What Is Coefficient of Skewness?
The coefficient of skewness measures the direction and strength of asymmetry. Positive values mean a longer right tail, negative values a longer left tail and zero means symmetry.
Enter your data to get Pearson’s first coefficient from the mode, Pearson’s second from the median, the quartile-based Bowley coefficient and the moment coefficients g1 and G1. A table says which one suits which situation and a histogram shows the shape.
Below are the formulas, a worked example and answers on why the coefficients differ.
How to Use the Coefficient of Skewness Calculator
- Enter your data.
- Read all four coefficients.
- Check the sign.
- Look at the histogram.
Coefficient of Skewness Formula
Pearson first
Pearson second
Bowley
Coefficient of Skewness Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Mean | |
| s | Sample standard deviation | |
| Q1, Q2, Q3 | Quartiles | |
| g1 | Moment skewness |
Coefficient of Skewness Worked Example
A data set with mean 5.5, median 4.5 and sample SD 3.3 has Pearson second coefficient 3 × (5.5 − 4.5) ÷ 3.3 = 0.91. The positive value shows a right tail pulled by large values such as 14.
With the example values loaded in the calculator above, the steps are:
- Mean = 5.5, median = 4.5, sample SD s = 3.2891.
- Pearson second = 3 × (5.5 − 4.5) ÷ 3.2891 = 0.912103.
- Quartiles: Q1 = 3.5, Q2 = 4.5, Q3 = 6.5; Bowley = (Q3 + Q1 − 2Q2) ÷ (Q3 − Q1) = 0.333333.
- Moment skewness g1 = m3 ÷ σ³ = 48 ÷ 31.228317 = 1.53707.
Result: Pearson second coefficient 3(mean − median)/s = 0.912103; Pearson first coefficient (mean − mode)/s = 0.456052; Bowley coefficient (quartile) = 0.333333.
How to Read a Skewness Coefficient
A positive value means a longer right tail, and a negative value means a longer left tail. For a data set with mean 5.5, median 4.5 and SD 3.3, the Pearson second coefficient is 3 × (5.5 − 4.5) ÷ 3.3 = 0.91, so the data lean to the right because of a few large values such as 14.
A common rule of thumb treats values between about −0.5 and 0.5 as roughly symmetric, and values beyond ±1 as strongly skewed. These are guides and not strict cut-offs, and the different coefficients use different formulas, so compare like with like.
Where Coefficient of Skewness Is Used
Checking symmetry
See whether data are lopsided before you choose the mean or the median.
Regression residuals
Check whether residuals are roughly symmetric.
Comparing coefficients
See the Pearson, Bowley and moment versions side by side.
Reports
Quote one coefficient and name it, since the numbers differ between formulas.
Common Coefficient of Skewness Mistakes to Avoid
- Name the coefficient you use. g1, G1, Pearson and Bowley give different numbers for the same data.
- Pearson’s first coefficient needs a clear single mode.
- Bowley’s coefficient uses quartiles, so outliers affect it less.
- Small samples give unstable skewness. Treat a value from 10 points with caution.
Coefficient of Skewness Terms and Definitions
- Skewness
- A measure of the asymmetry of a distribution.
Coefficient of Skewness FAQs
What is the coefficient of skewness?
It is a number that measures how lopsided a distribution is. Zero means symmetric, a positive value means a longer right tail, and a negative value means a longer left tail.
How do I calculate Pearson’s coefficient of skewness?
The second coefficient is 3 × (mean − median) ÷ standard deviation. The first is (mean − mode) ÷ standard deviation.
What does a negative skew mean?
The left tail is longer, so a few unusually small values pull the mean below the median.
Why do the coefficients differ?
They are built from different statistics. Pearson’s uses the mean and median or mode, Bowley’s uses quartiles and g1 uses the third moment. Each has its own scale, so values are not interchangeable.