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5.2.1: Practice Calculating z-scores

  • Page ID
    18031
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    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!

    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)

    Dr. MO (Taft College)


    5.2.1: Practice Calculating z-scores is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Michelle Oja.

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