Question

Give the z-score for a measurement from a normal distribution for the following:

a. 1 standard deviation above the mean

b. 1 standard deviation below the mean

c. Equal to the mean

d. 2.5 standard deviations below the mean

e. 3 standard deviations above the mean

Short answer

a. z = 1

b. z = -1

c. z = 0

d. z = -2.5

e. z = 3

Step-by-step solution

Given information

The variable is normally distributed.

Defining the random variable

Assume that xis a random variable that follows a normal distribution with a mean $\mu$ and standard deviation$\sigma$

Calculation of the z-score when 1 standard deviation above the mean

a.

z-score is defined as -

$z=\frac{x-\mu }{\sigma }$

For 1 standard deviation above the mean, then the z-score isrole="math" localid="1660273418117" $x=\mu +\sigma$

So,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{\mu +\sigma -\mu }{\sigma } \\ =\frac{\sigma }{\sigma } \\ =1 \end{gathered}$$

So, z = 1

Thus, the z-score for 1 standard deviation above the mean is 1.

Calculation of the z-score when 1 standard deviation below the mean

b.

For 1 standard deviation below the mean, then the z-score is$x=\mu +\sigma$

So,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{\mu -\sigma -\mu }{\sigma } \\ =\frac{-\sigma }{\sigma } \\ =-1 \end{gathered}$$

So, z -1

Thus, the z-score for 1 standard deviation below the mean is -1

Calculation of the z-score when 1 standard deviation equals to the mean

c.

For equal to the mean, then the z-score is $x=\mu$

So,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{\mu -\mu }{\sigma } \\ =\frac{0}{\sigma } \\ =0 \end{gathered}$$

So, z = 0

Thus, the z-score for equal to the mean is 0.

Calculation of the z-score when 2.5 standard deviations below the mean

d.

For 2.5 standard deviations below the mean, then the z-score is $x=\mu -2.5\sigma$,

So,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{\mu -2.5\sigma -\mu }{\sigma } \\ =\frac{-2.5\sigma }{\sigma } \\ =-2.5 \end{gathered}$$

So, z = 2.5

Thus, the z-score for 2.5 standard deviations below the mean is -2.5.

Calculation of the z-score when 3 standard deviations above the mean

e.

For 3 standard deviations above the mean, then the z-score is$x=\mu +3\sigma$

So,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{\mu +3\sigma -\mu }{\sigma } \\ =\frac{3\sigma }{\sigma } \\ =3 \end{gathered}$$

So, z = 3

Thus, the z-score for 3 standard deviations above the mean is 3.