Select Pearson’s second coefficient, mean, median or standard deviation to solve. Enter the other three values from the same data and unit basis.
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Coefficient of Skewness Formula
The calculator uses Pearson’s second coefficient of skewness, which estimates skewness from the mean, median, and standard deviation.
Rearranged formulas used to solve for the missing value:
- Sk = coefficient of skewness
- Mean = arithmetic average of the data set
- Median = middle value of the data set
- s = standard deviation
Select the value to solve, then enter the other three known summaries. The widget applies the displayed formula or its algebraic inverse. It does not calculate moment skewness from raw observations or verify that the summaries describe a realizable data set.
Standard deviation must be positive. With Sk = 0 and equal mean and median, any positive s satisfies the equation, so s is not uniquely determined. With unequal mean and median, Sk = 0 is inconsistent. A nonzero-Sk inverse must yield positive s.
How to Interpret the Skewness Coefficient
| Skewness coefficient | Mean–median relationship | What the coefficient establishes |
|---|---|---|
| Sk > 0 | Mean above median | The mean is greater than the median. This alone does not prove a longer right tail. |
| Sk = 0 | Mean equals median | The mean and median are equal. Equality alone does not prove symmetry. |
| Sk < 0 | Mean below median | The mean is less than the median. This alone does not prove a longer left tail. |
Illustrative Coefficient Magnitude Bands
| Absolute value of skewness | Literal magnitude interpretation |
|---|---|
| 0 ≤ |Sk| ≤ 0.5 | Mean–median gap is at most s/6; symmetry is not established. |
| 0.5 < |Sk| ≤ 1 | Mean–median gap is greater than s/6 and at most s/3. |
| |Sk| > 1 | Mean–median gap is greater than s/3; this is not a universal strength threshold. |
Example Calculations
Example 1: Calculate the skewness coefficient
Suppose the mean is 72, the median is 68, and the standard deviation is 12.
Pearson’s second coefficient is 1, indicating that the mean exceeds the median by s/3. It does not prove the complete distribution shape.
Example 2: Calculate the mean
Suppose the skewness coefficient is 0.6, the median is 50, and the standard deviation is 10.
The mean is 52.
FAQ
What does a positive coefficient of skewness mean?
A positive Pearson second coefficient means the mean is above the median. This often accompanies a right-tailed distribution, but the coefficient alone does not establish tail shape; inspect the data distribution.
What does a negative coefficient of skewness mean?
A negative Pearson second coefficient means the mean is below the median. This often accompanies a left-tailed distribution, but the coefficient alone does not establish tail shape; inspect the data distribution.
Can standard deviation be zero in this formula?
No. Standard deviation is in the denominator of the skewness formula, so it must be greater than 0. If the standard deviation is 0, all values are the same, and this coefficient of skewness is not defined.
