Question

Find the mean and standard deviation using short-cut method.x60662 63 64 65 66 67 68f 21 12 29 252 10456.

Answer 1

The given data is:

x i	60	61	62	63	64	65	66	67	68
f i	2	1	12	29	25	12	10	4	5

The formula to calculate the new values of the given data is,

y i = x i −A h

Here, A is the assumed mean and h is the height between the classes.

Substitute 64 for A and 1 for h in the above formula and make a table by using these values.

x i	f i	y i = x i −A h	( y i ) 2	f i y i	f i ( y i ) 2
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
	∑ i=1 n f i =100			∑ i=1 n y i f i =0	∑ i=1 n f i ( y i ) 2 =286

The formula to calculate the mean is,

x ¯ =A+ ∑ i=1 n y i f i ∑ i=1 n f i ×h x ¯ =A+ ∑ i=1 n y i f i N ×h (1)

Where, N is the sum of frequency.

Substitute 64 for A, 1 for h, 100 for N and 0 for ∑ i=1 n y i f i in equation (1).

x ¯ =64+ 0 100 ×1 =64

Therefore, the mean of the given data is 64.

The formula to calculate the variance is,

σ 2 = h 2 N 2 [ N ∑ i=1 n f i ( y i ) 2 − ( ∑ i=1 n f i y i ) 2 ](2)

Substitute 100 for N, 1 for h, 0 for ∑ i=1 n y i f i and 286 for ∑ i=1 n f i ( y i ) 2 in equation (2).

σ 2 = 1 ( 100 ) 2 [ 100( 286 )− ( 0 ) 2 ] = 1 10000 ( 28600 ) =2.86

Thus, the variance of the given data is 2.86.

The formula to calculate the standard deviation is,

S.D.= σ 2 (3)

Substitute 2.86 for σ 2 in equation (3).

S.D.= 2.86 =1.69

Thus, the standard deviation of the given data is given by 1.69.