Q. 14 - Elementary Statistics

Question

Regarding$z$-scores:
a. How is a $z$-score obtained?
b. What is the interpretation of a $z$-score?
c. An observation has a $z$-score of $2.9$. Roughly speaking, what is the relative standing of the observation?

Step-by-Step Solution

Answer

a.$z$-score is  obtained by $z=\frac{\left(x-\mu \right)}{\sigma }$

b. The interpretation of a$z$-score is that when it is $0$, it equals to mean, positive then above mean average and below mean average when negative.

c. The relative standing of the observation is  high.

Part (a) Step1: Given information

We need to find how is a$z$-score obtained .

Part (a) Step2: Explanation

A$z$-score is calculated using the formula $z=\frac{\left(x-\mu \right)}{\sigma }$, where$x$ is the raw score, population mean is $\mu$, and population standard deviation $\sigma$ in this formula. 

Part (b) Step1: Given information

We need to find the interpretation of a $z$-score 

Part (b) Step2: Explanation

The raw score is higher than the mean average if the $z$-score is positive. A $z$-score of $+1$, for example, means that the value is one standard deviation above the mean.

The raw score is below the mean average when the $z$-score is negative. A $z$-score of $-2$, for example, indicates that the data is two standard deviations below the mean.

A z-score of $0$ indicates that an element is equal to mean.

Part (c) Step1: Given information

We are given that  observation has a $z$-score of $2.9$ 

We need to find the relative standing of the observation .

Part (c) Step2: Explanation

Because normal distributions have around $95\%$ of values inside two standard deviations, the observation is relatively high.