Sample Kurtosis Calculator

Description

Calculate Pearson and Fisher excess kurtosis from sample central moments, with optional finite-sample bias correction and no normality-test claim.

Kurtosis uses a standardized fourth central moment to describe tail-related variation. Pearson kurtosis uses a normal-distribution reference value of three; excess kurtosis subtracts three. Optional finite-sample correction changes the estimator, not the input units.^[1]

Calculate Pearson and Fisher excess kurtosis from sample central moments, with optional finite-sample bias correction and no normality-test claim.

When to use this tool

Use this tool to calculate kurtosis under the units, normalization, and boundary conventions below.

How to use the tool and how it works

Supply values, biasCorrected in the field shapes below.
Run the calculation. Correct format, shape, range, or undefined-statistic errors before interpreting a result.
Read biasCorrected, count, excessKurtosis, pearsonKurtosis. Use the conventions below.

Compute normalized second and fourth sample central moments and form excess kurtosis m4/m2²−3. When biasCorrected is true, apply (n−1)[(n+1)g2+6]/[(n−2)(n−3)]. Return excess kurtosis and Pearson kurtosis equal to excess plus three.^[2]

Input field	Meaning and units	Accepted input
values	Values	Required. Array of Finite number.; minimum items 2; maximum items 512.
biasCorrected	Bias Corrected	Optional. Boolean; default false.

Output field	Interpretation	Returned shape
biasCorrected	Bias Corrected	Required. Boolean.
count	Requested count; the mechanics explain whether one draw expands into several returned records.	Required. Safe integer; minimum 2; maximum 512.
excessKurtosis	Excess Kurtosis	Required. Finite number.
pearsonKurtosis	Pearson Kurtosis	Required. Finite number.

json

{
  "input": {
    "values": [
      -2,
      -1,
      0,
      1,
      2
    ]
  },
  "output": {
    "biasCorrected": false,
    "count": 5,
    "excessKurtosis": -1.3,
    "pearsonKurtosis": 1.7
  }
}

Limitations and assumptions

Constant observations are rejected, and bias correction requires at least four observations. Uncorrected moments use n in their denominators. This is a shape statistic, not a normality test or a guarantee of reliable tail estimation.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

Choose the statistic according to the question and data conventions. Inspect the supplied series or matrix and the stated assumptions before comparing outputs from implementations with different defaults.

References

NIST: Kurtosis and excess kurtosis — NIST
Tool behavior and example: Sample Kurtosis Calculator — tool.io

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