For some data x_1,x_2,\ldots,x_nwith respective frequencies f_1,f_2,\ldots,f_n, the value of

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Solution

Q.
For some data $x_{1}, x_{2}, \dots, x_{n} w i t h r e s p e c t i v e f r e q u e n c i e s f_{1}, f_{2}, \dots, f_{n}, t h e v a l u e o f \sum_{i = 1}^{n} f_{i} (x_{i} - x)$ is equal to:

1 $n x$

2 1

3 $\sum f_{i}$

4 0

Ans.
(4)

$\sum_{i = 1}^{n} f_{i} (x_{i} - x) = f 1 (x 1 - x) + f 2 (x 2 - x) + \dots + f n (x n - x) = (f_{1} x_{1} + f_{2} x_{2} + \dots + f_{n} x_{n}) - (f_{1} x + f 2 x + \dots + f n x) = \sum_{i = 1}^{n} f_{i} x_{i} - x i = 1 n f i = \sum_{i = 1}^{n} f_{i} x_{i} - \frac{\sum_{i = 1}^{n} f_{i} x_{i}}{\sum_{i = 1}^{n} f_{i}} \cdot \sum_{i = 1}^{n} f_{i} = 0 x = \frac{\sum_{i = 1}^{n} f_{i} x_{i}}{\sum_{i = 1}^{n} f_{i}}$

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