More than mean and spread

A city reports its average household income is $80,000. But most households earn far less — a handful of millionaires pull the average up. The mean and variance alone can't capture this asymmetry.

What do you think?

Income distributions are typically skewed in which direction?

The moment hierarchy

We've seen the first two moments: the mean (1st moment) tells us where the distribution is centered, and the variance (2nd moment) tells us how spread out it is. The third and fourth moments describe the distribution's shape.

Skewness: the 3rd moment

Skewness

The skewness of $X$ is the standardized third central moment: $\text{Skew}(X) = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right]$ It measures the asymmetry of the distribution around its mean.

Skewness	Meaning	Example
$= 0$	Symmetric	Normal distribution
$> 0$	Right-skewed (long right tail)	Income, waiting times
$< 0$	Left-skewed (long left tail)	Exam scores (near-perfect clustering)

Why the cube? Squaring (variance) treats left and right deviations equally. Cubing preserves the sign — large positive deviations contribute positively, large negative deviations contribute negatively.

Explore how skewness affects distribution shape — watch mean, median, and mode diverge:

Skewness Explorer

Gamma(k, 1) family — skewness = 2/√k decreases as k increases

Shape k = 2.0 → Skewness = 1.41: 2.0

0.5 (very skewed)20 (nearly symmetric)

Mode = 1.0 Median = 0.96 Mean = 2.0

Skewness

1.414

Excess kurtosis

3.000

Mean − Median

1.039

1.41

Right-skewed: The long right tail pulls the mean above the median. Mode < Median < Mean. The orange-shaded area shows the heavy tail.

Kurtosis: the 4th moment

Kurtosis

The kurtosis of $X$ is the standardized fourth central moment: $\text{Kurt}(X) = E\left[\left(\frac{X - \mu}{\sigma}\right)^4\right]$ It measures the heaviness of the tails relative to the Normal distribution (kurtosis = 3).

The Normal distribution has kurtosis = 3. Excess kurtosis = kurtosis $- 3$ :

Excess Kurtosis	Type	Meaning
$= 0$	Mesokurtic	Normal-like tails
$> 0$	Leptokurtic	Heavier tails, more outliers
$< 0$	Platykurtic	Lighter tails, fewer outliers

High kurtosis doesn't mean "peaky." It means heavy tails, with more probability mass in extreme values. A distribution can be flat and still have high kurtosis if it has heavy tails.

Mold the distribution

Switch between distributions to see how skewness and kurtosis change. The exponential is right-skewed (skewness = 2); the uniform has light tails (kurtosis < 3):

Moments Visualizer

Distribution

Mean (μ): 5.0

010

Std Dev (σ): 1.50

0.53

Mean (1st)

5.00

Variance (2nd)

2.25

Skewness (3rd)

0.00

Kurtosis (4th)

3.00

Skewness = 0 → symmetric distribution · Kurtosis = 3 (mesokurtic, Normal baseline)

Common distributions compared

Distribution	Mean	Variance	Skewness	Kurtosis
Normal( $\mu, \sigma^2$ )	$\mu$	$\sigma^2$	0	3
Exponential( $\lambda$ )	$1/\lambda$	$1/\lambda^2$	2	9
Uniform( $a, b$ )	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$	0	1.8
Bernoulli( $p$ )	$p$	$p(1-p)$	$\frac{1-2p}{\sqrt{p(1-p)}}$	$3 + \frac{1-6p(1-p)}{p(1-p)}$

Why do higher moments matter?

What do you think?

Two portfolios each have mean return 10% and standard deviation 5%. Portfolio A has excess kurtosis = 0 (Normal tails). Portfolio B has excess kurtosis = 6 (very heavy tails). Which has more extreme crashes?

Practical interpretation

Positive skewness means the mean is pulled above the median because the right tail dominates. Negative skewness means the mean is pulled below the median because the left tail dominates. High kurtosis means rare events happen more often than a Normal model would predict.

Practice problems

A symmetric distribution always has skewness equal to what value? (whole number)

The Normal distribution has kurtosis equal to what value? (whole number)

The Exponential distribution has skewness equal to? (whole number)

Summary

Moment	Measures	Formula
1st: Mean	Center	$E[X]$
2nd: Variance	Spread	$E[(X - \mu)^2]$
3rd: Skewness	Asymmetry	$E[((X-\mu)/\sigma)^3]$
4th: Kurtosis	Tail weight	$E[((X-\mu)/\sigma)^4]$

Mean tells you where a distribution sits, variance tells you how spread out it is, skewness tells you which way it leans, and kurtosis tells you how extreme the tails are.

Test your understanding

If a distribution is left-skewed, is the mean above or below the median? (above or below)

A Uniform distribution has excess kurtosis (kurtosis - 3) equal to? (decimal to 1 place, e.g. -0.5)

What's next

Now that we can fully describe a distribution's shape, we'll meet the moment-generating function, a single formula that encodes all moments at once.