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The measure of variability or dispersion is the measure of scattered or the spread of the seperate scores around their central tendency.
According to -
Crow and Crow - The extent to which cases tend to gather around the average or central tendency or the extent to which they disperse themselves is called their variability or deviation.
Boring, Langfeld and Weld - Measure of variability tell us how widely the data is scattered about their mean.
Kinds of measures of variability or dispersion -
Range
Quartile Deviation
Mean Deviation
Standard Deviation
1. Range
1. Range
It is easiest to measure.
It is used when we need to know at a glance comparison of two or more groups for variability.
It is the difference between two extremes values.
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 -
Limitation of the Range -
It is based on two extreme values and tells nothing about the variation of the intermediate values. It is not an authentic measure of dispersion.
2. Quartile Deviation (Q)
2. Quartile Deviation (Q)
It is a refined method over range deviation.
Importance is given to the middle 50% of the value. Top and bottom 25% values each are ignored.
1st quartile (Q1) means upto 25th percentile.
3rd quartile (Q3) means upto 75th percentile.
Q = (Q3 - Q1) / 2
According to -
Skinner - Quartiles are the three points that divide a distribution into four equal parts.
Odell - The quartile deviation or semi-interquartile range is one half of the distance between the first and the third quartiles.
A. Calculation of quartile deviation in ungrouped data -
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 -
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 -
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 -
For symmetrical distribution -
Q3 - Median = Median - Q1
For non-symmetrical distribution -
For non-symmetrical distribution -
Q3 - Median > Median - Q1 (Positive skewed curve)
Q3 - Median < Median - Q1 (Negative skewed curve)
3. Mean Deviation (MD)
3. Mean Deviation (MD)
It is also known as Average Deviation (AD).
It is the simplest measure of variability that takes into account the fluctuation or variation of all the items in a series.
According to -
Garrett - It is the mean of the deviation of all the separate scores in the series taken from their mean.
A. Computation of Mean Deviation from ungrouped data -
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 -
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)
4. Standard Deviation (SD) (σ)
It is the most stable and reliable measure of variability as it employs mean for its computation.
According to -
Reichmann - Standard Deviation is also known as root mean square deviation. It is the square root of the mean value of all the deviation squared taken from the distribution mean.
James Drever - Standard Deviation is the square root of the means of the squares of individual deviation from the mean in a given series.
A. Computation of Standard Deviation from ungrouped data -
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 -
B. Computation of Standard Deviation from grouped data -
i) Main method -
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 -
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.
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