Skewness and Kurtosis - Probability Distributions

Definition

Skewness is a measure of the asymmetry of a probability distribution about its mean. A distribution is:

Positively skewed

if its right tail is longer or heavier,

Negatively skewed

if its left tail is longer or heavier,

Symmetric

if both sides are balanced.

Kurtosis is a measure of the shape of a probability distribution in terms of the peakedness and tail heaviness relative to a normal distribution. It indicates whether the data have:

a sharper peak and heavier tails,
a flatter peak and lighter tails,
or a shape similar to the normal distribution.

Mathematically, skewness and kurtosis are based on central moments of the distribution. If $X$ is a random variable with mean $\mu$ and standard deviation $\sigma$ , then:

Skewness

$\gamma_1 = \frac{E[(X-\mu)^3]}{\sigma^3}$

Kurtosis

$\gamma_2 = \frac{E[(X-\mu)^4]}{\sigma^4}$

Often, excess kurtosis is used: $\text{Excess kurtosis} = \gamma_2 - 3$ because the normal distribution has kurtosis equal to 3, so excess kurtosis for a normal distribution is 0.

Main Content

1. Skewness

Meaning and interpretation

Skewness measures the degree of asymmetry in a probability distribution.
If a distribution is symmetric, its left and right sides are mirror images, and skewness is approximately zero.
Positive skewness means the distribution has a long tail to the right. In such cases, most values lie on the lower side, but a few extreme large values pull the mean to the right.
Negative skewness means the distribution has a long tail to the left. Most values lie on the higher side, but a few extreme small values pull the mean to the left.

Relation among mean, median, and mode

In a positively skewed distribution: $\text{Mode} < \text{Median} < \text{Mean}$
In a negatively skewed distribution: $\text{Mean} < \text{Median} < \text{Mode}$
In a symmetric distribution: $\text{Mean} = \text{Median} = \text{Mode}$

Example of skewness

Consider exam scores where most students score between 40 and 70, but a few score near 100. This distribution is likely positively skewed.
Consider retirement age data where most people retire around 60–65, but some retire very early due to illness. This may create left skewness.

ASCII illustration

  Symmetric:        Positive skew:      Negative skew:
       /\                /\                 /\
      /  \              /  \               /  \
     /    \            /    \             /    \
    /      \          /      \           /      \
  --/--------\--    --/--------\____   ____/--------\--

2. Kurtosis

Meaning and interpretation

Kurtosis describes the shape of the tails and the central peak of a distribution.
A distribution with high kurtosis tends to have a sharper peak and heavier tails, meaning more frequent extreme deviations.
A distribution with low kurtosis tends to be flatter with lighter tails, meaning fewer extreme values.

Types of kurtosis

Leptokurtic: More peaked than a normal distribution and has heavier tails. Extreme values are more common.
Mesokurtic: Similar to the normal distribution. Excess kurtosis is approximately zero.
Platykurtic: Flatter than a normal distribution and has lighter tails. Extreme values are less common.

Interpretation of excess kurtosis

Excess kurtosis > 0: Leptokurtic
Excess kurtosis = 0: Mesokurtic
Excess kurtosis < 0: Platykurtic

Example of kurtosis

A stock market return distribution may be leptokurtic because extreme gains or losses happen more frequently than expected under a normal distribution.
A uniform-like set of values may be platykurtic because the values are spread evenly and not concentrated around a sharp peak.

ASCII illustration

  Leptokurtic:      Mesokurtic:        Platykurtic:
        /\               /\               ____
       /  \             /  \            __/    \__
      /    \           /    \          /          \
     /      \         /      \        /            \

3. Probability Distributions and Shape Analysis

Why skewness and kurtosis matter in distributions

Probability distributions describe how probabilities are assigned to possible values of a random variable.
Two distributions can have the same mean and variance but very different shapes.
Skewness and kurtosis help capture these shape differences, making distribution analysis more complete.

Connection with the normal distribution

The normal distribution is symmetric, so its skewness is 0.
Its kurtosis is 3, and excess kurtosis is 0.
Many statistical models assume normality, but real data often deviate from it.
Skewness and kurtosis help identify whether a distribution is far from normal and whether transformations may be needed.

Role in data analysis

If a distribution is strongly skewed, analysts may apply transformations such as logarithm, square root, or Box-Cox transformation.
If kurtosis is high, it suggests the presence of outliers or heavy tails, which may require robust methods.
In hypothesis testing and regression, non-normality can affect confidence intervals and p-values.

Example comparing two distributions

Distribution A: symmetric, moderate tails, resembles a normal curve.
Distribution B: same mean as A but right-skewed with heavy tails.
Without skewness and kurtosis, both may seem similar based on mean and variance alone, but their practical behavior is very different.

Working / Process

1. Collect the distribution data or define the probability model

Start with either a sample dataset or a theoretical probability distribution.
Identify the random variable and determine its mean $\mu$ and standard deviation $\sigma$ .
For sample data, compute sample mean and sample standard deviation first.

2. Compute central moments

Find deviations from the mean: $(X-\mu)$ .
Raise these deviations to the third power for skewness and fourth power for kurtosis.
Take the expected value of these powered deviations.
Standardize them by dividing by $\sigma^3$ or $\sigma^4$ so the measures are dimensionless.

3. Interpret the results

If skewness is near zero, the distribution is approximately symmetric.
If skewness is positive, the distribution is right-skewed; if negative, left-skewed.
If excess kurtosis is positive, the distribution has heavier tails than normal.
If excess kurtosis is negative, the distribution is flatter than normal.
Use the interpretation to decide whether the data behave normally or whether transformations or robust methods are needed.

Advantages / Applications

Describes distribution shape more completely

Mean and variance alone do not fully describe a distribution.
Skewness and kurtosis reveal asymmetry and tail behavior, giving a deeper understanding of the data.

Useful in real-world decision-making

In finance, skewness and kurtosis help measure investment risk and the chance of extreme losses.
In quality control, they help detect unusual production behavior or outliers.
In economics, they help interpret income, wealth, and consumption distributions.

Supports statistical modeling and inference

These measures help test whether a dataset is approximately normal.
They guide analysts in choosing suitable transformations and estimation methods.
They are valuable in comparing probability distributions and evaluating model fit.

Summary

Skewness shows whether a distribution is symmetric or tilted to one side.
Kurtosis shows how heavy-tailed or peaked a distribution is compared with normal.
Both measures help explain the shape of probability distributions beyond mean and variance.
Important terms to remember: skewness, kurtosis, positive skew, negative skew, leptokurtic, mesokurtic, platykurtic, excess kurtosis