Question

5. x 9293 97 98 102 104 |109132

Answer 1

The given data is:

x i	92	93	97	98	102	104	109
f i	3	2	3	2	6	3	3

Now, make a table by using discrete frequency.

x i	f i	f i x i	x i − x ¯	( x i − x ¯ ) 2	f i ( x i − x ¯ ) 2
92	3	276	−8	64	192
93	2	186	−7	49	98
97	3	291	−3	9	27
98	2	196	−2	4	8
102	6	612	2	4	24
104	3	312	4	16	48
109	3	327	9	81	243
	∑ i=1 n f i =22	∑ i=1 n x i f i =2200			∑ i=1 n f i ( x i − x ¯ ) 2 =640

The formula to calculate the mean when discrete frequency is given is,

x ¯ = ∑ i=1 n x i f i ∑ i=1 n f i x ¯ = 1 N ∑ i=1 n x i f i (1)

Where, N is the sum of discrete frequency.

Substitute 22 for N and 2200 for ∑ i=1 n x i f i in equation (1).

x ¯ = 2200 22 =100

Therefore, the mean of the given data is 100.

The formula to calculate the variance is,

σ 2 = 1 N ∑ i=1 n f i ( x i − x ¯ ) 2 (2)

Substitute 22 for N and 640 for ∑ i=1 n f i ( x i − x ¯ ) 2 in equation (2).

σ 2 = 640 22 =29.09

Thus, the variance of the given data is 29.09.