Question

8. Classes0-1010-20 20-3030-4040-50Frequencies1516

Answer 1

The given data is:

Classes	0-10	10-20	20-30	30-40	40-50
frequencies	5	8	15	16	6

The midpoint of the intervals is calculated by adding first and last terms and dividing it by 2. 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 25 for A and 10 for h in the above formula and make a table by using these values.

Classes	Frequencies, f i	Midpoint, x i	y i = x i −A h	( y i ) 2	f i y i	f i ( y i ) 2
0-10	5	5	−2	4	−10	20
10-20	8	15	−1	1	−8	8
20-30	15	25	0	0	0	0
30-40	16	35	1	1	16	16
40-50	6	45	2	4	12	24
	∑ i=1 n f i =50				∑ i=1 n y i f i =10	∑ i=1 n f i ( y i ) 2 =68

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 25 for A, 10 for h, 50 for N and 10 for ∑ i=1 n y i f i in equation (1).

x ¯ =25+ 10 50 ×10 =25+2 =27

Therefore, the mean of the given data is 27.

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 50 for N, 10 for h, 10 for ∑ i=1 n y i f i and 68 for ∑ i=1 n f i ( y i ) 2 in equation (2).

σ 2 = ( 10 ) 2 ( 50 ) 2 [ 50( 68 )− ( 10 ) 2 ] = 1 25 ( 3400−100 ) = 3300 25 =132

Thus, the variance of the given data is 132.