Measure how lopsided a list of numbers is: sample and population skewness as Excel and SPSS print them, a D'Agostino p-value, and a check on how much one extreme value drives the result.
Skewness Calculator - G1, g1, b1 and D'Agostino Test
Measure how lopsided a list of numbers is: sample and population skewness as Excel and SPSS print them, a D'Agostino p-value, and a check on how much one extreme value drives the result.
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Skewness calculator with D'Agostino's test and all three textbook formulas
Why does Excel give two different skewness values for the same column? Because SKEW and SKEW.P use different formulas, and a third one lives in Minitab and R. This calculator takes a list of numbers and returns all three, marks the one your software prints, and adds what a bare number cannot tell you: a p-value from D'Agostino's test, the standard error SPSS reports, the quartile-based Bowley measure, and how much the result depends on a single extreme value.
Twelve incomes, four numbers
Sample data: 3200 3500 3800 4000 4200 4500 5000 5500 6200 7000 9000 14000. These are the four figures the calculator puts front and center for that list.
Read together: the list is highly skewed to the right, the asymmetry is statistically significant, one value supplies a large share of it, and even a log scale leaves a clear right lean.
Data, sample or population, alpha: the form in order
- Data - at least 3 numbers, up to 5,000, separated by spaces, commas or new lines. A column pasted from a spreadsheet works. Use a dot for decimals and no thousands separators.
- Sample or population - sample gives G1, the figure Excel SKEW, SPSS and SAS print. Population gives g1, as Excel SKEW.P. All three formulas appear in the table either way.
- Significance level - the threshold for D'Agostino's test: 0.05, 0.01 or 0.10. The test needs at least 8 values.
- Read the verdict, the p-value, the standard error, the table of measures and any note about an extreme value or a log scale.
Sign, size and whether it is real
The sign tells you which tail is longer. Positive means values stretch out toward the high end, as with incomes, house prices or reaction times; negative means the long tail points low, as with scores on an easy test where most people are near the top. The size is usually read with Bulmer's rule of thumb: below 0.5 in absolute value the shape counts as approximately symmetric, from 0.5 to 1 as moderately skewed, above 1 as highly skewed. The calculator applies it to the value shown to three decimals, so a figure displayed as 0.500 is never called symmetric.
Whether the asymmetry is real is a separate question. Skewness from a small sample wobbles a lot: the standard error is 0.752 for 8 values and still 0.637 for 12. D'Agostino's test turns the skewness into a Z score that is close to normal from 8 values on, the same test scipy runs as skewtest. For 2 8 0 4 1 9 9 0 it gives Z = 0.446 and p = 0.6554, matching the scipy documentation.
Which formula your software uses
Joanes and Gill compared the three estimators and listed where each one appears. The values are for the income list above.
| Measure | Incomes | Formula | Printed by |
|---|---|---|---|
| G1 | 1.999 | g1 × √(n(n - 1)) / (n - 2) | Excel and Google Sheets SKEW, SPSS, SAS |
| g1 | 1.740 | m3 / m21.5, moments divided by n | Excel SKEW.P, scipy default |
| b1 | 1.527 | m3 / s3, with the sample SD | Minitab, R package e1071 default |
| Bowley | 0.347 | (Q3 + Q1 - 2 × median) / (Q3 - Q1) | robust summaries; limited to -1 and 1 |
| Pearson's second | 1.052 | 3 × (mean - median) / SD | introductory textbooks |
The three moment formulas converge as the sample grows, since the correction factors approach 1. With 12 values they still differ by almost half a unit, which is why a skewness figure should always say which formula produced it.
Things skewness does that surprise people
Known distributions have fixed values. A sample drawn from any of these will scatter around the figure in the third column, more widely the smaller it is.
| Distribution | Skewness formula | Value | Typical data |
|---|---|---|---|
| Normal, any mean and SD | 0 | 0.000 | symmetric by construction |
| Uniform (flat) | 0 | 0.000 | symmetric, yet far from normal |
| Exponential | 2 | 2.000 | waiting times, lifetimes |
| Chi-square, k = 4 | √(8/k) | 1.414 | shrinks as k grows |
| Chi-square, k = 50 | √(8/k) | 0.400 | already close to symmetric |
| Poisson, mean 4 | 1/√λ | 0.500 | counts of rare events |
| Poisson, mean 25 | 1/√λ | 0.200 | larger counts, milder skew |
| Lognormal, sigma = 1 | (eσ² + 2)√(eσ² - 1) | 6.185 | incomes, prices, sizes |
Where a skewness figure gets asked for
Students checking a descriptive-statistics table, where SPSS prints skewness with its standard error and a ratio above about 2 is the classroom signal of asymmetry. Analysts deciding whether a variable needs a log scale before a t-test or a regression; the calculator tries the log for positive, right-leaning data. Quality engineers watching a process whose measurements should sit symmetrically around a target, where a growing skew hints at tool wear or a hard limit. Anyone reporting a mean who wants to know whether the median would describe the typical value better.
Four questions about skewness
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