Calculate population excess kurtosis g2 and adjusted Excel KURT estimate G2; requires at least four values and non-zero spread.
The Kurtosis Calculator reports population excess kurtosis g2 and the adjusted excess-kurtosis estimate G2 associated with Excel KURT. It requires at least four accepted finite values and non-zero spread. If the list is shorter or all values are identical, the calculator shows a validation message and does not display an undefined result.
Population excess kurtosis is based on the fourth central moment divided by the square of the second central moment, then shifted by three. The adjusted G2 applies a finite-sample correction. Both are dimensionless summaries of the entered values; they are not a complete description of a distribution and are not a formal test.
Values may be separated by commas or whitespace. Invalid and non-finite tokens are identified as ignored, and only accepted values enter the formulas. Extreme observations can strongly influence fourth powers, so inspect the raw list before interpreting a surprising coefficient. This page does not remove outliers or fit a probability distribution.
Let m₂ = Σ(xᵢ − x̄)² / n and m₄ = Σ(xᵢ − x̄)⁴ / n. Population excess kurtosis is g2 = m₄ / m₂² − 3. The subtraction of three defines the excess form used on this page.
For n ≥ 4 and non-zero spread, the adjusted value is G2 = (n − 1) / [(n − 2)(n − 3)] × [(n + 1)g2 + 6]. This is algebraically equivalent to the standard bias-adjusted excess-kurtosis formula used by Excel KURT.
The calculation is undefined when m₂ is zero because standardization divides by spread, and the adjusted formula also requires four observations. No special numeric substitute is assigned in either case; the page reports why a result is unavailable.
The first result row is population excess kurtosis g2. The second is adjusted G2, labelled with Excel KURT so the finite-sample correction is explicit. Their values can differ because G2 applies a correction based on n; report the chosen label rather than only saying “kurtosis.”
For the default list 4, 8, 15, 16, 23, 42, population g2 is approximately −0.2401 and adjusted G2 is approximately 1.7997. The difference is produced by the adjustment formula at n = 6, not by a change to the observations. Both are calculations on the supplied list.
The page does not decide whether the data are normally distributed. A coefficient near a reference value is not a hypothesis test, and a single fourth-moment summary cannot replace examination of the observations or a method designed for distributional inference.
Like other standardized central-moment coefficients, g2 does not carry the measurement unit: adding a constant leaves deviations unchanged, while multiplying every value by a common non-zero scale changes numerator and denominator by matching powers. The adjusted G2 also depends on the count, so the same population coefficient can yield a different adjusted value at a different sample size.
The correction can be checked directly from the displayed g2 formula. For n = 6, its multiplier is 5 / (4 × 3), and the bracket is 7g2 + 6. Substituting g2 ≈ −0.2401 gives approximately 1.7997. This is an arithmetic check on the adjustment, not a test of whether the dataset follows a particular distribution.
These excess-kurtosis outputs are descriptive calculations from the accepted finite values. They do not test a distributional hypothesis, establish a population property, or measure uncertainty.
For research use, verify the formula convention and sample-size requirements in the analysis plan. This page does not fit a distribution or determine whether a particular model is appropriate.