Skewness is a measure of the asymmetry of a dataset or distribution. This value can be positive or negative. A negative skew typically indicates that the tail is on the left side of the distribution. A positive value typically indicates that the tail is on the right.
Kurtosis is simply a measure of the “tailedness” of a dataset or distribution. The kurtosis formula used by this calculator is identical to the formula used in Excel, which finds what is known as excess kurtosis.
To find the skewness and kurtosis of a dataset, simply enter the comma-separated values in the box below, then click the “Calculate” button.
What formulas are being used in this? I’m trying to make my own calculator for these values and I can’t get them to match.
Hi Pax…
When you calculate skewness and kurtosis by hand, you usually get different answers from online calculators.
This isn’t because you’re doing the math wrong — it’s because calculators use corrected (“unbiased”) sample formulas, not the pure population formulas.
Here’s exactly what almost all calculators are using:
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First, some notation
You have data values:
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x1, x2, …, xn
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The mean is:
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x̄ = (1/n) * Σ(xi)
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The sample variance uses (n−1):
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s² = (1/(n−1)) * Σ(xi − x̄)²
“`
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Skewness Formula Calculators Use
They use the corrected sample skewness, also called Fisher’s skewness:
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Skewness = [ n / ((n−1)(n−2)) ] * [ Σ(xi − x̄)³ / s³ ]
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This correction factor:
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n / ((n−1)(n−2))
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is the reason your results won’t match if you use the population formula.
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Kurtosis Formula Calculators Use
This one confuses people because there are two steps.
### Step 1 — Compute the “biased” kurtosis (b₂):
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b2 = Σ(xi − x̄)⁴ / (n * s⁴)
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### Step 2 — Apply Fisher’s correction:
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Kurtosis = [(n−1)/((n−2)(n−3))] * [ (n+1)*b2 − 3*(n−1) ] + 3
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And then most calculators report “excess kurtosis”, which is:
“`
Excess Kurtosis = Kurtosis − 3
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That’s why sometimes your answer will differ by exactly 3.