Z-Score

Find the Z-score for any value in a dataset. Convert raw scores to standard scores with this free Z-Score Calculator.

Data Set

Mean

Standard Deviation

Result

What is a Z-Score?

A Z-Score (or standard score) describes the position of a raw score in terms of its distance from the mean, measured in standard deviations. It allows you to normalize data and compare values from different distributions.

  • Z = 0: The score is exactly at the mean.
  • Z > 0: The score is above the mean.
  • Z < 0: The score is below the mean.
  • Z = 1: The score is 1 standard deviation above the mean.

Z-Score Formula

The statistical formula for calculating a Z-score is:

Z=x−μσZ = \frac{x - \mu}{\sigma}

Where:

  • xx = The raw value being evaluated
  • μ\mu (mu) = The population mean
  • σ\sigma (sigma) = The population standard deviation

Why use Z-Scores?

Z-scores are incredibly powerful for standardization. Imagine trying to compare a student's SAT score with their ACT score. Since the scales are completely different (SAT out of 1600, ACT out of 36), you can't compare the raw numbers.

By converting both to Z-scores, you can see which score is statistically "better" relative to the population of test-takers.

Example Problem

Scenario: A student scores 85 on a history exam. The class mean was 75 and the standard deviation was 5. How well did they do?

  1. Identify values:

    • x=85x = 85
    • μ=75\mu = 75
    • σ=5\sigma = 5

  2. Apply Formula: Z=85−755Z = \frac{85 - 75}{5} Z=105Z = \frac{10}{5} Z=2.0Z = 2.0

Result: The student's score is 2.0 standard deviations above the mean. This places them in approximately the 97.7th percentile, meaning they scored better than roughly 97.7% of the class.

Interpreting Z-Scores in a Normal Distribution

In a standard normal distribution (Empirical Rule / 68-95-99.7 Rule):

  • 68% of values fall between Z = -1 and Z = +1.
  • 95% of values fall between Z = -2 and Z = +2.
  • 99.7% of values fall between Z = -3 and Z = +3.

A Z-score less than -3 or greater than +3 is considered an outlier or extremely rare event.

Frequently Asked Questions

Q: Can a Z-score be negative? A: Yes. A negative Z-score simply means the data point is below the average. For example, a Z-score of -1.5 means the value is 1.5 standard deviations below the mean.

Q: What is a "good" Z-score? A: It depends on context. In test scores, a high positive Z-score is good. In golf or race times, a low (often negative) Z-score is better. generally, anything beyond +2 or -2 is statistically significant.