If you are examining the height of a population, how would the variation be affected if everyone is wearing platform shoes that are exactly 3 inches tall?

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Answer 1 · 2015-10-22T16:27:54

The variation will not be affected in any way.

Let's calculate the variations in both cases:

Case 1 - without platform shoes

Let the set of heights be $H_1={x_1;x_2;...;x_n}$ $x_i in QQ_{+}$

Let's calculate the mean and variation:

Mean $bar(x)_1=1/n*Sigma_{i=1}^{n} x_i$

Variation: $sigma_1^2=1/n*Sigma_{i=1}^n (x_i-bar(x)_1)$

Case 2 - with platform shoes

Let now the set of heights be $H_2={x_1+3;x_2+3;...;x_n+3}$ $x_i in QQ_{+}$

Let's calculate mean and variation:

Mean: $bar(x)_2=1/n*Sigma_{i=1}^{n} (x_i+3)=1/n*(Sigma_{i=1}^{n} x_i+3n)=1/n*Sigma_{i=1}^{n} x_i+1/n*3n=bar(x)_1+3$

So we can see that the mean is increased by 3.

Variation:

$sigma_2^2=1/n*Sigma_{i=1}^n (x_i+3-(bar(x)_1+3))=$

$=1/n*Sigma_{i=1}^n(x_i+3-bar(x)_1-3)=1/n*Sigma_{i=1}^n(x_i-bar(x)_1)=sigma_1^2$

So we see, that the variation remains unchanged if the same number is added to (or substracted from) all data in a set.