Definition · Plain-language
Z-score
A z-score (standard score) measures how many standard deviations a data point lies above or below the mean of its distribution, using the formula z = (x − μ) / σ.
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The formula and what it standardises
A z-score converts a raw value into a count of standard deviations from the mean. The formula is z = (x − μ) / σ: subtract the population mean μ from the value x to find how far above or below the mean it is, then divide by the standard deviation σ to express that distance in standard-deviation units. The subtraction centres the data on zero; the division rescales it so that one unit equals one standard deviation. This is why z-scores are also called standardised scores — they strip away the original units and allow values from different distributions to be compared on a common scale.
The standard normal distribution
Standardising a normal distribution produces the standard normal distribution: a normal curve with a mean of 0 and a standard deviation of 1. Z-scores are the values along its horizontal axis. Because the shape is fixed, the area under the curve between any two z-scores is known, and that area gives the proportion of cases — or the probability — falling in that range. Under the empirical rule, roughly 68% of values lie within z = ±1, about 95% within z = ±2 (more precisely ±1.96), and about 99.7% within z = ±3. This fixed geometry is what makes the z-score so useful for probability.
Reading a z-table
A z-table (standard normal table) gives the cumulative area under the standard normal curve to the left of a given z-score — that is, the probability of a value being less than or equal to it. To use one, find the z-score to the first decimal place down the left-hand column, then the second decimal across the top row; the cell where they meet is the cumulative probability. For example, z = 1.96 corresponds to about 0.975, meaning roughly 97.5% of values fall below it. To find the area to the right, subtract the table value from 1, since the total area under the curve equals 1.
Interpreting and using z-scores
A z-score of +2 means a value lies two standard deviations above the mean; −1.5 means one and a half below. Because the sign carries direction and the magnitude carries distance, z-scores make it easy to judge how unusual a value is and to compare scores measured in different units — a test mark against a reaction time, say. They also flag potential outliers: values beyond roughly ±3 are rare in a normal distribution. When the population mean and standard deviation are unknown and estimated from a sample, the analogous standardised value is usually a t-score rather than a z-score.
Key facts
At a glance
- Definition: number of standard deviations a value lies from the mean
- Formula: z = (x − μ) / σ
- Also called: standard score or standardised score
- Standard normal: has mean 0 and standard deviation 1
- Sign: positive is above the mean, negative is below, zero equals the mean
- Use: compares values across distributions and finds probabilities via a z-table
Common misconceptions
What people often get wrong
Often heard: A z-score can only be calculated for data that follow a normal distribution.
Actually: The z-score formula z = (x − μ) / σ applies to any distribution. However, converting a z-score into a probability using the standard normal curve assumes approximate normality.
Often heard: A negative z-score means the data point or the calculation is wrong.
Actually: A negative z-score is perfectly valid — it simply means the value lies below the mean. About half of all values in a symmetric distribution have negative z-scores.
Often heard: A higher z-score always means a better or larger result.
Actually: A z-score only measures distance from the mean in standard deviations. Whether high is good depends on context — a high z-score for an error rate or a reaction time is undesirable.
Going deeper
