Z-Score Calculator

A z-score (also called a standard score) tells you how many standard deviations a value is above or below the mean of its distribution. A z-score of 0 means the value equals the mean, a positive z-score means the value is above the mean, and a negative z-score means it is below the mean. Because z-scores are standardized, they let you compare values from different data sets or units on the same scale.

The z-score formula is z = (x − μ) / σ, where x is the raw score, μ (mu) is the population mean, and σ (sigma) is the population standard deviation. Subtract the mean from your raw score, then divide the result by the standard deviation. For example, a score of 85 with a mean of 75 and a standard deviation of 10 gives z = (85 − 75) / 10 = 1.0, meaning the score is exactly one standard deviation above the mean.

A z-score tells you how unusual or typical a value is relative to the rest of the distribution, expressed purely in standard deviations from the mean. A z-score near 0 indicates a typical, average value, while a z-score beyond ±2 or ±3 indicates an increasingly rare or extreme value. This makes z-scores useful for spotting outliers, standardizing test scores, and comparing measurements across different scales.

A z-score is converted to a percentile using the cumulative distribution function (CDF) of the standard normal distribution, often written Φ(z). This calculator computes Φ(z) with a numerical approximation of the error function and multiplies by 100, so a z-score of 1.0 converts to roughly the 84th percentile — meaning the value scores higher than about 84% of the distribution. Negative z-scores convert to percentiles below 50, and positive z-scores convert to percentiles above 50.

For a two-tailed 90% confidence interval, the critical z-value is 1.645. For 95% confidence it is 1.960, for 99% confidence it is 2.576, and for 99.5% confidence it is 2.807. These values come from the points on the standard normal curve that leave the specified percentage of area in the middle, with the remaining area split equally between the two tails.

A positive z-score means the raw value is above the mean of the distribution, while a negative z-score means the raw value is below the mean. The magnitude of the z-score (ignoring the sign) shows how far the value is from the mean in standard deviations — a z-score of −2 is just as far from the mean as a z-score of +2, but on the opposite side.

There is no universally 'good' z-score — it depends on context. In many practical settings, a z-score between −2 and +2 is considered normal or typical (covering about 95% of a normal distribution), while values beyond ±2 or ±3 are considered unusual or outliers. Whether a high or low z-score is 'good' depends on what is being measured, for example a high z-score is desirable for a test score but may be undesirable for a defect rate.