5.2.1: Practice Calculating z-scores
- Page ID
- 18031
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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!
Your turn!
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.
- Answer
-
\(z_112\) = 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
- Answer
-
The IQ scores are 135 (for \(z\) = 2.19) and 99 (\(z\) = -0.06) .
Contributors and Attributions
Foster et al. (University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus)