Mean Squares

TL;DR

• Computes the average squared deviation of observations from their group mean.
• Calculate a mean square for each group, then compare those mean squares to assess whether groups differ substantially.
• Commonly used as part of hypothesis testing to judge whether observed differences are likely meaningful.

Definition

The term “mean squares” refers to a statistical calculation used to determine the difference between two sets of data. This calculation is typically used in hypothesis testing, where it helps researchers determine whether there is a significant difference between the two sets of data.

Explanation

To compute a mean square for a group:

1. Find the group’s mean.
2. For each observation, compute the square of its difference from the group mean.
3. Sum those squared differences and divide the sum by the number of observations in the group.

After computing mean squares for the groups being compared, the researcher compares them; a large difference between mean squares suggests a significant relationship between the grouping variable and the measured outcome.

Examples

Study time and test grades

Suppose a researcher studies the relationship between time spent studying and test grades. The sample yields these group averages:

• Average grade for students who studied more than 4 hours: 87
• Average grade for students who studied less than 4 hours: 73

The mean square for the group that studied more than 4 hours is written as:

$$\frac{(87-87)^2 + (90-87)^2 + (85-87)^2 + \dots + (78-87)^2}{n}$$

The mean square for the group that studied less than 4 hours is written as:

$$\frac{(73-73)^2 + (70-73)^2 + (68-73)^2 + \dots + (79-73)^2}{n}$$

Once the mean squares for both groups are calculated, they are compared: a large difference suggests a significant relationship between study time and test grades.

Use cases

• Comparing the average incomes of different demographic groups.
• Evaluating the effectiveness of different marketing campaigns.

Related terms

• Hypothesis testing