Question

If M is the mean of n observations x₁ - k, x₂ - k, x₃ - k, _ _ _, x_n - k, where k is any real number, then what is the mean of x₁, x₂, x₃, _ _ _, x_n?
1. M
2. M + k
3. M - k
4. kM

Answer 1

Correct Answer - Option 2 : M + k

Concept:

For the observations \(\rm x_1 ,x_2 ,x_3 ,..........,x_n\)

Mean (\(\rm\overline x\)) = \(\rm {x_1 +x_2 + x_3 +..........+x_n\over n}\)

Calculation:

Given mean of the n observation is M

M = \(\rm x_1 - k +x_2 - k+ x_3 - k +..........+x_n-k\over n\)

M = \(\rm (x_1 +x_2 + x_3 +..........+x_n)-nk\over n\)

M = \(\rm {x_1 +x_2 + x_3 +..........+x_n\over n}-{nk\over n}\)

M = \(\rm {x_1 +x_2 + x_3 +..........+x_n\over n}\) - k

M + k = \(\rm {x_1 +x_2 + x_3 +..........+x_n\over n}\)

∴ The required mean of x1, x2, x3, _ _ _, xn = M + k