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Z Z-score Calculator

Calculate the standard score (z-score) for any raw value given a mean and standard deviation, or convert a z-score to a cumulative probability (p-value) using the normal distribution.

Enter the raw score, population mean, and standard deviation to calculate the z-score.

Z-score Formula

z = (x − μ) / σ

Where x is the raw score, μ is the population mean, and σ is the standard deviation. A positive z-score indicates the value is above the mean; negative means below.

How to Calculate a Z-score

  1. 1
    Identify the Values
    Collect the raw score (x), the mean (μ) of the distribution, and the standard deviation (σ).
  2. 2
    Subtract the Mean
    Compute the difference x − μ. A positive result means x is above average; negative means below.
  3. 3
    Divide by Std Deviation
    Divide the difference by σ to express the distance in standard deviation units.
  4. 4
    Interpret the Result
    A z-score of 0 is at the mean. Scores between −1 and +1 are within one standard deviation (68% of normal data).

Worked Example

A student scored 85 on an exam where the class mean is 72 and the standard deviation is 8.

z = (85 − 72) / 8
z = 13 / 8
z = 1.625
The student scored 1.625 standard deviations above the mean.

Frequently Asked Questions

A z-score (standard score) measures how many standard deviations a data point is from the mean of a distribution. A z-score of 0 means the value equals the mean; ±1 means one standard deviation away.

That depends on context. In quality control, scores beyond ±3 are considered outliers. In standardized testing, a z-score above +1 (top 84%) is generally considered good. For confidence intervals, ±1.96 corresponds to the 95% CI.

Use the cumulative distribution function (CDF) of the standard normal distribution. P(Z < z) gives the one-tailed p-value. Use the Z-score to Probability tab above for instant results.

The standard normal distribution is a normal (bell-curve) distribution with mean 0 and standard deviation 1. Any normal distribution can be converted to it by computing z-scores, allowing use of universal probability tables.

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