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UPSC NDA I 2021 Mathematics Solved Paper

Question : 61 of 120

Marks: +1, -0

\text{If } M \text{ is the mean of } n \text{ observations } x_{1} - k, x_{2} - k, x_{3} - k, \ldots, x_{n} - k \text{, where } k \text{ is any real number, then what is the mean of } x_{1}, x_{2}, x_{3}, \ldots, x_{n}?

Solution:

Concept:

The mean of a data set changes by the same constant added or subtracted from each observation.

Explanation:

Let the mean of the original

n

observations

x_1, x_2, \dots, x_n

\overline{x}

The given mean

M

is of the observations

x_1 - k, x_2 - k, \dots, x_n - k

By definition,

M = \frac{(x_1 - k)+(x_2 - k)+\dots+(x_n - k)}{n}

Simplify the numerator:

(x_1+x_2+\dots+x_n) - nk

M = \frac{x_1+x_2+\dots+x_n}{n} - \frac{nk}{n} = \overline{x} - k

Hence

\overline{x} = M + k

Therefore, the mean of

x_1, x_2, \dots, x_n

M + k

Answer:

M + k

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