Tail weight of a list of numbers in four formulas, from Excel KURT to Pearson's b2, with the Anscombe-Glynn test, a normality check and the benchmark distribution the tails resemble.
Kurtosis Calculator - Excess Kurtosis, Excel KURT and Tests
Tail weight of a list of numbers in four formulas, from Excel KURT to Pearson's b2, with the Anscombe-Glynn test, a normality check and the benchmark distribution the tails resemble.
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Excess kurtosis, tail weight and four formulas in one place
Tail weight, not peak height: that is what kurtosis measures, and it is the part of a data set most summaries skip. This calculator takes a list of numbers and returns the sample excess kurtosis G2 that Excel KURT and SPSS print, the population value g2, Pearson's original b2 and the version R's e1071 package uses, together with the standard error, the Anscombe-Glynn test of the tails, the D'Agostino-Pearson normality test and the benchmark distribution your data resemble. For the twenty numbers 0 to 19, G2 is exactly -1.200, the value of a flat uniform spread, and the tail test gives p = 0.0880.
What the form asks, and what comes back
- Data - at least 4 numbers and up to 5,000, separated by spaces, commas or new lines, with a dot for decimals. A column pasted from a spreadsheet is fine.
- Sample or population - sample puts G2 in the main tile, the value Excel KURT, Google Sheets, SPSS and SAS report. Population puts g2 there, the plain moment ratio minus 3. All four formulas appear in the table regardless.
- Significance level - 0.05, 0.01 or 0.10, the threshold for the tail test.
- Read the shape word (leptokurtic above zero, platykurtic below), the nearest benchmark distribution, both p-values, the standard error and any note about a single far value.
Kurtosis is about the tails
Kurtosis is the average fourth power of standardized distances from the mean. Raising distances to the fourth power makes values near the mean almost irrelevant: a point one standard deviation out contributes 1, a point three out contributes 81. So the number is driven by how often values land far from the center, that is, by the tails. For a normal distribution the average is 3, which is why most software subtracts 3 and reports excess kurtosis, zero for a bell curve.
Older textbooks described high kurtosis as a "sharper peak". Peter Westfall's critique showed that this reading does not hold: distributions with a flat top can have huge kurtosis if their tails are heavy, and a sharply peaked shape can have low kurtosis if its tails are cut off. The calculator therefore describes the result in terms of tails only.
Two names come with the sign. Leptokurtic means excess kurtosis above zero: occasional values far out supply a large share of the variance, as with returns on investments, insurance claims or measurement errors with rare glitches. Platykurtic means below zero: the data have a hard floor and ceiling or split into clusters, as with rounded ratings or a uniform spread. Mesokurtic, exactly zero, is the normal case.
Kurtosis needs more data than any other common summary. The standard error under normality is about √(24/n) for large samples, so with 20 values it is still 0.992. That is why the tests start at 20 values here, as scipy recommends for its kurtosis test.
Excess kurtosis of common distributions
The calculator names the nearest of six of these benchmarks. The rest help place a result that falls between them.
| Distribution | Excess kurtosis | Formula or note |
|---|---|---|
| Two equal spikes (coin flip, p = 0.5) | -2.000 | the lowest possible value |
| Uniform | -1.200 | flat, hard limits on both sides |
| Semicircle | -1.000 | rounded top, no tails at all |
| Triangular | -0.600 | sum of two uniform values |
| Normal | 0.000 | the reference, b2 = 3 |
| Poisson, mean 4 | 0.250 | 1/λ |
| Student t, 10 degrees of freedom | 1.000 | 6/(ν - 4), infinite for ν of 4 or less |
| Logistic | 1.200 | bell-shaped, slightly heavier tails |
| Chi-square, 10 degrees of freedom | 1.200 | 12/k |
| Laplace (double exponential) | 3.000 | two exponential tails back to back |
| Gamma, shape 2 | 3.000 | 6/k |
| Lognormal, sigma = 0.5 | 5.898 | e4σ² + 2e3σ² + 3e2σ² - 6 |
| Exponential | 6.000 | also Student t with 5 degrees of freedom |
Four formulas, and which program prints which
Joanes and Gill compared the estimators in use and traced them to the software. Values below are for the list 2 3 3 4 4 4 5 5 6 12 3 4 4 5 4 3 4 5 4 1, twenty values with one far out.
| Measure | Value | Normal gives | Printed by |
|---|---|---|---|
| G2, sample excess kurtosis | 9.212 | 0 | Excel and Google Sheets KURT, SPSS, SAS |
| g2, excess kurtosis | 6.780 | 0 | scipy kurtosis default, many textbooks |
| b2, kurtosis | 9.780 | 3 | Pearson's original measure, Stata |
| b2 - 3 with the sample SD | 5.826 | 0 | R package e1071 default, Minitab |
Four programs, four numbers from 5.826 to 9.212, all correct for their definition. A kurtosis figure without its formula is not comparable, and a value near 3 may be either a perfectly normal b2 or a heavy-tailed excess.
Relationships worth knowing
How to read your number
The bands below are placed between benchmark distributions rather than at invented cut-offs. The George and Mallery rule, which accepts G2 between -2 and 2 before parametric tests, is printed separately in the result.
| Excess kurtosis | Reading | Typical cause |
|---|---|---|
| below -1.2 | lighter than uniform | two clusters, or values piled at both limits |
| -1.2 to 0 | lighter than normal | capped scales, rounded ratings, evenly spread values |
| 0 to 3 | between normal and Laplace | most measured data; check the test before calling it non-normal |
| 3 to 6 | heavy tails | waiting times, returns, a few extreme readings |
| above 6 | heavier than exponential | usually one or two values; check them first |
Worked readings: 0 to 19 gives G2 -1.200, tail test Z = -1.706, p = 0.0880, and the normality test K² = 2.910, p = 0.2334. The twenty-value list with the 12 gives G2 9.212, Z = 3.666, p = 0.0002, and K² = 28.368. Four values, 1 2 3 40, give G2 3.963 with a standard error of 2.619, which is why no test is reported for them.
The standard errors and both tests assume normal data. A significant result says the tails are unlikely under normality; it does not say which distribution the data do follow, and a non-significant one on a small sample is weak evidence of normality.
Working G2 out by hand for 1, 2, 3 and 40
Four values are enough to follow every step, and they show why the choice of formula matters.
Squares: 110.25, 90.25, 72.25, 812.25; their mean m2 = 271.25.
Fourth powers: 12,155.06, 8,145.06, 5,220.06, 659,750.06; their mean m4 = 171,317.56.
b2 = m4 / m2² = 2.328, so g2 = b2 - 3 = -0.672.
G2 = ((n + 1) × g2 + 6) × (n - 1) / ((n - 2)(n - 3)) = (5 × -0.672 + 6) × 3 / 2 = 3.963.
The single 40 supplies 96.3% of the fourth-power sum. And the two formulas disagree even on the sign: the population value says lighter tails than normal, Excel KURT says heavier. With four values the correction multiplies by 3/2 and adds a large constant, so neither figure means much, and the standard error of 2.619 says as much.
Kurtosis questions with worked answers
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