In one of Sollivan’s statistics sections, the standard deviation of the heights of all students was 3.9 inches. The standard deviation of the heights of males was 3.4 inches and the standard deviation of females was 3.3 inches. Why is the standard deviation of the entire class more than the standard deviation of the males and females considered separately?

Verified step by step guidance

1. Recall that the standard deviation measures the spread or variability of data around the mean within a group.
2. Understand that when combining two groups (males and females) into one larger group (the entire class), the overall variability depends not only on the variability within each group but also on the differences between the group means.
3. Recognize that if the average height of males and females differ, this difference contributes additional variability to the combined data set, increasing the overall standard deviation.
4. Mathematically, the total variance of the combined group can be expressed as the weighted average of the variances within each group plus the variance due to the difference between the group means.
5. Therefore, the standard deviation of the entire class is larger than the standard deviations of males and females separately because it includes both the within-group variability and the variability caused by the difference in group means.

Key Concepts

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of data values. A higher standard deviation indicates that data points are spread out over a wider range, while a lower standard deviation means they are closer to the mean.

Within-Group vs. Between-Group Variation

Within-group variation refers to variability inside each subgroup (e.g., males or females), while between-group variation captures differences between subgroup means. The total variation in combined data includes both within-group and between-group variations.

Effect of Group Means on Overall Standard Deviation

When combining groups with different means, the overall standard deviation can increase because the spread includes differences between group averages. This between-group difference adds to the total variability, making the combined standard deviation larger than that of individual groups.