# 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?

Oct 22, 2015

The variation will not be affected in any way.

#### Explanation:

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} ${x}_{i} \in {\mathbb{Q}}_{+}$

Let's calculate the mean and variation:

Mean ${\overline{x}}_{1} = \frac{1}{n} \cdot {\Sigma}_{i = 1}^{n} {x}_{i}$

Variation: ${\sigma}_{1}^{2} = \frac{1}{n} \cdot {\Sigma}_{i = 1}^{n} \left({x}_{i} - {\overline{x}}_{1}\right)$

Case 2 - with platform shoes

Let now the set of heights be H_2={x_1+3;x_2+3;...;x_n+3} ${x}_{i} \in {\mathbb{Q}}_{+}$

Let's calculate mean and variation:

Mean: ${\overline{x}}_{2} = \frac{1}{n} \cdot {\Sigma}_{i = 1}^{n} \left({x}_{i} + 3\right) = \frac{1}{n} \cdot \left({\Sigma}_{i = 1}^{n} {x}_{i} + 3 n\right) = \frac{1}{n} \cdot {\Sigma}_{i = 1}^{n} {x}_{i} + \frac{1}{n} \cdot 3 n = {\overline{x}}_{1} + 3$

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

Variation:

${\sigma}_{2}^{2} = \frac{1}{n} \cdot {\Sigma}_{i = 1}^{n} \left({x}_{i} + 3 - \left({\overline{x}}_{1} + 3\right)\right) =$

$= \frac{1}{n} \cdot {\Sigma}_{i = 1}^{n} \left({x}_{i} + 3 - {\overline{x}}_{1} - 3\right) = \frac{1}{n} \cdot {\Sigma}_{i = 1}^{n} \left({x}_{i} - {\overline{x}}_{1}\right) = {\sigma}_{1}^{2}$

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