Find the mean, variance and standard deviation using shortcut method
Height in cms          No. of Children
70-75                                     3
75-80                                     4
80-85                                     7
85-90                                     7
90-95                                   15
95-100                                   9
100-105                                 6
105-110                                 6
110-115                                 3

Solution:

Let us draw the frequency table based on the given problem

Class	Frequency fi	Mid-point (xi)	yi = (xi - 92.5)/5	yi²	fiyi	fiyi²
70 - 75	3	72.5	- 4	16	- 12	48
75 - 80	4	77.5	- 3	9	- 12	36
80 - 85	7	82.5	- 2	4	- 14	28
85 - 90	7	87.5	- 1	1	- 7	7
90 - 95	15	92.5	0	0	0	0
95 - 100	9	97.5	1	1	9	9
100 - 105	6	102.5	2	4	12	24
105 - 110	6	107.5	3	9	18	54
110 - 115	3	112.5	4	16	12	48
	60				6	254

Mean,

x = A + [∑⁹_{i = 1} f_iy_i]/N × h

= 92.5 + 6/60 × 5

= 92.5 + 0.5

= 93

Variance,

(σ ²) = h² / N² [N ∑⁹_{i = 1}(f_iy_i)² - (∑⁹_{i = 1}f_iy_i)²]

= (5)² / (60)² [60 × 254 - (6)²]

= 25/3600 × 15204

= 105.58

Standard deviation,

(σ) = √ 105.58

= 10.27

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NCERT Solutions Class 11 Maths Chapter 15 Exercise 15.2 Question 9

Find the mean, variance and standard deviation using shortcut method Height in cms          No. of Children  70-75     3  75-80     4 80-85       7  85-90      7  90-95         15  95-100        9  100-105        6  105-110        6  110-115        3

Summary:

The mean, variance, and standard deviation for the given data are 93, 105.58, and 10.27