Question

If x_i’s are the midpoints of the class intervals of grouped data, f_i’s are the corresponding frequencies and `barx` is the mean, then `sum(f_ix_i - barx)` is equal to ______.

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Answer 1

If x_i’s are the midpoints of the class intervals of grouped data, f_i’s are the corresponding frequencies and `barx` is the mean, then `sum(f_ix_i - barx)` is equal to 0.

Explanation:

Mean (x) = `"Sum of all the observations"/"Number of observations"`

x = `(f_1x_1 + f_2x_2 + ...... + f_nx_n)/(f_1 + f_2 + ...... + f_n)`

x = `(sumf_ix_i)/(sumf_i), sum f_i` = n

x = `(sumf_ix_i)/n`

nx = `sumf_ix_i`   ...(1)

`sum(f_ix_i - x) = (f_1x_1 - x) + (f_2x_2 - x) + .... + (f_nx_n - x)`

`sum(f_ix_i - x) = (f_1x_1 + f_2x_2 + .... + f_nx_n) - (x + x +  .... n  "times")`

`sum(f_ix_i - x) = sumf_ix_i - nx`

`sum(f_ix_i - x) = nx - nx`   ...(From equation 1)

`sum(f_ix_i - x)` = 0