SigmaCalc

Kurtosis Calculator

How likely are extreme outliers, compared to a normal distribution?

Dataset (comma / space / new-line separated)

Excess kurtosis

-0.141

mesokurtic (similar to a normal distribution)

What you entered

Dataset: 10 values

1
Mean
Σx ÷ 10= 12.7
2
Second moment: average of (x−mean)²
Σ(x−mean)² ÷ n= 17.81
3
Fourth moment: average of (x−mean)⁴
Σ(x−mean)⁴ ÷ n= 907.0217
4
Kurtosis: fourth moment ÷ (second moment)²
907.0217 ÷ 17.81²= 2.8595
5
Excess kurtosis: subtract 3 (normal distribution baseline)
2.8595 − 3= -0.1405
6
Interpretation
excess kurtosis = -0.1405= mesokurtic (similar to a normal distribution)

Result

Excess kurtosis: -0.141

Excess kurtosis of -0.141 — this dataset is mesokurtic (similar to a normal distribution).

Excess kurtosis and tail weight

Excess kurtosis compares tail weight against a normal distribution, which sits at zero. Negative means lighter tails and a flatter peak; positive means heavier tails.

Dataset	Excess kurtosis	Shape
2, 4, 4, 4, 5, 5, 7, 9	-0.219	Mesokurtic - similar to normal
10, 12, 23, 23, 16, 23, 21, 16	-1.398	Platykurtic - lighter tails
1, 2, 3, 4, 5	-1.300	Platykurtic - lighter tails
5, 10, 15, 20, 25, 30	-1.269	Platykurtic - lighter tails
70, 75, 80, 85, 90, 95, 100	-1.250	Platykurtic - lighter tails
3, 7, 7, 19, 24, 30, 32, 45	-1.107	Platykurtic - lighter tails

Evenly spaced datasets like rows three to five come out strongly platykurtic, near -1.25, because a uniform spread has no tails to speak of - the theoretical value for a continuous uniform distribution is -1.2. Only row one, which has a genuine cluster around 4 and a straggler at 9, lands close to normal. Kurtosis is about tail weight rather than peakedness, despite the common description: a high value means extreme values are more likely than a normal distribution predicts, which matters a great deal in finance and risk work. This calculator reports excess kurtosis, so a normal distribution is 0 rather than 3.

Peaks, tails, and the number 3

Kurtosis measures how concentrated a distribution is in its tails versus its center — a normal distribution has a raw kurtosis of exactly 3, so this calculator reports 'excess kurtosis' (kurtosis − 3), where 0 means normal-like tail behavior.

Why extreme events matter more here

Positive excess kurtosis (leptokurtic) means more extreme outliers than a normal distribution would predict — this is a big deal in finance and risk modeling, where underestimating tail risk (a 'fat tail') has historically preceded major market blowups.

Frequently asked questions

What does excess kurtosis of 0 vs 3 vs −1 mean?

Excess kurtosis 0 (mesokurtic): normal-distribution-like tails. Positive (leptokurtic, e.g. +3): heavier tails, more extreme outliers than normal — stock returns often show this. Negative (platykurtic, e.g. −1): lighter tails, fewer extremes — like a uniform distribution.

Why do financial analysts care about kurtosis?

Standard risk models assume normal distributions (kurtosis = 3). Real market returns have excess kurtosis of 5–10+, meaning 'once in a century' crashes happen far more often than the normal distribution predicts. Ignoring fat tails was a factor in the 2008 financial crisis.

How is kurtosis different from skewness?

Skewness measures asymmetry (lopsidedness). Kurtosis measures tail weight (likelihood of extreme values). A perfectly symmetric distribution can have any kurtosis. Use both together: skewness tells you which direction extremes lean; kurtosis tells you how extreme they get.