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z-scores or standard scores

The z-score defined 

A z-score indicates how far a raw score lies from the arithmetic mean of a normally distributed set of scores. It expresses this distance in standard‑deviation units. Thus, a z-score of 1.0 means the score is one standard deviation above the mean, while a z-score of –1.0 means it is one standard deviation below the mean.

z-scores are typically plotted along the x‑axis of a normal distribution—the familiar bell‑shaped curve.

CALCULATE z-scores by using the calculator at the bottom of this page.

Here's an image of the normal curve with z-scores below the curve.



In APA style, z-scores are reported in lowercase italics. The uppercase Z refers to a different statistic.

A z-score is calculated by subtracting the mean (M) from the raw score (X) and dividing the result by the standard deviation (SD):

z = (X - M) / SD

Example:

If a test score is 60, the mean is 50, and the standard deviation is 10, then:

     [ 60 - 50 = 10 ]

     [ 10 / 10 = 1.0 ]

The z-score is 1.0, meaning the score is one standard deviation above the mean.

Most z-scores fall between –3.0 and +3.0, although scores beyond this range are possible.


Related posts

Mean

Standard Deviation

Normal curve or distribution

Statistics A to Z (Link to other statistics)

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Post Author

Geoffrey W. Sutton, Professor Emeritus of Psychology at Evangel University, holds a master’s degree in counseling and a PhD in psychology from the University of Missouri-Columbia. His postdoctoral work encompassed education and supervision in forensic and neuropsychology and psychopharmacology. As a licensed psychologist, he conducted clinical and neuropsychological evaluations and provided psychotherapy for patients in various settings, including schools, hospitals, and private offices. During his tenure as a professor, Dr. Sutton taught courses on psychotherapy, assessment, and research. He has authored over one hundred publications, including books, book chapters, and articles in peer-reviewed psychology journals. 

His website is https://suttong.com 

You can find Dr. Sutton's books on   AMAZON    and  GOOGLE. 

Many publications are free to download at ResearchGate   and Academia  

Find chapters and essays on Substack. [ @GeoffreyWSutton ]


z-score calculator

Enter raw scores separated by commas (e.g., 60, 50, 55, 65):



Results

Mean: -

Standard Deviation: -

z-scores

Raw score (X) z-score
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