Arithmetic Mean — Continuous Series

In a continuous (grouped) series, observations are grouped into class intervals (e.g., 0–10, 10–20, …), each with a stated frequency $f$. Because the individual values inside a class are not known separately, each class is represented by its mid-point $x = \dfrac{\text{lower limit} + \text{upper limit}}{2}$, and the mean is then computed exactly as for a discrete series, treating the mid-points as the $x$ values.

Direct method:

Assumed-mean method: choose a convenient mid-point as $A$, compute $d = x - A$ for every class, and apply

Step-deviation method: since class mid-points in a continuous series are always equally spaced by the class width $h$, this method applies especially naturally here:

$\bar{x} = A + \dfrac{\Sigma fd'}{N} \times h, \qquad d' = \dfrac{x-A}{h}$ …