Statistical & Aggregation
Sample & Population Skewness Formulas (SKEW vs SKEW.P Third Standardized Moment)
Measure distribution asymmetry and tail direction using both the adjusted Fisher-Pearson sample skewness (SKEW, requiring n >= 3) and population skewness (SKEW.P) built from SUMPRODUCT, AVERAGE, STDEV.S, and STDEV.P.
Right-tail vs left-tail asymmetry and small-sample bias correction
Symmetric distributions have zero skewness, positive skewness (> 0) indicates an elongated right tail (such as enterprise deal sizes or latency spikes), and negative skewness (< 0) indicates a heavy left tail (such as leveraged drawdown returns).
Population skewness SKEW.P averages cubed population Z-scores (1 / n) * SUM(((x_i - mean) / sigma_p)^3), whereas sample skewness SKEW applies the Fisher-Pearson unbiasedness multiplier [n / ((n - 1) * (n - 2))] * SUM(((x_i - mean) / s)^3), which requires at least n = 3 observations and non-zero dispersion.
| Observation (B2:B6) | Value | Deviation (x - 16) | Cubed Deviation (x - 16)^3 |
|---|---|---|---|
| Obs 1 | 10 | -6 | -216 |
| Obs 2 | 12 | -4 | -64 |
| Obs 3 | 13 | -3 | -27 |
| Obs 4 | 15 | -1 | -1 |
| Obs 5 (Right outlier) | 30 | +14 | +2,744 (Sum = +2,436 -> SKEW = 1.9594) |
Adjusted Fisher-Pearson Sample Skewness (SKEW)
Computes Excel’s exact sample SKEW across B2:B6 while guarding n < 3 or zero standard deviation.
=LET(nObs, COUNT(B2:B6), sd, STDEV.S(B2:B6), IF(OR(nObs < 3, sd = 0), "Undefined Skewness", LET(mu, AVERAGE(B2:B6), ROUND((nObs / ((nObs - 1) * (nObs - 2))) * SUMPRODUCT(((B2:B6 - mu) / sd) ^ 3), 4))))Population Third Standardized Moment (SKEW.P)
Computes population SKEW.P across B2:B6 using population standard deviation STDEV.P.
=LET(nObs, COUNT(B2:B6), sdp, STDEV.P(B2:B6), IF(OR(nObs < 1, sdp = 0), "Undefined Skewness", LET(mu, AVERAGE(B2:B6), ROUND(SUMPRODUCT(((B2:B6 - mu) / sdp) ^ 3) / nObs, 4))))Try the example
Use the original values or change them to test your understanding.