How Do You Estimate the Mean From Grouped Data?

To estimate the mean from grouped data, multiply each class midpoint by its frequency, add these products, and divide by the total frequency:

$$\text{estimated mean}=\frac{\sum fm}{\sum f}$$

Here, $f$ is the frequency and $m$ is the midpoint of a class interval.

The Method

Grouped data gives intervals rather than individual values, so the exact mean cannot usually be calculated. The midpoint of each interval is used as a representative value for every observation in that class.

For each class, calculate:

$$m=\frac{\text{lower boundary}+\text{upper boundary}}{2}$$

Suppose the times taken by students to complete a task are grouped as follows:

Therefore,

$$\text{estimated mean}=\frac{410}{20}=20.5\text{ minutes}$$

The calculation is a weighted mean: each midpoint is weighted by the number of observations in its class. A class with frequency $7$ therefore contributes seven copies of its midpoint, not one. This explains why the numerator is $\sum fm$ rather than $\sum m$.

The answer is an estimate because the actual values within each interval are unknown. For example, the three observations in the first class are treated as though they were all equal to $5$, although they may have been different.

A common misconception is to average the class midpoints without considering frequency. This is incorrect because classes containing more observations must have more influence on the mean.

IB Exam Technique

This is SL 4.3: Mean, median, and mode, studied by both SL and HL Mathematics: Applications and Interpretation students. On any GDC-active paper, show the midpoint or (fm) calculation before giving the result, use the population statistics setting unless told otherwise, retain full precision, and round only the final answer. Always call the result an estimated mean, not an exact mean.

On a GDC, enter midpoints in one list and frequencies in a second, then run one-variable statistics using the frequency list. The displayed mean should match the manual calculation, providing a useful check.