Arithmetic Mean — Computation for Individual, Discrete and Continuous Series

The arithmetic mean ($\bar X$) is the value obtained by dividing the sum of all observations by their number. Its method of computation depends on how the data is organised.

Individual series (data listed item by item, ungrouped):

• Direct method:

• Short-cut (assumed mean) method — useful when the values are large or inconvenient to add directly. An assumed mean $A$ is chosen (ideally close to the likely mean), and the deviation $d = X - A$ is found for every item:

Discrete series (distinct values $X$, each with frequency $f$):

• Direct method:

• Short-cut method, with $d=X-A$:

Continuous (grouped) series (class intervals, each with mid-value $m$ and frequency $f$): the same two formulas apply using the class mid-value in place of $X$, and a third, quicker method is commonly used —

• Step-deviation method. Take an assumed mean $A$ (usually the mid-value of the class with the highest frequency) and a common class width $h$; compute $d' = \dfrac{m-A}{h}$ for every class, which reduces the deviations to small whole numbers:

All three methods, applied correctly to the same data, always give the identical final mean — the short-cut and step-deviation methods only make the arithmetic lighter, they do not change the answer. …