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Statistics

Quartile Deviation

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September 6, 2015
Marjankhan

Definition

The quartile deviation is half the difference between the third quartile and the first quartile of a frequency distribution, or simply distribution. Mathematically, quartile deviation would be represented as follows;

clip_image002[5]

Quartile deviation is also known as semi-interquartile range. Here, the difference between the third and first quartiles is called interquartile range. The interquartile range may be taken as measure of dispersion (i.e. the extent to which the values are spread out from the average).

Quartile deviation only clicks once you have worked a dozen of them by hand. The Statistics Made Simple Practice Questions workbook has graded drills on every measure of spread, each with a fully worked solution.
Statistics Made Simple
Every measure of spread, worked by hand.
Range, quartile deviation, mean absolute deviation, variance and standard deviation — explained from scratch in the 246-page textbook, then drilled in 1,569 practice questions with full answers.

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Characteristics of Quartile Deviation

i. The quartile deviation is simple to understand and easy to calculate.

ii. As a measure of variation, it is superior to the range because it is not affected by extreme values.

iii. The values of quartile deviation might be the same for two dissimilar distributions provided the quartiles are the same.

iv. It is not utilized in algebraic manipulation.

v. It is of the use only in a case where one wants to study the dispersion of items in the middle or the main body of the series. This usually happens in a frequency distribution where distribution tends to be intense in the middle or the main body of the series and the distribution of items towards the extremes are not of much significance.

Example

We will find the semi-interquartile range or the quartile deviation for the distribution of weights of 120 students in following Table 19;

Table 19

Weight (lb)

Frequency (f)

Class Boundaries

Cumulative Frequency

110 – 119

1

109.5 – 119.5

0

120 – 129

4

119.5 – 129.5

5

130 – 139

17

129.5 – 139.5

22

140 – 149

28

139.5 – 149.5

50

150 – 159

25

149.5 – 159.5

75

160 – 169

18

159.5 – 169.5

93

170 – 179

13

169.5 – 179.5

106

180 – 189

6

179.5 – 189.5

112

190 – 199

5

189.5 – 199.5

117

200 – 209

2

195.5 – 209.5

119

210 – 219

1

209.5 – 219.5

120

∑f = n = 120

clip_image004[4]

clip_image006[4]

clip_image008[4]

clip_image010[4]

clip_image012[4]

clip_image014[4]

clip_image016[4]

clip_image018[4]

Putting the values of clip_image020[6] and clip_image022[6] in the formula, we get the following;

clip_image024[4]

clip_image026[4]

clip_image028[4]

Explanation

We are aware of the fact that 50% of the values lie between clip_image030[4] and clip_image032. The range i.e. Median clip_image034 Q.D, also contains approximately 50% of the values. In a symmetrical distribution, the quartile are equi-distant from the median and the quartile deviation measures the distance from the median to the lower quartile or the distance from the median to the upper quartile. Therefore, the in a symmetrical distribution, if we measure quartile deviation below and above the median than it will includes the central 50% of the values of the distribution. We always need to remember that, in a distribution where there is not complete symmetry, quartile deviation measures the average distance from the quartiles to the median.

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