Statistics Definitions > Within Mean Square

## What is Within Mean Square?

Within Mean Square (WMS) is an estimate of the population variance. It is based on the average of all variances *within* the samples. **Within Mean **is a weighted measure of how much a (squared) individual score varies from the sample mean score (Norman & Streiner, 2008). The notation for within mean square error is MS_{within} or you may sometimes see it written as:

Instead of σ the symbol for the population mean, you might also see “s” for the sample mean in the notation.

The most common formula is:

Where:

- “df” is the total degrees of freedom. To calculate this, subtract the number of groups from the overall number of individuals.
- SS
_{within}is the sum of squares within groups. The formula is: degrees of freedom for each individual group (n-1) * squared standard deviation for each group.

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## What is Within Mean Square used for?

Within Mean Square is used to calculate an F ratio in a one way ANOVA. The total sum of squares (SS) is the sum of both the **within mean square** and the **between mean square** (BMS). In a hypothesis test, the ratio BMS/WMS follows the shape of an F Distribution. If the ratio exceeds an F value for the test, it shows that there is a significant difference in your results.

## Between vs. Within

The two terms sound similar, but there is a notable difference between the two terms.

**Between group variation** (sometimes called among group variation) is how much variation there is due to interaction between samples. The mean square between is therefore the mean square among sample means.

**Within group variation **is how much variation there is due to differences within individual samples, so the mean square within is defined as the mean square within samples.

## References

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

Norman, D. & Streiner, G. (2008) Biostatistics: The Bare Essentials. PMPH-USA.

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