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Z-Score Calculator

Distance of a score from the mean in standard deviations.

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Z-Score

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Overview

The z-score, also known as a standard score, quantifies the distance of a data point from the population mean in units of standard deviation. This transformation allows researchers to compare scores from different distributions by placing them on a common scale with a mean of zero and a standard deviation of one.

Symbols

Variables

z = Z-Score, x = Raw Score, \mu = Mean, \sigma = Std. Deviation

Z-Score
Raw Score
Mean
Std. Deviation

Apply it well

When To Use

When to use: Apply z-scores when you need to standardize raw data to compare observations across different scales or to find the probability of a value occurring within a normal distribution. It is most effective when the population mean and standard deviation are known and the data is approximately normally distributed.

Why it matters: In psychological assessment, z-scores allow clinicians to compare a patient's performance across varied tests, such as IQ and memory, despite their original differing point scales. They are essential for identifying clinical outliers and determining whether a result is statistically significant within a population.

Avoid these traps

Common Mistakes

  • Flipping x and mu.
  • Negative z-scores are normal; they just mean below the mean.

One free problem

Practice Problem

A student takes a standardized intelligence test with a mean (mu) of 100 and a standard deviation (SD) of 15. If the student's raw score (x) is 130, what is their calculated z-score?

Raw Score130
Mean100
Std. Deviation15

Solve for:

Hint: Subtract the mean from the raw score and then divide by the standard deviation.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Statistics by David Freedman, Robert Pisani, Roger Purves
  2. Wikipedia: Standard score
  3. Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). SAGE Publications.
  4. Standard score. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Standard_score
  5. GCSE Psychology / Mathematics — Statistics