What this calculator does
Kurtosis measures tail weight. This calculator reports excess kurtosis, which subtracts 3 so that a normal distribution sits at zero. Positive values mean heavier tails than normal, negative values mean lighter.
The common description of kurtosis as peakedness is misleading and worth unlearning. What the statistic actually responds to is the extremes: deviations are raised to the fourth power, so values far from the mean dominate completely. A uniform data set like 1 through 10 gives −1.2, not because it has a flat peak but because it has no tails at all.
The formula
Each value is converted to a z-score, raised to the fourth power and summed, then scaled by a factor involving n that makes the estimate unbiased. A further term subtracts the baseline so that a normal distribution reads zero rather than three. At least four data points are required, and the fourth power makes the statistic even more outlier-sensitive than skewness.
| Term | Meaning |
|---|---|
| Excess kurtosis | Kurtosis minus 3, so that a normal distribution reads zero. |
| Leptokurtic | Positive excess kurtosis: heavier tails and more extreme values than normal. |
| Platykurtic | Negative excess kurtosis: lighter tails, with values confined closer to the centre. |
| Mesokurtic | Excess kurtosis near zero, matching a normal distribution. |
The inputs explained
| Field | What to enter |
|---|---|
| Data (comma or space separated) | Your data, comma or space separated. At least four values are needed, and the statistic is very unreliable below about fifty. |
When to use it
Assessing financial risk
Returns data is typically leptokurtic, meaning extreme moves happen more often than a normal model predicts, which is exactly what risk models need to account for.
Checking a normality assumption
Together with skewness, kurtosis is a standard first check on whether normal-theory methods are reasonable.
Detecting outlier-driven data
A very high kurtosis usually signals that a few extreme points are dominating the data set.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does kurtosis look like for different data?
Uniform, mildly heavy-tailed and outlier-driven data compared.
| Data set | Excess kurtosis (G₂) | Shape | Sample standard deviation |
|---|---|---|---|
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | -1.200 | Platykurtic: lighter tails than normal | 3.028 |
| 2, 4, 4, 4, 5, 5, 7, 9, 3, 6 | 0.7141 | Leptokurtic: heavier tails than normal | 2.025 |
| 5, 5, 5, 5, 6, 6, 6, 6, 1, 11 | 4.022 | Leptokurtic: heavier tails than normal | 2.413 |
Questions
What does high kurtosis mean?
That the data has heavier tails than a normal distribution: extreme values occur more often than a normal model would predict. It does not mean the distribution has a sharper peak, despite that being the usual textbook description.
Is kurtosis about peakedness?
Not really, and this is a well-known misconception. Because deviations are raised to the fourth power, values near the mean contribute almost nothing and values far away contribute almost everything. The statistic is essentially a measure of tail weight and outlier presence.
What is excess kurtosis?
Kurtosis minus 3. A normal distribution has a raw kurtosis of 3, so subtracting it puts the normal case at zero and makes positive and negative values directly interpretable as heavier or lighter tails. This calculator reports the excess form.
How many data points do I need?
Formally four, but kurtosis is the least stable of the common summary statistics. Estimates from under fifty points are very noisy, and from under twenty they are close to meaningless. Treat small-sample kurtosis with considerable caution.
For which way the data leans, see the skewness calculator. For the full summary, see the descriptive statistics calculator.