Question

Consider the following three measurements: 0, 4, and 12. Find the z-score for each measurement if they are from a population with the following mean and standard deviation equal to

a.µ = 2 and σ = 1

b.µ = 4 and σ = 2

c.µ = 8 and σ = 2

d.µ = 8 and σ = 8

Short answer

a. 0 = -2, 4 = 2, 12 = 10

b. 0 = -2, 4 = 0, 12 = 4

c. 0 = -4, 4 = -2, 12 = 2

d. 0 = -1, 4 = -0.5, 12 = 0.5

Step-by-step solution

Finding the z-score for 0, 4, 12 when mean is 2 and σ = 1

$\begin{array}{c}\text{z-scorefor0=}\frac{\text{Value}-\mu }{\text{σ}}\\ =\frac{0-2}{1}\\ =-2\end{array}$

$\begin{array}{c}\text{z-scorefor4=}\frac{\text{Value}-\text{μ}}{\text{σ}}\\ \text{=}\frac{\text{4}-\text{2}}{\text{1}}\\ \text{=2}\end{array}$

$\begin{array}{c}\text{z-scorefor12=}\frac{\text{Value}-\mu }{\text{σ}}\\ \text{=}\frac{\text{12}-\text{2}}{\text{1}}\\ \text{=10}\end{array}$

When the mean is 2 and the standard deviation is 1, z- scores for 0, 4, 12 are -2, 2, 10 respectively.

Calculating the z-score for 0, 4, 12 with mean is 4 and σ 2

$\begin{array}{c}\text{z-scorefor0=}\frac{\text{Value}-\mu }{\text{σ}}\\ \text{=}\frac{\text{0}-\text{4}}{\text{2}}\\ \text{=}-\text{2}\end{array}$

$\begin{array}{c}\text{z-scorefor4=}\frac{\text{Value}-\mu }{\text{σ}}\\ \text{=}\frac{\text{4}-\text{4}}{\text{2}}\\ \text{=0}\end{array}$

$\begin{array}{c}\text{z-scorefor12=}\frac{\text{Value}-\mu }{\text{σ}}\\ \text{=}\frac{\text{12}-\text{4}}{\text{2}}\\ \text{=4}\end{array}$

When the mean is 4 and the standard deviation is 2, the z-scores for 0, 4, and 12 are -2, 0, and 4 respectively.

Computing the z-score for 0, 4, 12 where the mean is 8 and the standard deviation is 2

$\begin{array}{c}\text{z-scorefor0=}\frac{\text{Value}-\text{μ}}{\text{σ}}\\ \text{=}\frac{\text{0}-\text{8}}{\text{2}}\\ \text{=}-\text{4}\end{array}$

$\begin{array}{c}\text{z-scorefor4=}\frac{\text{Value}-\text{μ}}{\text{σ}}\\ \text{=}\frac{\text{4}-\text{8}}{\text{2}}\\ \text{=}-\text{2}\end{array}$

$\begin{array}{c}\text{z-scorefor12=}\frac{\text{Value}-\text{μ}}{\text{σ}}\\ \text{=}\frac{\text{12}-\text{8}}{\text{2}}\\ \text{=2}\end{array}$

Here the z-scores are 0 = -4, 4 = -2, 12 = 2.

Enumerating the z-score for 0, 4, 12 where the mean is 8 and the standard deviation is 8

$\begin{array}{c}\text{z-scorefor0=}\frac{\text{Value}-\text{μ}}{\text{σ}}\\ \text{=}\frac{\text{0}-\text{8}}{\text{8}}\\ \text{=}-\text{1}\end{array}$

$\begin{array}{c}\text{z-scorefor4=}\frac{\text{Value}-\text{μ}}{\text{σ}}\\ \text{=}\frac{\text{4}-\text{8}}{\text{8}}\\ \text{=}-\text{0.5}\end{array}$

$\begin{array}{c}\text{z-scorefor12=}\frac{\text{Value}-\text{μ}}{\text{σ}}\\ \text{=}\frac{\text{12}-\text{8}}{\text{8}}\\ \text{=0.5}\end{array}$

Therefore, the z-scores are -1, -0.5 and 0.5.