Z-Score Calculator

A z-score tells you how many standard deviations a data point sits above or below the mean, using the formula z = (x - mean) / stdDev. A z of 0 equals the mean exactly; z = 1.0 is one standard deviation above; z = -2.0 is two standard deviations below. Because z-scores are dimensionless, they let you compare values from distributions with different scales or units.

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Converting raw values to z-scores maps any normally distributed variable onto this common scale, which is why z-scores pair directly with standard normal probability tables and p-values.

The one-tailed p-value is the probability of getting a result as extreme as yours in one direction (the upper tail when z >= 0, the lower tail when z < 0). The two-tailed p-value covers both directions and equals 2 x (1 - CDF(|z|)), so it is always larger. Use two-tailed when you care about deviations in either direction, one-tailed only when the direction was specified before collecting data.

At z = 1.96 the two-tailed p-value is 0.05, making it the boundary for the conventional 95% significance threshold. In practical terms, about 95% of values in any normal distribution fall within -1.96 to +1.96 standard deviations of the mean, so a result beyond that range is unlikely to be due to chance alone.

The calculator uses the Abramowitz and Stegun 7.1.26 approximation of the error function, with a maximum absolute error of roughly 1.5e-7. All output values are rounded to 6 decimal places. That level of precision is more than sufficient for statistical coursework, quality control, and most research applications.