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 .

Mean DeviationStandard Deviation
Uses absolute value to assure that final result is positiveSquares to make all values positive, then takes positive square root.
Using absolute value makes it less suitable for further mathematical treatment.Most suited measure of dispersion for further mathematical treatment.
Makes use of all values in the distribution.Makes use of all values in the distribution.
Can be calculated about any measure of central tendencyCalculated using arithmetic mean.

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

x:1234
f:2141

Solution:

STEP 1: To find Mean

xffx
122
212
3412
414
Total∑f = 8∑fx = 20

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

STEP 2: To find Standard Deviation

xfx-\bar x(x-\bar x)^2f(x-\bar x)^2
12-1,52.254.50
21-0.50.250.25
340.50.251
411.52.252.25
N=8\sum f(x-\bar x)^2=8

\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

x:811121416
f:22522

(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

x:610121416
f:22522

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

xfd={x-12\over2}fd
∑f=∑fd=

\bar x=

dfd-\bar d(d-\bar d)^2f(d-\bar d)^2
-31
-12
03
12
32
N=\sum f=\sum f(d-\bar d)^2

𝜎𝑑2 =

𝜎𝑑 =

Coefficient of Variation

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

Example 1: Calculate the coefficient of variation

x:12345
f:12221

(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

About 0,

  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

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