Kurtosis is a statistic that describes how much probability sits in the tails of a distribution compared with a normal distribution. It helps you understand how often extreme values or outliers are likely to occur. In data analysis, this matters because two data sets can have the same mean and standard deviation but very different risks of extreme observations.
Kurtosis is best understood as tail weight, not just how tall or pointy the center looks.
Key Facts
- Population kurtosis: β2 = μ4 / σ^4, where μ4 is the fourth central moment.
- Excess kurtosis: γ2 = β2 - 3.
- A normal distribution has β2 = 3 and excess kurtosis γ2 = 0.
- Leptokurtic distributions have positive excess kurtosis and heavier tails than normal.
- Platykurtic distributions have negative excess kurtosis and lighter tails than normal.
- Higher kurtosis usually means extreme values are more likely, but it does not necessarily mean the center peak is taller.
Vocabulary
- Kurtosis
- Kurtosis is a measure of how heavy the tails of a distribution are relative to a normal distribution.
- Excess kurtosis
- Excess kurtosis is kurtosis minus 3, so a normal distribution has a value of 0.
- Leptokurtic
- A leptokurtic distribution has heavier tails and a higher chance of extreme values than a normal distribution.
- Mesokurtic
- A mesokurtic distribution has tail weight similar to a normal distribution.
- Platykurtic
- A platykurtic distribution has lighter tails and fewer extreme values than a normal distribution.
Common Mistakes to Avoid
- Calling kurtosis only peakedness is wrong because kurtosis mainly describes tail weight and the likelihood of extreme values.
- Assuming high kurtosis always means a taller center is wrong because different distributions can have similar centers but very different tails.
- Forgetting to subtract 3 for excess kurtosis is wrong because ordinary kurtosis and excess kurtosis use different reference points.
- Comparing kurtosis without considering outliers is wrong because a few extreme observations can strongly affect the fourth moment.
Practice Questions
- 1 A distribution has population fourth central moment μ4 = 48 and standard deviation σ = 2. Find its kurtosis β2 and excess kurtosis γ2.
- 2 Data Set A has excess kurtosis 1.8, Data Set B has excess kurtosis 0, and Data Set C has excess kurtosis -0.7. Classify each as leptokurtic, mesokurtic, or platykurtic.
- 3 Two distributions have the same mean and standard deviation. One has much heavier tails than the other. Explain which one has greater kurtosis and why this matters for predicting outliers.