Statistical Analysis
Sample & Population Excess Kurtosis (KURT Fourth-Moment Tail Risk Formula)
Compute unbiased Fisher sample excess kurtosis (KURT) and population excess kurtosis from standardized fourth moments to detect fat-tailed (leptokurtic) vs thin-tailed (platykurtic) series.
Why fourth-moment excess kurtosis subtracts 3 and requires small-sample bias correction
A Gaussian normal distribution has a raw standardized fourth moment E[((X - mu) / sigma)^4] equal to 3. Excess kurtosis subtracts 3 so a normal reference curve scores 0, positive values (> 0, leptokurtic) flag heavy outlier tails and sharp peaks, and negative values (< 0, platykurtic) indicate uniform or bounded light-tailed behavior.
On finite samples of size n >= 4, dividing sum(((x_i - mean) / s)^4) by n understates tail weight. Fisher’s unbiased sample estimator (matching Excel KURT) scales the fourth-power standardized sum by n*(n + 1) / ((n - 1)*(n - 2)*(n - 3)) and subtracts 3*(n - 1)^2 / ((n - 2)*(n - 3)).
| Series Regime | Excess Kurtosis | Tail Interpretation | Example Return Profile |
|---|---|---|---|
| Platykurtic (< 0) | -1.2000 | Lighter tails than normal | Evenly spaced 5-point ladder [10, 12, 14, 16, 18] |
| Mesokurtic (= 0) | 0.0000 | Normal Gaussian tail benchmark | Standard bell-curve baseline |
| Leptokurtic (> 0) | +4.3536 | Fat tails / outlier shock present | Shocked series [10, 12, 14, 16, 40] |
Portable Unbiased Sample Excess Kurtosis (KURT) Formula
Evaluate n, sample mean, and sample standard deviation STDEV inside LET, sum the fourth powers of standardized z-scores with SUMPRODUCT, and apply Fisher’s finite-sample degree-of-freedom multipliers (returning N_LT_4 when fewer than 4 observations exist).
=LET(cnt, COUNT(B2:B6), mu, AVERAGE(B2:B6), sd, STDEV(B2:B6), IF(cnt < 4, "N_LT_4", IF(sd = 0, 0, ROUND((cnt * (cnt + 1) / ((cnt - 1) * (cnt - 2) * (cnt - 3))) * SUMPRODUCT(((B2:B6 - mu) / sd) ^ 4) - (3 * (cnt - 1) ^ 2) / ((cnt - 2) * (cnt - 3)), 4))))Population Excess Kurtosis Formula
When the range represents an entire population census rather than a sample, standardize by STDEVP, divide the fourth-power sum by n, and subtract 3 directly.
=LET(cnt, COUNT(B2:B6), mu, AVERAGE(B2:B6), sdp, STDEVP(B2:B6), IF(sdp = 0, 0, ROUND(SUMPRODUCT(((B2:B6 - mu) / sdp) ^ 4) / cnt - 3, 4)))Try the example
Use the original values or change them to test your understanding.