If x̄ represents the mean of n observations x₁ , x₂ , ..., x_n , then value of $\sum_{i=1}^{n}(x_{i}-\overline{x})$ is:
a. -1
b. 0
c. 1
d. n - 1

Solution:

It is given that

$\\\overline{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}\\\\It\: can\: be\: written\: as\\\\\sum_{i=1}^{n}x_{i}=n\overline{x}....(1)\\\\\sum_{i=1}^{n}(x_{i}-\overline{x}) =\sum_{i=1}^{n}x_{i}-\sum_{i=1}^{n}\overline{x}\\\\=n\overline{x} -\overline{x}\sum_{i=1}^{n}1(using \: equation (1))\\\\ =n\overline{x}-\overline{x}n(As\sum_{i=1}^{n}1=n)$

= 0

Therefore, the value is 0.

✦ Try This: If the mean of the observations : x + 4, x + 8, x + 12, x + 16, x + 20 is 8, the mean of the last three observations is

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 14

NCERT Exemplar Class 9 Maths Exercise 14.1 Problem 13

If x̄ represents the mean of n observations x₁ , x₂ , ..., x_n , then value of $\sum_{i=1}^{n}(x_{i}-\overline{x})$ is: a. -1, b. 0, c. 1, d. n - 1

Summary:

In statistics, the mean for a given set of observations is equal to the sum of all the values of a collection of data divided by the total number of values in the data. If x̄ represents the mean of n observations x₁ , x₂ , ..., x_n, then the value is 0

☛ Related Questions:

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