Range and Quartile

IntroductionThe extent to which the numerical data in any distribution differs from each other is known as dispersion. Objectives of measuring dispersion are: to test the reliability of an average, to serve as a basis for control of variability, etc.The properties of an ideal measure of dispersion are: it is not affected by fluctuation of samples, stable and definite, rigidly defined, capable of further algebraic treatment, not affected by extreme terms, etc.Dispersion can be measured in two ways:    Absolute Measure    Relative MeasureAbsolute measures are range, inter-quartile range, quartile deviation, mean deviation and standard deviation. Relative measures comprise coefficient of range, coefficient of quartile deviation, coefficient of mean deviation and coefficient of standard deviation.For measuring the dispersion from the spread of values we use range and interquartile range and quartile deviation. Range is the difference between the largest and the smallest value of the distribution. Range can be calculated in all the three types of series individual series, discrete series and continuous series. The relative measurement of coefficient of range is calculated by dividing the difference between the largest and the smallest values by the sum of the largest and the smallest value. The interquartile range is calculated as the difference between the 3rd quartile and the 1st quartile of the distribution and the quartile deviation is the average of the difference of the 3rd quartile and 1st quartile. The coefficient of quartile deviation is calculated by dividing the difference between 3rd quartile and 1st quartile by the sum of the two. There are various merits and demerits associated with range and quartile.

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• Q1

Why should we measure dispersion about some particular value? Do the range and quartile deviation measure dispersion about some value?

Marks:2

We should measure dispersion about some particular value because measure of dispersion about some particular value gives a better idea about the dispersion of values within the range.

Range or quartile deviation does not measure dispersion about some value. Range and quartile deviation do not take into account the distribution of values.

• Q2

Write down the formulae used for measuring inter-quartile range for individual and continuous series.

Marks:2

Formula used for the computing inte-rquartile range for individual series is:

Formula used for computing inter-quartile range for continuous series is:

• Q3

The yield of wheat per acre for 10 districts of a state is as under:

 District 1 2 3 4 5 6 7 8 9 10 Yield of wheat 12 10 15 19 21 16 18 9 25 10

Find out:

(i) Range and Coefficient of Range
(ii) Quartile Deviation and its coefficient

Marks:3

In order to calculate the quartile and median, the data needs to be arranged in ascending order of magnitude:

 Yield of wheat (in tones) 9 10 10 12 15 16 18 19 21 25

(i) Range and Coefficient of Range:
Range = L – S
= 25 – 9
= 16 tons

(ii) Quartile Deviation and its Coefficient :

• Q4

Define Range. Write down the formula for measuring coefficient of range.

Marks:2

Range is the difference between the largest and the smallest value in the distribution. It is determined by two extreme values of observation.

Range = L – S

To compare the variability of two or more distributions given in different units of measurement, we cannot use absolute measure but we need a relative measure which is independent of the units of measurement. The relative measure is known as ‘Coefficient of Range’, and denoted by the formula: