frac{-32.93}{2.8^3} = \frac{-32.93}{21.95} = -1.50 .$$

Negative skew of $-1.50$: four unremarkable small gains and one large loss — exactly the shape a short-volatility or option-selling strategy produces, and exactly what the negative sign is built to flag.

Worked example 2: kurtosis, calm days vs. one crash day

Take the same five points. Fourth powers of the deviations: $1.4^4 = 3.84$, three more at $3.84$, and $(-5.6)^4 = 983.4$. Sum $= 3.84 \times 4 + 983.4 = 998.8$, averaging to $\frac{998.8}{5} = 199.76$.

$$\text{kurt} = \frac{199.76}{2.8^4} = \frac{199.76}{61.47} = 3.25 .$$

Excess kurtosis $= 3.25 - 3 = 0.25$ — only mildly fat-tailed, despite one loss being four times any other observation, because with $n=5$ the sample is tiny and noisy. Now imagine the same shape repeated over 250 trading days — four calm gains for every one sharp loss, all year — and the excess kurtosis climbs into the range of 3 to 5, the territory real daily equity index returns actually occupy. The lesson: a single extreme point barely dents kurtosis in a small sample, but a pattern of extremes dominates it in a large one.

What this means in practice

• Value-at-Risk models that assume normality systematically understate tail risk for return series with positive excess kurtosis — the true 1st-percentile loss is worse than a normal model predicts, because the normal distribution has thin tails by construction.
• Negative skew in a strategy's return stream (small steady gains, occasional large loss) is the statistical signature of tail risk — it shows up in strategies that are effectively short a crash, like covered calls or credit spreads, even when the Sharpe ratio looks attractive.
• Both statistics are unstable with small samples — a single new data point can swing skewness or kurtosis by a large amount, so treat them as directional signals over long histories, not precise numbers from a month of daily data.