Kurtosis Calculator Formula

Understand the math behind the kurtosis calculator. Each variable explained with a worked example.

Use the Kurtosis Calculator

Formulas Used

Kurtosis

kurtosis = m4 / pow(m2, 2)

Excess Kurtosis

excess_kurtosis = m4 / pow(m2, 2) - 3

Variables

VariableDescriptionDefault
v1Value 12
v2Value 24
v3Value 34
v4Value 46
mnDerived value= (v1 + v2 + v3 + v4) / 4calculated
m2Derived value= (pow(v1 - mn, 2) + pow(v2 - mn, 2) + pow(v3 - mn, 2) + pow(v4 - mn, 2)) / 4calculated
m4Derived value= (pow(v1 - mn, 4) + pow(v2 - mn, 4) + pow(v3 - mn, 4) + pow(v4 - mn, 4)) / 4calculated

How It Works

How to Calculate Kurtosis

Formula

Kurtosis = m4 / m2^2

where m4 is the fourth central moment and m2 is the second central moment (variance).

Excess Kurtosis = Kurtosis - 3

  • Excess kurtosis > 0: Leptokurtic (heavier tails than normal)
  • Excess kurtosis = 0: Mesokurtic (similar to normal)
  • Excess kurtosis < 0: Platykurtic (lighter tails than normal)

    • The normal distribution has a kurtosis of 3, so subtracting 3 gives excess kurtosis relative to normal.

    Worked Example

    Compute the kurtosis of the values 2, 4, 4, 6.

    v1 = 2v2 = 4v3 = 4v4 = 6
    1. 01Mean = (2 + 4 + 4 + 6) / 4 = 4
    2. 02Deviations: -2, 0, 0, 2
    3. 03m2 = (4 + 0 + 0 + 4) / 4 = 2
    4. 04m4 = (16 + 0 + 0 + 16) / 4 = 8
    5. 05Kurtosis = 8 / 4 = 2
    6. 06Excess Kurtosis = 2 - 3 = -1 (platykurtic)

    Ready to run the numbers?

    Open Kurtosis Calculator