# Measures of Dispersion – 2 (Statistics for Psychologists)

## Standard Deviation

If $x_i | f_i , i=1, 2, \cdots, n$ is the frequency distribution, then standard deviation, denoted by Greek letter σ (sigma) is given by

$\sigma=\sqrt{\frac1N \sum_i f_i (x_i-\bar x)^2}$

where $\bar x$ is the arithmetic mean of the distribution and ∑f =N .

Standard Deviation is the most used measure of dispersion – and most suitable for further mathematical treatment.

## Variance

The square of standard deviation is called variance, denoted by $\sigma^2$

$\sigma^2 = \frac1N\sum_i f_i(x-\bar x)^2$

Example 1: Find the Standard Deviation and Variance

Solution:

STEP 1: To find Mean

$\bar x=\frac{20}8=2.5$

STEP 2: To find Standard Deviation

$\sigma^2=\frac1N\sum f(x-\bar x)^2=\frac88=1$

$\sigma=\sqrt{\frac1N\sum f(x-\bar x)^2}=\sqrt{1}=1$

Example 2: Find the Standard Deviation and Variation

(Try it yourself!)

In fact, if the numbers are big, you can simplify the calculation!

1. Choose any A (preferably median of observations or the observation with highest frequency) and h
2. Put $d={x-A\over h}$
3. Calculate 𝜎d
4. Then 𝜎𝑥 = h 𝜎d

Example 3

(Try it yourself with A = 12, h = 2)

$\bar x=$

𝜎𝑑2 =

𝜎𝑑 =

## Coefficient of Variation

Coefficient of Variation = $C.V.=100\times\frac{\sigma}{\bar x}$

Example 1: Calculate the coefficient of variation

(Try it yourself)

## Moments

The rth moment of a distribution centred at any number A is given by

$\mu_r'=\frac1N\sum f(x-A)^r$

1. 𝜇0′ = 1
2. 𝜇1′ is the arithmetic mean.

The rth  moment about $\bar x$ (i.e., $A = \bar x$) is called rth central moment

$\mu_r=\frac1N\sum f(x-\bar x)^r$

1. 𝜇0 = 1
2. 𝜇1 = 0
3. 𝜇2 = 𝜎2