Arithmetic Mean — Grouped Data

When data is grouped into class intervals (for example, wages grouped as ₹100–120, ₹120–140, and so on), we no longer know each individual observation — only how many observations fall into each interval. To compute a mean, every observation in a class is assumed to lie at that class's midpoint, called the class mark:

Direct Method. Once the class marks are found, the same $\sum fx / \sum f$ idea from discrete data applies directly, using the class marks in place of individual values:

Assumed Mean Method. When the class marks are large or awkward numbers, the direct method involves heavy multiplication. Instead, one class mark is chosen as a trial value, the assumed mean $A$ (usually a class mark near the middle of the distribution, ideally one with a reasonably high frequency, so the deviations stay small), and the deviation $d_i = x_i - A$ is computed for every class:

Step-Deviation Method. When every class interval has the same width $h$, all the deviations $d_i$ are exact multiples of $h$. Dividing each deviation by $h$ turns them into small, easy-to-multiply "steps" $u_i$: