Arithmetic Mean — Computation, Merits and Limitations

Arithmetic mean (AM), commonly just called the "mean" or "average," is the sum of all observations divided by their number. It is by far the most widely used measure of central tendency in business and economic statistics.

1. Individual series — direct method:

where $\Sigma x$ is the sum of all observations and $n$ is the number of observations.

2. Discrete series — direct method:

3. Continuous series — direct method: replace each class interval by its mid-point $m$, then apply the discrete-series formula:

Shortcut (assumed mean) method — useful when values are large or inconvenient to sum directly. Choose any value $A$ (an "assumed mean," usually a value near the middle of the data or a mid-point of a middle class), and compute deviations $d = x - A$ (individual/discrete) or $d = m - A$ (continuous):

Step-deviation method — for continuous series with equal class width $h$, divide the deviation by $h$ to keep the arithmetic small: $d' = \dfrac{m - A}{h}$

All three methods on the SAME data must give the IDENTICAL mean — this is exactly the cross-check used to dual-solve every arithmetic-mean numerical in this chapter.

Combined mean of two (or more) groups — when a group of $n_1$ observations has mean $\bar{X}_1$ and a second group of $n_2$ observations has mean $\bar{X}_2$, the combined mean of both groups taken together is:

Weighted mean — when observations do not carry equal importance, each value $x$ is multiplied by its own weight $w$ before averaging:

Merits of arithmetic mean:

• Rigidly (precisely) defined by a formula — leaves no room for subjective judgement.
• Based on ALL observations in the data set, unlike median or mode.
• Capable of further algebraic treatment — combined mean, weighted mean, and later dispersion measures are all built on it.
• Easy to understand and compute.

Limitations of arithmetic mean: …