# 5.2.1: Practice Calculating z-scores


Example $$\PageIndex{1}$$

Assume the following scores represent a sample of statistics classes taken by five psychology professors: 2, 3, 5, 5, 6. If the standard deviation is 1.64, what is the z-score for each of these professor's?

Solution

The mean is 4.2 stats classes taken ($$\displaystyle \bar{X} = \dfrac{\sum X}{N} = \dfrac{21}{5} = 4.2$$).

$z_2=\dfrac{x-\overline{X}}{s} = \dfrac{2-4.2}{1.64} = \dfrac{-2.2}{1.64} = -1.34 \nonumber$

$z_3=\dfrac{x-\overline{X}}{s} = \dfrac{3-4.2}{1.64} = \dfrac{-1.2}{1.64} = -0.73 \nonumber$

$z_Both5=\dfrac{x-\overline{X}}{s} = \dfrac{5-4.2}{1.64} = \dfrac{0.80}{1.64} = 0.49 \nonumber$

$z_6=\dfrac{x-\overline{X}}{s} = \dfrac{6-4.2}{1.64} = \dfrac{1.8}{1.64} = 1.10 \nonumber$

PS You might want to practice calculating the standard deviation yourself to make sure that you haven't forgotten how!

Exercise $$\PageIndex{1}$$

Calculate $$z$$-scores for the three IQ scores provided, which were taken from a population with a mean of 100 and standard deviation of 16: 112, 109, 88.

$$z_112$$ = 0.75

$$z_109$$ = 0.56

$$z_88$$ = -0.75

This time, you'll get the z-score and will need to find the IQ scores.  Remember, you can do this with the z-score formula that you used above and to algebra to find $$x|), or you an use the other z-score formula. Exercise \(\PageIndex{2}$$

Use the $$z$$-scores provided to find two IQ scores taken from a population with a mean of 100 and standard deviation of 16:

$$z$$ = 2.19

$$z$$ = -0.06

The IQ scores are 135 (for $$z$$ = 2.19) and 99 ($$z$$ = -0.06) .