Measures of Variability

The measure of variability or dispersion is the measure of scattered or the spread of the seperate scores around their central tendency. 

According to - 

Kinds of measures of variability or dispersion - 

1. Range 

Range = Highest score - Lowest score

E.g. - Range of 8, 20, 15, 17, 21, 14, 9, 16, 7, 11, 13, 14, 20, 21, 23, 28, 30
Range = 30 - 7 = 23

Limitation of the Range - 

2. Quartile Deviation (Q) 

Q = (Q3 - Q1) / 2

According to - 

A. Calculation of quartile deviation in ungrouped data - 

Q = (Q3 - Q1) / 2 

where, Q1 is N/4 th term
and Q3 is 3N/4 th term. 

E.g.- Q of 17, 15, 20, 25, 18, 22, 19, 20
Numbers in ascending order = 15, 17, 18, 19, 20, 22, 25, 28
Q1 = N/4 = 8/4 = 2nd term = 17
Q3 = 3N/4 = 3×8 / 4 = 6th term = 22
Q = (22-17) / 2
= 5/2
= 2.5

B. Calculation of quartile deviation in grouped data - 

Q = (Q3 - Q1) / 2 

where, Q1 = L + [(N/4 - cfb) / f] × i 

and Q3 = L + [(3N/4 - cfb) / f] × i

where, L is Actual lower limit of the quartile class
N is the total of frequencies
cfb is cumulative frequency of class below quartile class
f is frequency of quartile class
i is the class interval. 

Quartile class is the class in which N/4 and 3N/4 lies. 

Let's understand this with an example -

Here, N/4 = 50/4 = 12.5
It lies in the class 28-30
Thus, Q1 quartile class is 28-30 

and, 3N/4 = 3 × 50 / 4 = 37.5
It lies in the class 37-39
Thus, Q3 quartile class is 37-39

Q1 = L + [(N/4 - cfb) / f] × i
= 27.5 + [(12.5 - 11) / 7] × 3
= 27.5 + 0.64
= 28.14

Q3 = L + [(3N/4 - cfb) / f] × i
= 36.5 + [(37.5 - 37) / 8] × 3
= 36.5 + 0.187
= 36.69

Q = (Q3 - Q1) / 2
= (36.69 - 28.14) / 2
= 8.55 / 2
= 4.275
= 4.28

C. Interpretation of Quartile Deviation - 

In distribution where we prefer median as a measure of Central tendency, the quartile deviation is also preferred as measure of dispersion.

Median and quartile deviation both are not suitable to algebraic operations because both donot consider all the values of the given distribution. 

For symmetrical distribution - 

Q3 - Median = Median - Q1

For non-symmetrical distribution - 

Q3 - Median > Median - Q1 (Positive skewed curve) 

Q3 - Median < Median - Q1 (Negative skewed curve)

3. Mean Deviation (MD) 

According to - 

A. Computation of Mean Deviation from ungrouped data - 

MD = ∑|x| / N

where x = X - M, deviation from mean
and N = Total number of scores

E.g. - Mean Deviation of 15, 10, 6, 8, 11

Mean of the series = (15 + 10 + 6 + 8 + 11) / 5
= 50/5 = 10

MD = ∑|x| / N
= 12/5
= 2.4

B. Computation of Mean Deviation from grouped data - 

MD = ∑|fx| / N 

where, x = X - M, deviation from mean
f is the respective frequency
N is the total of all frequencies 

Let's understand it with an example -

Mean = M = A + (∑fd / N) × i
= 72 + (5 / 25) × 5
= 72 + 1
= 73 

and, Mean Deviation = MD = ∑|fx| / N
= 140/25
= 5.6 

Because this method ignores the importance of plus and minus sign, it doesn't fulfill the assumptions of algebraic properties and hence it is not possible to use this method in higher statistics.

4. Standard Deviation (SD) (σ) 

It is the most stable and reliable measure of variability as it employs mean for its computation. 

According to - 

A. Computation of Standard Deviation from ungrouped data - 

SD = √(∑x² / N)

where x = X - M, deviation from mean
N is total number of values

Mean of the scores = (10 + 6 + 9 + 8 + 7) / 5
= 40 / 5
= 8 

SD = √(∑x² / N)
= √(10 / 5)
= √2
= 1.414 

B. Computation of Standard Deviation from grouped data - 

i) Main method - 

SD = √(∑fx² / N)

where, x = X - M, deviation from mean
f is the respective frequency
N is the total of all frequencies

Mean = M =  A + (∑fd / N) × i
= 116 + (-8 / 24) × 3
= 116 - 1
= 115 

SD = √(∑x² / N)
= √(1074 / 24)
= √44.75
= 6.69 

In this method we have made use of mean, we can also solve it with short-cut method where mean is not used.

ii) Short-Cut Method - 

SD = i × √[(∑fd² / N) - (∑fd / N)²] 

where, d = (X - A) / i and i is the class interval
f is the respective frequency
N is the total of all frequencies

SD = i × √[(∑fd² / N) - (∑fd / N)²]
= 3 × √[(122 / 24) - (-8 / 24)²]
= 3 × √[(122 / 24) - (64 / 24×24)²]
= 3/24 × √(122 × 24 - 64)
= 1/8 × √2864
= 53.51 / 8
= 6.69 


Short-Cut method is generally used when the value of mean is in fraction, and if mean is a whole number than we generally use the main method.