Question

The yield of wheat and rice per acre for 10 districts of a state is as under: district 1 2 3 4 5 6 7 8 9 10 wheat 12 10 15 19 21 16 18 9 25 10 rice 22 29 12 23 18 15 12 34 18 12 calculate for each crop, (i) range (ii) q.D. (iii) mean deviation about mean (iv) mean deviation about median

Answer 1

(i) Range (Wheat) = 16, Range (Rice) = 22

(ii) QD (Wheat) = 4.75 QD (Rice) = 6.25

(iii) Mean deviation about median (wheat) = 4.3 (Rice) = 5.7

Step-by-step explanation:

yield of wheat per acre : 12 10 15 19 21 16 18 9 25 10

yield of rice per acre :  22 29 12 23 18 15 12 34 18 12

(i) Range :

a. Wheat :

Highest value of the distribution (H) : 25

Lowest value of the distribution (L) : 9

Range = H - L

Range = 25 - 9 = 16

b. Rice :

Highest value of the distribution (H) : 34

Lowest value of the distribution (L) : 12

Range = 34 - 12 = 22

(ii) Quartile deviation (QD) :

a. Wheat :

Arrange the data of wheat in ascending order :

9 10 10 12 15 16 18 19 21 25

$Q_{1}=\frac{n+1}{4}\text{th term}$

$Q_{1}=\frac{10+1}{4}$

$Q_{1}=\frac{11}{4}$

Q1 = 2.75 th term

2nd item + 0.75 (size of 3rd item - size of 2nd item)

= 10 + 0.75(10 - 10)

= 10 + 0

Q1 = 10

$Q_{3}=\frac{3(n+1)}{4}\text{th term}$

$Q_{3}=\frac{3(10+1)}{4}$

$Q_{3}=\frac{33}{4}\text{th term}$

Q3 = 8.25 th term

Size of 8th item + 0.25 (size of 9th item - size of 8th item)

= 19 + 0.25 (21 - 19)

= 19 + 0.25(2)

= 19 + 0.50

Q3 = 19.50

$QD=\frac{Q_{3}-Q_{1} }{2}$

$QD=\frac{19.50-10}{2}$

$QD=\frac{9.50}{2}$

QD of wheat yield = 4.75

b. Rice :

Arrange the data of rice in ascending order :

12 12 12 15 18  18 22 23 29 34

$Q_{1}=\frac{n+1}{4}\text{th term}$

$Q_{1}=\frac{10+1}{4}$

$Q_{1}=\frac{11}{4}$

Q1 = 2.75 th term

2nd item + 0.75 (size of 3rd item - size of 2nd item)

= 12 + 0.75(12 - 12)

= 12 + 0

Q1 = 12

$Q_{3}=\frac{3(n+1)}{4}\text{th term}$

$Q_{3}=\frac{3(10+1)}{4}$

$Q_{3}=\frac{33}{4}\text{th term}$

Q3 = 8.25 th term

Size of 8th item + 0.25 (size of 9th item - size of 8th item)

= 23 + 0.25 (29 - 23)

= 23 + 0.25(6)

= 23 + 1.5

Q3 = 24.50

$QD=\frac{Q_{3}-Q_{1} }{2}$

$QD=\frac{24.5-12}{2}$

$=\frac{12.50}{2}$

QD = 6.25

(iii) Mean deviation about median

a. Data of wheat in ascending order :

9 10 10 12 15 16 18 19 21 25

Median = $\frac{15+16}{2}$ = 15.5

x          |x - median|

9             6.5

10            5.5

10            5.5

12            3.5

15             0.5

16            0.5

18            2.5

19             3.5

21            5.5

25            9.5  

155          43  

Mean deviation about median = $\frac{43}{10}$ = 4.3

b. Data of rice in ascending order :

12 12 12 15 18  18 22 23 29 34

Median of the data = $\frac{18+18}{2}$ = 18

x        | x-median |

12         6

12         6

12         6

15         3

18         0

18         0

22        4

23        5

29        11

34        16  

195      57

Mean deviation about median = $\frac{57}{10}$  = 5.7

(i) Range (Wheat) = 16, Range (Rice) = 22

(ii) QD (Wheat) = 4.75 QD (Rice) = 6.25

(iii) Mean deviation about median (wheat) = 4.3 (Rice) = 5.7

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