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Kurtosis

Definition

Kurtosis is a standardized fourth-moment measure of a distribution. For a random variable with mean μ, positive finite variance σ² and finite fourth moment, Pearson kurtosis is E[(X − μ)⁴]/σ⁴. Excess kurtosis subtracts 3, making the ordinary normal distribution's value zero instead of three. Because deviations are raised to the fourth power, the measure is particularly sensitive to extreme observations. It is often discussed in terms of tail behavior, but it is not a complete description of tails, a direct tail probability or simply the height of a distribution's peak.

What it means

Variance summarizes squared deviations from the mean, while kurtosis compares fourth-power deviations with squared variance. Standardization removes ordinary units, allowing comparison across linear changes of measurement scale. Two distributions can have the same variance but different kurtosis because they allocate probability differently between central and more extreme values. Conversely, identical kurtosis does not imply identical distributions. Historical descriptions in terms of peakedness are incomplete: peak shape alone does not determine the fourth standardized moment, and a single kurtosis number cannot identify a distribution's full shape.

How it works

To calculate an unadjusted sample version, find the sample mean, then compute m₂ = Σ(xᵢ − x̄)²/n and m₄ = Σ(xᵢ − x̄)⁴/n. Pearson sample kurtosis is m₄/m₂², and excess is that ratio minus 3. This formula uses n in both moments; inserting an n − 1 variance changes the estimator. Software may apply a sample-size adjustment and may return Pearson or excess kurtosis by default. State the convention and missing-value policy. If all observations are identical, m₂ is zero and the standardized ratio is undefined.

Three practical examples

• Distribution description: summarize how strongly large deviations contribute to a measured dataset, alongside plots and other summaries.
• Residual diagnostics: investigate unusually large model errors and whether a proposed error distribution is plausible.
• Simulation studies: compare estimator behavior under distributions with different fourth-moment properties. A calculated kurtosis is a descriptive input to investigation, not a rule for automatically deleting observations or choosing one model.

Think of it this way

Think of a scoring rule that gives a deviation of four units 256 fourth-power points, compared with only one point for a deviation of one unit. That strong weighting makes a few large deviations influential. Dividing by squared variance puts the score on a common scale, but it does not turn it into a percentage or tell you which observations are erroneous. Extreme values may be meaningful parts of the process being studied.

Worked example

Use observations 1, 2, 3, 4 and 5. Their mean is 3, giving deviations −2, −1, 0, 1 and 2. The second central moment is (4 + 1 + 0 + 1 + 4)/5 = 2. The fourth central moment is (16 + 1 + 0 + 1 + 16)/5 = 6.8. Unadjusted Pearson kurtosis is 6.8/2² = 1.7, and unadjusted excess kurtosis is 1.7 − 3 = −1.3. A common adjusted excess estimator for n > 3 is [(n − 1)/((n − 2)(n − 3))][(n + 1)g₂ + 6]. Here it gives (4/6)[6(−1.3) + 6] = −1.2. Different software outputs can therefore be correct under different conventions. Multiplying every observation by 10 and adding 7 leaves both unadjusted kurtosis values unchanged.

Illustrative case study

Fictional case study: Alder Monitoring sees a sharp rise in reported residual kurtosis after a software update. Analysts first verify that one dashboard returns excess kurtosis while another returns Pearson kurtosis. After correcting that reporting mismatch, a remaining increase is traced to a small set of genuine large errors concentrated in one operating condition. The team investigates that condition rather than removing the observations to make the summary look normal. It reports the calculation convention and supplements the number with residual plots and exceedance counts.

Common misconceptions and mistakes

Common mistakes include comparing Pearson and excess values without adjusting by three, assuming a value of three proves normality, and interpreting kurtosis as only peakedness. A nonnormal distribution can share the normal distribution's kurtosis. Some heavy-tailed distributions have no finite fourth moment, so population kurtosis is undefined or infinite even though a finite sample produces a numerical value. Small samples can yield unstable estimates. A high value does not specify a unique tail-decay law or reveal whether one or both tails contain the unusual observations.

Business use

Businesses can use kurtosis to flag distribution features worth investigating, especially in residual monitoring and measurement quality. Record whether values are adjusted and whether they use the Pearson or excess convention. Examine sensitivity to individual observations and compare relevant groups rather than combining incompatible processes indiscriminately. If decisions concern specific thresholds, estimate those exceedance probabilities directly with appropriate uncertainty instead of treating kurtosis as a substitute. When using variance-based uncertainty formulas, check whether the assumed distribution has the moments those formulas require.

People also ask

Is the normal distribution's kurtosis zero or three?

Zero under excess kurtosis and three under Pearson kurtosis.

Does changing units change kurtosis?

A nonzero linear rescaling and shift do not change the standardized population measure or the corresponding moment-ratio sample version.

Does a finite sample value prove population kurtosis exists?

No. The underlying fourth moment may still diverge.

Check your understanding

Why do −1.3 and −1.2 both appear in the example?

Answer: The first is the unadjusted excess moment ratio. The second applies a stated sample-size adjustment. The difference reflects estimator conventions, not a different dataset.

Related terms

• Central moment
• Variance
• Skewness
• Heavy-tailed distribution
• Outlier
• Leptokurtic
• excess kurtosis

Learn next

compare datasets with the same mean and variance but different fourth moments, then examine how a single extreme observation changes sample kurtosis. Pair the calculation with plots and direct tail summaries to avoid overinterpreting one shape statistic.

• Variance

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Sources

Sources: NIST/SEMATECH e-Handbook, Measures of Skewness and Kurtosis, https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm. SciPy, kurtosis documentation, https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kurtosis.html. KNIME, Data Science Glossary, https://www.knime.com/blog/data-science-glossary. Numerical example and fictional case are original illustrations.

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