Z-scores

A Brief Explanation of Z-Scores: A z-score is a standard score that is used by researchers to add focus and clarity to data. Z-scores indicate how many standard deviations a a raw score is from the mean. The mean is fixed at zero and standard deviations are fixed at 1. For example, suppose the mean test score for a sample is 80 with a standard deviation of 12 and you scored a 98 on that test. Your z-score is +1.5, indicating that you scored 1.5 standard deviations above the mean. If a z-score is close to zero the corresponding raw score is close to the mean for the test. If a z-score is -2 the corresponding raw score is 2 standard deviations below the mean.

contributed by Helen Knudsen

Impact of converting raw scores to z-scores

When each raw score is converted to a z score:

1. The distribution of standard scores is similar in shape to the distribution of raw scores 2. The mean of the distribution of z scores will always equal 0, regardless of the value of the mean in the raw score distribution. 3. Both the variance of the distribution and the standard deviation of z scores always equals 1.

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It is helpful to see what this looks like in a side-by-side distribution of raw and standard scores. Notice that the mean raw scores are 6.0; whereas the standard score is set at a mean of zero. And whereas the standard deviation from the raw scores was 3.18, when converted to standard scores, the standard deviation is 1.00 (Hinkle, Wiersma, & Jurs, 2003, p. 72).

In sum, calculating a z score for each raw score in a distribution will transform the original distribution of scores into one with identical shape but a mean of 0 and a standard deviation of 1 (Hinkle, Wiersma, & Jurs, 2003, p. 71).

References: Hinkle, D.E., Wiersma, W., & Jurs, S.G. (2003). Applied statistics for the behavioral sciences (5th edition). Boston, M.A.: Houghton Mifflin Company.

contributed by Emily Kilbourn

Calculating Z-scores using SPSS

I, among others were having a hard time calculating Z-scores using the SPSS program. Amy, Michelle and I brainstormed last week, but had no luck. The book is vague in terms of how to approach it. Thanks for the guidance, Frank.

When calculating Z-scores on SPSS, follow these directions:

1) Once you have the data entered in SPSS, click on "Analyze", "Descriptive Statistics", "Descriptives".

2) Move the variable over that you want to analyze.

3) Click on the small box that states, "Save standardized values as variables".

4) Click on "Options" if you would like to calculate mean, median, mode, etc. in addition to Z-scores.

5) Click "OK".

6) The Z-scores will appear in a separate column in the data editor.

contributed by Chris Longo