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
Using the example values (press “Load example” above to try them):
- 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 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
Checking for heavy tails
Seeing whether extreme values are more common than a normal model expects.
Testing normality
Using Jarque-Bera as a quick check before methods that assume normal data.
Matching software
Reproducing the value that Excel or another package reports.
Describing distribution shape
Adding tail behaviour to skewness in a summary.
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.