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Question
If xi’s are the midpoints of the class intervals of grouped data, fi’s are the corresponding frequencies and `barx` is the mean, then `sum(f_ix_i - barx)` is equal to ______.
Options
0
–1
1
2
Solution
If xi’s are the midpoints of the class intervals of grouped data, fi’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
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