# Arithmetic, Geometric and Harmonic Mean (Statistics for Psychologists)

These notes are primarily posted for BSc Psychology (Research Methodology) and BSc Clinical Psychology (Statistics) students of Martin Luther Christian University, Shillong. Ideas are borrowed mainly from Gupta & Kapoor (link) (Textbook) and Sahoo (link).

## Means

### Arithmetic mean

Arithmetic mean is defined as the sum of all observations in a data series divided by the count of all observations in that series. Mathematically,

Arithmetic mean denoted by $\bar{x}$  of $n$ observations $x_1,x_2,\cdots,x_n$ is given by $\bar{x}={x_1+x_2+\cdots+x_n\over n}={\sum_{i=1}^n x_i\over n}$

The symbol ∑ (capital sigma) is a Greek letter; but here, it is used to simplify the way we write summations.
$\sum_{i=1}^n x_i$ means $x_1+x_2+\cdots+x_n\over n$ and is read as “summation i=1  to n, $x_i$”.

Example 1: Find the arithmetic mean of 1,3,4,3,5,6,1,1
Solution: Note that there are 8 numbers, hence, n=8. Then,
$\bar{x}={1+3+4+3+5+6+1+1\over8}=3$.

One of the ways we write the data more concisely is using the idea of ‘frequency’ of a particular number. Here’s the definition of arithmetic mean, in case of ‘frequency distribution’:

Arithmetic mean denoted by $\bar{x}$  of $n$ observations $x_i$ with frequencies $f_i$ is given by
$\bar{x}={f_1x_1+f_2x_2+\cdots+f_nx_n\over f_1+f_2+\cdots+f_n}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}$

Example 2:

Solution:

Thus, the arithmetic mean,

$\bar{x}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}={299\over73}=4.09$

Example 3: Note Example 1. The given data can be re-written more neatly as:

Can you find the average using the second formula?

Thus, arithmetic mean, (answer will be the same as Example 1)

$\bar{x}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}=$_______

If the data is given in ranges, we take the midpoint of each range as our x for calculation.

Example 4: Calculate the arithmetic mean of the marks from the following table:

Solution:

Thus, arithmetic mean,

$\bar{x}={\sum_{i=1}^n f_ix_i\over \sum_{i=1}^n f_i}=$_______

Example 5: Calculate the arithmetic mean from the following table:

#### Weighted Arithmetic Mean

Weighted Arithmetic mean denoted by $\bar{x}$  of $n$ observations xi with weights wi is given by

$\bar{x}={w_1x_1+w_2x_2+\cdots+w_nx_n\over w_1+w_2+\cdots+w_n}={\sum_{i=1}^n w_ix_i\over \sum_{i=1}^n w_i}$

Example 6: Find the simple and weighted arithmetic mean of the first 6 natural numbers, the weight being the corresponding numbers.

Solution: The first 6 natural number are 1, 2, 3, 4, 5, 6.

Weighted arithmetic mean =

#### Properties of Arithmetic Mean

1. Sum of the  $x_i - \bar{x}$ is zero. i.e., $\sum_{i=1}^n{f_i [x_i-\bar x]}=0$
2. The sum of the squared of the deviations of a set of values is minimum when taken about mean.
3. (Mean of Composite Series) If $\bar{x}_i$ (i=1,2,3,…k) are the means of -series of sizes ni (i=1,2,3,…k) respectively, then the mean  of the composite series obtained on combining these series is given by the formula

$\bar x = {n_1 \bar x_1+ n_2 \bar x_2 +\cdots+ n_k \bar x_k \over n_1+\cdots+n_k}={\sum_i n_i\bar x_i\over\sum_i n_i}$

The last property might be useful in future, in your research and entrance examinations for higher studies!

#### Merits and Demerits of Arithmetic Mean

Remember the discussion about characteristics of an ideal central measure? Let’s see how ‘good’ arithmetic mean is.

Remember requisites for ideal central tendency we discussed earlier? Turns our arithmetic mean is not ‘ideal’ – it has its own merits and demerits. That’s why we have different central tendencies, which will be discussed below.

There are two more mathematical means which are useful in a few cases, called Geometric Mean and Harmonic mean. But Arithmetic mean is the easiest to understand, easiest to calculate and most widely used. Hence, if we simply say ‘mean’ of a distribution, we are referring to Arithmetic Mean.

## Geometric and Harmonic Mean

Consider a distribution with n observations, x1, x2, x3, …, xn.

Geometric mean is the nth root of their product.

$G = [x_1\cdot x_2 \times \cdots \cdot x_n]^\frac1n$

and Harmonic mean, $H = \frac1{\frac1n(\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n})}$

Both are rigidly defined, but mathematically hard to calculate and understand. Also, is one of the observations is zero, then G becomes zero and H becomes impossible to calculate!