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 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
Contents
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
To copy z-scores from a spreadsheet to SPSS keeping the values, follow these directions:
1) Copy the z-scores from the spreadsheet
2) Go to "paste special" and go down to "values only."
3) The z-score values will appear in SPSS.
contributed by Paula Connolly
Video Tutorial: How to Calculate and Interpret Z-Scores in SPSS
contributed by Sandra Peña
Z scores In statistics, it measures how many standard deviations a data point is from the mean, transforming data into a standard normal distribution to compare values from different datasets, identify outliers, and determine probabilities using a z-table. A positive z-score means the value is above the mean, while a negative one means it's below.
Formula
z = (X – μ) / σ X: The raw data point. μ (mu): The population mean. σ (sigma): The population standard deviation.
Key Uses
Standardization: Converts any dataset to a standard scale, allowing for comparisons (e.g., comparing a test score in one class to another). Probability: Helps find the percentage of data falling below or above a certain point using a z-table. Outlier Detection: Values with large z-scores (e.g., beyond ±2 or ±3) are often considered outliers.
Interpretation Example
If a student scores 1150 on the SAT (mean=500, std dev=100), their z-score is (1150-500)/100 = 1.5. This means their score is 1.5 standard deviations above the average. Looking up z=1.5 on a z-table shows about 93.32% of scores are below it, so only 6.68% scored higher.
Key Uses in Education
Comparing Subjects Outliers Identification Percentiles Composite Scores Program Effectiveness Evaluation
‘ ‘contributed by Jerry Velez’ ’
