Calculation of Arithmetic Mean for Grouped Data

For grouped data the observations carry frequencies. There are two cases.

Discrete series

Each distinct value $X$ occurs with frequency $f$.

Direct method — multiply each value by its frequency, total the products, and divide by the total frequency:

Example. Plots come in three sizes — 100, 200, 300 sq. m — with 200, 50 and 10 plots respectively.

The same two shortcuts apply, now weighting each deviation by its frequency. Assumed mean: $\bar{X} = A + \dfrac{\sum fd}{\sum f}$. Step deviation (with $d' = \dfrac{X-A}{c}$): $\bar{X} = A + \dfrac{\sum fd'}{\sum f} \times c$.

Continuous series

Here the data is grouped into class intervals (which may be exclusive like 0–10, 10–20; inclusive like 0–9, 10–19; or unequal like 0–20, 20–50). The procedure is exactly the same as a discrete series, except that the mid-point $m$ of each class replaces the value $X$.

Example. Marks of students by class: