Skewness & Kurtosis Calculator
Compute population and sample skewness and excess kurtosis from any dataset. Enter values separated by commas, spaces, or line breaks — get distribution shape analysis with full step-by-step derivation instantly.
Calculating...
Enter a dataset (min 4 values) and click Calculate to analyse distribution shape.
How Skewness & Kurtosis Are Calculated
Sample formulas add correction factors so estimates are unbiased when working with a subset of data.
What Is Skewness?
Skewness measures the asymmetry of a probability distribution around its mean. A perfectly symmetric distribution — like the normal distribution — has a skewness of zero. When the right tail is longer (pulled by high-value outliers), skewness is positive. When the left tail is longer (pulled by low-value outliers), skewness is negative.
Real-world examples of right-skewed data include income distributions (most people earn modest amounts, a few earn millions), property prices, and the number of social media followers. Left-skewed data appears in scenarios where most values are high but a few are very low — such as the age at death in a healthy population, or exam scores in a class where almost everyone does well.
The rule of thumb: if skewness is between -0.5 and 0.5, the distribution is approximately symmetric. Between -1 and -0.5, or between 0.5 and 1, it is moderately skewed. Beyond ±1 it is highly skewed, and the mean is no longer a reliable measure of centre — the median is more representative.
What Is Kurtosis?
Kurtosis measures the weight of a distribution's tails relative to the normal distribution. This calculator reports excess kurtosis (also called Fisher's kurtosis), which subtracts 3 from the raw kurtosis so that the normal distribution scores exactly 0. That makes the result intuitive: positive excess kurtosis means heavier-than-normal tails, negative means lighter tails.
Leptokurtic (γ₂ > 0)
Heavy tails, sharp central peak. More prone to extreme outliers. Examples: financial returns, seismic data.
Mesokurtic (γ₂ ≈ 0)
Tail weight similar to the normal distribution. Many natural phenomena fall here.
Platykurtic (γ₂ < 0)
Light tails, flat peak. Fewer extreme outliers than normal. Examples: uniform distributions, bounded measurements.
High kurtosis in financial returns (leptokurtosis) is one reason the 2008 financial crisis was so severe — models based on normal distributions severely underestimated the probability of extreme market moves.
Population vs Sample Formulas
Both skewness and kurtosis come in population and sample versions. Population formulas assume you have every member of the group; sample formulas apply a correction factor to account for the fact that a sample's values cluster closer to the sample mean than to the true population mean.
| Formula | Population | Sample (Fisher-Pearson) |
|---|---|---|
| Skewness | m3 / sigma^3 | [N/((N-1)(N-2))] x sum[(xi-xbar)/s]^3 |
| Kurtosis | m4/sigma^4 - 3 | [(N(N+1))/((N-1)(N-2)(N-3))] x sum^4 - [3(N-1)^2/((N-2)(N-3))] |
| Min N | 2 | 4 (kurtosis has N-3 in denominator) |
| Used by | Census data, full populations | Excel SKEW() and KURT(), SPSS, SAS |
The sample formulas are unbiased estimators — they correct for the tendency of sample-based calculations to underestimate spread. For large datasets (N > 100), population and sample results differ very little. For small datasets, the difference can be substantial.
Skewness & Kurtosis Reference Table
Verify your calculations with these worked examples. All values are rounded to 4 decimal places.
| Dataset | N | Pop γ₁ | Sample G₁ | Pop γ₂ | Sample G₂ | Shape |
|---|---|---|---|---|---|---|
| 2, 4, 4, 4, 5, 5, 7, 9 | 8 | 0.6563 | 0.8186 | -0.2188 | 0.9406 | Moderately right-skewed, mesokurtic |
| 1, 2, 3, 4, 5, 6, 7 | 7 | 0.0 | 0.0 | -1.25 | -1.2 | Symmetric, platykurtic |
| 1, 1, 1, 2, 3, 8 | 6 | 1.4771 | 2.0224 | 0.5838 | 4.2002 | Highly right-skewed, leptokurtic |
| 2, 7, 8, 9, 9, 9 | 6 | -1.4771 | -2.0224 | 0.5838 | 4.2002 | Highly left-skewed, leptokurtic |
Frequently Asked Questions
What is skewness in statistics?
Skewness is a measure of asymmetry. It quantifies how much a distribution's shape deviates from a perfect mirror image around its mean. Zero skewness means the left and right tails are equally long. Positive skewness means the right tail is longer; negative skewness means the left tail is longer. Mathematically, skewness is the standardised third central moment: the average of the cubed deviations from the mean, divided by the cube of the standard deviation.
What does positive vs negative skewness mean?
Positive skewness (right skew) means the distribution has a long right tail. The mean is greater than the median, which is greater than the mode. Income, wealth, and many biological measurements are right-skewed. Negative skewness (left skew) is the opposite: a long left tail, with the mean below the median. Life expectancy in healthy populations is often left-skewed because most people live to old age, with some dying young pulling the mean down.
What is excess kurtosis and why subtract 3?
Raw kurtosis is the standardised fourth central moment: the average of the fourth-power deviations from the mean, divided by σ⁴. The normal distribution has a raw kurtosis of exactly 3. Subtracting 3 produces excess kurtosis, which centres the normal distribution at zero. This makes interpretation easier — positive excess kurtosis means heavier tails than normal, zero means normal-like tails, and negative means lighter tails. Excel's KURT() function returns excess kurtosis.
What is the difference between leptokurtic, mesokurtic, and platykurtic?
These three terms describe tail weight. A leptokurtic distribution (excess kurtosis > 0) has heavier tails than normal — extreme values occur more frequently than you would expect. Student's t-distribution is leptokurtic. A mesokurtic distribution (excess kurtosis ≈ 0) has tail weight similar to the normal distribution. A platykurtic distribution (excess kurtosis < 0) has lighter tails — fewer extreme values. The uniform distribution is platykurtic. Note that kurtosis describes tails, not the sharpness of the peak — that's a common misconception.
Why does the sample skewness formula use N-2 in the denominator?
The Fisher-Pearson sample skewness formula uses N/((N-1)(N-2)) as a correction factor rather than a simple 1/N. This is analogous to Bessel's correction for variance (using N-1 instead of N). Without the correction, skewness calculated from a sample systematically underestimates the population skewness. The N-2 factor compensates for this bias and ensures the estimate is consistent as sample size grows. Similarly, sample kurtosis uses N-3 in its denominator correction.
Can a distribution be symmetric but not normal?
Yes. Skewness = 0 only means the distribution is symmetric — it tells you nothing about tail weight or shape beyond that. The uniform distribution is symmetric with skewness 0 but is highly platykurtic. The t-distribution with 30 degrees of freedom is symmetric but leptokurtic. You need both skewness and kurtosis together to characterise a distribution's shape, and even together they do not uniquely determine the distribution — they are descriptive summaries, not complete descriptions.
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