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Question
If `barx` represents the mean of n observations x1, x2, ..., xn, then value of `sum_(i = 1)^n (x_i - barx)` is ______.
Options
–1
0
1
n – 1
Solution
If `barx` represents the mean of n observations x1, x2, ..., xn, then value of `sum_(i = 1)^n (x_i - barx)` is 0.
Explanation:
The formula of the mean `(barx)` is:
`barx = (sum_(i = 1)^n x_i)/n`
`sum_(i = 1)^n x_i = nbarx` ...(I)
Here, n is total number of observation.
The value of `sum_(i = 1)^n (x_i - barx)` is calculated as follows:
`sum_(i = 1)^n(x_i - barx) = sum_(i = 1)^n x_i - sum_(i = 1)^n barx`
Now, from equation (I), we get
`sum_(i = 1)^n (x_i - barx) = nbarx - sum_(i = 1)^n barx`
= `nbarx - barx sum_(i = 1)^n 1`
= `nbarx - nbarx`
= 0
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