Quartiles

There are three quartiles called, first quartile, second quartile and third quartile. There quartiles divides the set of observations into four equal parts. The second quartile is equal to the median. The first quartile is also called lower quartile and is denoted by{Q_1}. The third quartile is also called upper quartile and is denoted by{Q_3}. The lower quartile {Q_1} is a point which has 25% observations less than it and 75% observations are above it. The upper quartile {Q_3} is a point with 75% observations below it and 25% observations above it.

Quartile for Individual Observations (Ungrouped Data):

\begin{gathered} {Q_1} = {\text{Value of }}{\left( {\frac{{n +  1}}{4}} \right)^{th}}{\text{item}} \\ {Q_2} = {\text{Value of 2}}{\left( {\frac{{n  + 1}}{4}} \right)^{th}}{\text{item}} = {\text{Value of }}{\left( {\frac{{n + 1}}{2}}  \right)^{th}}{\text{item}} = Median \\ {Q_3} = {\text{Value of 3}}{\left( {\frac{{n  + 1}}{4}} \right)^{th}}{\text{item}} \\ \end{gathered}

Quartile for a Frequency Distribution (Discrete Data):
 

\begin{gathered} {Q_1} = {\text{Value of }}{\left( {\frac{{n +  1}}{4}} \right)^{th}}{\text{item}}\left( {n = \sum f} \right) \\ {Q_2} = {\text{Value of 2}}{\left( {\frac{{n  + 1}}{4}} \right)^{th}}{\text{item}} = {\text{Value of }}{\left( {\frac{{n +  1}}{2}} \right)^{th}}{\text{item}} = Median \\ {Q_3} = {\text{Value of 3}}{\left( {\frac{{n  + 1}}{4}} \right)^{th}}{\text{item}} \\ \end{gathered}

Quartile for Grouped Frequency Distribution:

\begin{gathered} {Q_1}{\text{ = }}l + \frac{h}{f}\left(  {\frac{n}{4} - c} \right){\text{ }}\left( {n = \sum f} \right) \\ {Q_2} = l + \frac{h}{f}\left( {\frac{{2n}}{4}  - c} \right) = l + \frac{h}{f}\left( {\frac{n}{2} - c} \right) = Median \\ {Q_3} = l + \frac{h}{f}\left( {\frac{{3n}}{4}  - c} \right) \\ \end{gathered}

Example:

The wheat production (in Kg) of 20 acres is given as: 1120, 1240, 1320, 1040, 1080, 1200, 1440, 1360, 1680, 1730, 1785, 1342, 1960, 1880, 1755, 1720, 1600, 1470, 1750, and 1885. Find the quartile deviation and coefficient of quartile deviation.

Solution:

After arranging the observations in ascending order, we get
1040, 1080, 1120, 1200, 1240, 1320, 1342, 1360, 1440, 1470, 1600, 1680, 1720, 1730, 1750, 1755, 1785, 1880, 1885, 1960.
            {Q_1} = {\text{Value of }}\left( {\frac{{n +  1}}{4}} \right)th{\text{ item}}
                   =  {\text{Value of }}\left( {\frac{{20 + 1}}{4}} \right)th{\text{ item}}
                   =  {\text{Value of (5}}{\text{.25)}}th{\text{ item}}
                   =  5th{\text{ item}} + 0.25(6th{\text{ item}} - 5th{\text{ item}}) = 1240 + 0.25(1320  - 1240)
            {Q_1} = 1240 + 20 = 1260
            {Q_3} = {\text{Value of }}\frac{{{\text{3(n +  1)}}}}{{\text{4}}}th{\text{ item}}
                   =  {\text{Value of }}\frac{{{\text{3(20 + 1)}}}}{{\text{4}}}th{\text{ item}}
                   =  {\text{Value of (15}}{\text{.75)}}th{\text{ item}}
                   =  15th{\text{ item}} + 0.75(16th{\text{ item}} - 15th{\text{ item}}) = 1750 +  0.75(1755 - 1750)
            {Q_3} = 1750 + 3.75 = 1753.75

Example:

Calculate the quartile deviation and coefficient of quartile deviation from the data given below:

Maximum Load
(short-tons)

Number of Cables

9.3 - 9.7

2

9.8 - 10.2

5

10.3 - 10.7

12

10.8 - 11.2

17

11.3 - 11.7

14

11.8 - 12.2

6

12.3 - 12.7

3

12.8 - 13.2

1

Solution:

The necessary calculations are given below:

Maximum Load
(short-tons)

Number of Cables
(f)

Class
Boundaries

Cumulative
Frequencies

9.3 - 9.7

2

9.25 - 9.75

2

9.8 - 10.2

5

9.75 - 10.25

2 + 3 = 7

10.3 - 10.7

12

10.25 - 10.75

7 + 12 = 19

10.8 - 11.2

17

10.75 - 11.25

19 + 17 = 36

11.3 - 11.7

14

11.25 - 11.75

36 + 14 = 50

11.8 - 12.2

6

11.75 - 12.25

50 + 6 = 56

12.3 - 12.7

3

12.25 - 12.75

56 + 3 = 59

12.8 - 13.2

1

12.75 - 13.25

59 + 1 = 60

            {Q_1} = {\text{Value of }}\left( {\frac{n}{4}}  \right)th{\text{ item }} = {\text{Value of }}\left( {\frac{{60}}{4}}  \right)th{\text{ item = }}15th{\text{ item}}
            {Q_1} lies in the class 10.25 - 10.75
            \therefore           {Q_1} = l +  \frac{h}{f}\left( {\frac{n}{4} - c} \right)
            Where l = 10.25, h = 0.5, f  = 12, n/4 = 15 and c = 7
            \therefore           {Q_1} = 10.25 +  \frac{{0.5}}{{12}}\left( {15 - 7} \right) = 10.25 + 0.33 = 10.58

            {Q_3} = {\text{Value of }}\left( {\frac{{3n}}{4}}  \right)th{\text{ item }} = {\text{Value of }}\left( {\frac{{3 \times 60}}{4}}  \right)th{\text{ item = 4}}5th{\text{ item}}
            {Q_3} lies in the class 11.25 - 11.75
            \therefore           {Q_3} = l +  \frac{h}{f}\left( {\frac{{3n}}{4} - c} \right)
            Where l = 11.25, h = 0.5, f  = 14, 3n/4 = 45 and c = 36
            \therefore           {Q_1} = 11.25 +  \frac{{0.5}}{{14}}\left( {45 - 36} \right) = 11.25 + 0.32 = 11.57

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