Find the mean and standard deviation using shortcut method
x_i 60  61  62  63  64  65  66  67  68
f_i   2    1  12  29  25  12  10    4    5

Solution:

Let us draw the frequency table based on the given problem.

xi	fi	fi = (xi - 64)/1	yi2	fiy1	fiyi2
60	2	- 4	16	- 8	32
61	1	- 3	9	- 3	9
62	12	- 2	4	- 24	48
63	29	- 1	1	- 29	29
64	25	0	0	0	0
65	12	1	1	12	12
66	10	2	4	20	40
67	4	3	9	12	36
68	5	4	16	20	80
	100	220		0	286

Mean,

x = A + [ ∑⁹_{i = 1} f_i y_i ] / N × h

= 64 + 0 / 100 x 1

= 64

Variance,

(σ ²) = h² / N² ∑⁹_{i = 1}(x_i - x)²

= 1/100² [100 × 286 - 0]

= 2.86

Standard deviation,

(σ) = √ 2.86

= 1.69

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NCERT Solutions Class 11 Maths Chapter 15 Exercise 15.2 Question 6

Find the mean and standard deviation using shortcut method xi 60  61  62  63  64  65  66  67  68 fi   2    1  12  29  25  12  10    4    5

Summary:

The mean and standard deviation for the given data are 64 and 1.69