NCERT Class XI Mathematics - Statistics - Solutions

Question : 31

Total: 34

Given that

−

is the mean and σ2 is the variance of n observations x1,x2, ......., xn. Prove that the mean and variance of the observations ax1,ax2,ax3,......, axn are ax and a2σ2, respectively, (a ≠ 0).

Solution:

Here

−

x1+x2+x3+...+xn

Σx

Also ,

x12+x22+x32+...+xn2

Σx2

New mean =

ax1+ax2+ax3+...+axn

a(x1+x2+x3+...+xn)

= a

−

Also , σ2 =

n(x12+x22+...+xn2)−(x1+x2+...+xn)2

∴ New variance =

n(a2x12+a2x22+a2x32+...+a2xn2)−(ax1+ax2+ax3+...+axn)2

a2[

n(x12+x22+x32+...+xn2)−(x1+x2+x3+...+xn)2

]=a2σ2

.
Hence proved

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