Question

(1 point BONUS) Show that the sum of the z-scores of a data set has to be zero. That is, show that

Answer 1

We want to show that the sum of the z- scores of a data set has to be zero.

We have,

$\sum_{i=1}^{n}z_{i}=\sum_{i=1}^{n}\frac{x_{i}-\bar{x}}{s}$

ie $\sum_{i=1}^{n}z_{i}=\frac{1}{s}\sum_{i=1}^{n}(x_{i}-\bar{x})$

Since,

$\sum_{i=1}^{n}x_{i}=n\bar{x}$

We have

,$\sum_{i=1}^{n}z_{i}=\frac{1}{s}(n\bar{x}-n\bar{x})$

$\sum_{i=1}^{n}z_{i}=\frac{1}{s}(0)$

$\sum_{i=1}^{n}z_{i}=0$

Hence the sum of the z scores of a data set is zero