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If $x i' s$ are the mid points of the class intervals of grouped data, $f i' s$ are the corresponding frequencies and $x$ is the mean, then from the following choices, the value of $∑ (f i x i − x ¯)$ is:

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0

− 1

1

2

answer is A.

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Detailed Solution

It is given that,

x i' s

are the mid points of the class intervals of grouped data.

f i' s

are the frequencies and

x

is the mean.
We have to find the value of

∑ (f i x i − x ¯)

.
The formula to find the mean is,

x ¯ = ∑ f i x i n

So,

n x ¯ = ∑ f i x i

. ……(1)
The given summation

∑ (f i x i − x ¯)

can be written as

∑ f i x i − ∑ x ¯

∑ (f i x i − x ¯) = (f 1 x 1 − x ¯) + (f 2 x 2 − x ¯) + ... + (f n x n − x ¯)

= (f 1 x 1 + f 2 x 2 + ... + f n x n) − (x ¯ + x ¯ + ... + n times)

= ∑ f i x i − n x ¯

∑ (f i x i − x ¯) = ∑ f i x i − n x ¯

Substitute the value of equation (1) within above equation,

∑ (f i x i − x ¯) = n x ¯ − n x ¯

= 0

The value of

∑ (f i x i − x ¯)

0

.
Hence option 1) is correct.

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