What Is Kurtosis?
Kurtosis describes the weight of a distribution's tails relative to a normal distribution. Excess kurtosis subtracts 3, the kurtosis of a normal distribution, so a normal curve scores 0, heavy-tailed data score above 0 and light-tailed data score below 0.
Packages differ in which formula they use. This calculator reports the moment-based excess kurtosis (type 1), the adjusted version used by Excel's KURT (type 2), the standard error of kurtosis and the Jarque-Bera statistic, which combines skewness and kurtosis into one test of normality.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as whether kurtosis measures peakedness and what counts as a large value.
How to Use the Kurtosis Calculator
- Enter at least four values; reliable kurtosis needs far more, ideally 50 or more.
- Pick which definition to feature. Match your software or textbook.
- Read the interpretation and test results, then compare with the histogram.
Kurtosis Formula
Moment kurtosis and excess kurtosis (type 1)
Adjusted excess kurtosis (type 2, Excel KURT)
Standard error of kurtosis
Jarque-Bera statistic
Kurtosis Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| g₂ | Excess kurtosis (type 1) | Moment-based, minus 3 |
| G₂ | Adjusted excess kurtosis (type 2) | As in Excel KURT |
| n | Sample size | Needs at least 4 values |
| SE | Standard error of kurtosis | For judging significance |
| JB | Jarque-Bera statistic | Normality test |
Kurtosis Worked Example
With the example values loaded in the calculator above, the steps are:
- Mean = 4.6; m₂ = 7.30667; m₄ = Σ(x − x̄)⁴ ÷ n = 272.158.
- Kurtosis = m₄ ÷ m₂² = 5.0978; excess kurtosis g₂ = 5.0978 − 3 = 2.0978.
- Adjusted G₂ = [(n − 1) ÷ ((n − 2)(n − 3))] × [(n + 1)g₂ + 6] = 3.5507.
- SE of kurtosis = 1.1209; z = 3.168.
Result: Excess kurtosis (G₂) = 3.5507; Kurtosis (not excess) = 6.5507.
How to Read the Kurtosis Result
The sample data cluster between 2 and 7 but include 12 and −1. Excess kurtosis is 3.55, and the standard error of 1.12 gives z = 3.17. Positive excess kurtosis means heavier tails than a normal curve, which is to say more extreme values than the bell shape would predict.
Kurtosis is built from fourth powers of the distances from the mean, so the two outliers dominate it. Check whether the high value comes from a genuine long tail or from one or two entry errors before reading anything into it. The Jarque-Bera statistic, 4.16 here, combines skewness and kurtosis in a single normality check.
How to interpret kurtosis
Kurtosis is about tails, not about how sharp the peak looks. Two distributions can have the same kurtosis with different peaks. A single extreme outlier can raise sample kurtosis dramatically, so a high value is often a signal to check for outliers or data errors.
| Name | Excess kurtosis | Tails | Example |
|---|---|---|---|
| Platykurtic | below 0 | Lighter than normal | Uniform (−1.2) |
| Mesokurtic | about 0 | Normal-like | Normal |
| Leptokurtic | above 0 | Heavier than normal | Laplace (+3), Student t, stock returns |
Where kurtosis matters
- Finance and risk: heavy tails mean extreme gains and losses are more common than a normal model assumes.
- Quality control: unusual kurtosis can indicate a mixture of processes or measurement issues.
- Checking assumptions: a quick screen before t-tests, ANOVA or regression, together with skewness and a Q-Q plot.
- Data cleaning: very high kurtosis often points to outliers.
Cautions with sample kurtosis
Kurtosis uses fourth powers, so it is extremely sensitive to individual extreme values and needs large samples. With fewer than about 30 observations, treat the number as a rough indication. The Jarque-Bera test is a large-sample test and tends to be unreliable for small n, where tests such as Shapiro-Wilk are preferred.
| Tool | Function | Reports |
|---|---|---|
| Excel / Google Sheets | KURT | G₂ (excess) |
| SPSS, SAS | Kurtosis | G₂ (excess) |
| Minitab | Kurtosis | b₂ (excess) |
| SciPy | kurtosis(fisher=True) | g₂ (excess) |
| Pandas | .kurt() | G₂ (excess) |
Where Kurtosis Is Used
Tail weight
Measure how heavy the tails are compared with a normal distribution.
Software comparison
See the type 1, 2 and 3 definitions side by side.
Normality check
Use the Jarque-Bera statistic together with skewness.
Risk and finance
Check whether returns have heavier tails than a bell curve.
Common Kurtosis Mistakes to Avoid
- Check whether your source reports kurtosis or excess kurtosis. They differ by 3.
- Look for outliers before interpreting high kurtosis.
- Combine with skewness: |skew| < 2 and |excess kurtosis| < 7 is a lenient guide some researchers use for structural equation modelling.
- Use a Q-Q plot or Shapiro-Wilk test for small-sample normality checks.
Kurtosis Terms and Definitions
- Excess kurtosis
- Kurtosis minus 3, so the normal distribution scores 0.
- Fourth central moment
- The average of fourth powers of deviations from the mean.
- Leptokurtic
- Having heavier tails than a normal distribution.
Kurtosis FAQs
What is kurtosis?
A measure of how heavy the tails of a distribution are compared with a normal distribution, reflecting the tendency to produce extreme values.
What is excess kurtosis?
Kurtosis minus 3. It is zero for a normal distribution, positive for heavy-tailed distributions and negative for light-tailed ones.
Does high kurtosis mean a sharp peak?
Not exactly. Kurtosis measures tail weight. Peaked shapes often have heavy tails, but the statistic is driven by the tails.
What is a normal range for kurtosis?
Excess kurtosis between about −1 and 1 is close to normal; many texts accept −2 to +2 as acceptable for approximate normality.
Why is my kurtosis different in Excel?
Excel’s KURT reports the adjusted excess kurtosis G₂. Other tools use g₂ or b₂. All three are shown here.
How many data points do I need?
Kurtosis is unstable in small samples. Use 50 or more where possible and always inspect the histogram.
What does the Jarque-Bera test do?
It tests whether skewness and kurtosis together match a normal distribution. A small p-value suggests the data are not normal. It works best for large samples.
What is the difference between kurtosis and excess kurtosis?
Excess kurtosis equals kurtosis minus 3, so a normal distribution scores 0. Software often reports excess kurtosis; Excel KURT does. The sample data gives 3.55 excess, or 6.55 not excess.
What does high kurtosis mean?
Heavier tails, meaning more extreme values than a normal distribution would give. It is about tail weight and outliers, not how pointed the peak looks.
How do I test whether kurtosis is significant?
Divide the excess kurtosis by its standard error. A ratio beyond about ±2 suggests the tails differ from normal. The sample data gives 3.55 ÷ 1.12 = 3.17.
What is the Jarque-Bera statistic?
It combines skewness and kurtosis into one normality test statistic. Larger values mean a stronger departure from normality.