Statistics Definitions > Pearson’s Coefficient of Skewness

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## What is Pearson’s Coefficient of Skewness?

Karl Pearson developed two methods to find skewness in a sample.

- Pearson’s Coefficient of Skewness #1 uses the mode. The formula is:

Where = the mean, Mo = the mode and s = the standard deviation for the sample.

**See**: Pearson Mode Skewness. - Pearson’s Coefficient of Skewness #2 uses the median. The formula is:

Where = the mean, Mo = the mode and s = the standard deviation for the sample.

It is generally used when you don’t know the mode.

**Sample problem:** Use Pearson’s Coefficient #1 and #2 to find the skewness for data with the following characteristics:

- Mean = 70.5.
- Median = 80.
- Mode = 85.
- Standard deviation = 19.33.

**Pearson’s Coefficient of Skewness #1 (Mode)**:

Step 1: Subtract the mode from the mean: 70.5 – 85 = -14.5.

Step 2: Divide by the standard deviation: -14.5 / 19.33 = -0.75.

**Pearson’s Coefficient of Skewness #2 (Median)**:

Step 1: Subtract the median from the mean: 70.5 – 80 = -9.5.

Step 2: Multiply Step 1 by 3: -9.5(3) = -28.5

Step 2: Divide by the standard deviation: -28.5 / 19.33 = -1.47.

**Caution**: Pearson’s first coefficient of skewness uses the mode. Therefore, if the mode is made up of too few pieces of data it won’t be a stable measure of central tendency. For example, the mode in both these sets of data is 9:

1 2 3 4 5 6 7 8 9 9.

1 2 3 4 5 6 7 8 9 9 9 9 9 9 9 9 9 9 9 9 10 12 12 13.

In the first set of data, the mode only appears twice. This isn’t a good measure of central tendency so you would be cautioned *not* to use Pearson’s coefficient of skewness. The second set of data has a more stable set (the mode appears 12 times). Therefore, *Pearson’s coefficient of skewness will likely give you a reasonable result.*

## Interpretation

In general:

- The direction of skewness is given by the sign.
- The coefficient compares the sample distribution with a normal distribution. The larger the value, the larger the distribution differs from a normal distribution.
- A value of zero means no skewness at all.
- A large negative value means the distribution is negatively skewed.
- A large positive value means the distribution is positively skewed.

## References

Beyer, W. H. CRC Standard Mathematical Tables, 31st ed. Boca Raton, FL: CRC Press, pp. 536 and 571, 2002.

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.

Lindstrom, D. (2010). Schaum’s Easy Outline of Statistics, Second Edition (Schaum’s Easy Outlines) 2nd Edition. McGraw-Hill Education

“Cook, T. (2005). Introduction to Statistical Methods for Clinical Trials (Chapman & Hall/CRC Texts in Statistical Science) 1st Edition. Chapman and Hall/CRC

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