Z-score Calculator
Compute z-scores, convert between z-scores and probabilities, and find probabilities between z-scores.
Z-score from raw score
The z-score is (x - μ) / σ, where x is the raw score, μ is the mean, and σ is the standard deviation.
Z-score:
Interpretation:
Equation used:
Z-score and Probability Converter
Convert between z-scores and cumulative probabilities from the standard normal distribution.
Probability for z-score:
Z-score for probability:
Probability between Two Z-scores
Calculate the area under the normal curve between two z-scores.
Probability between:
Area:
What a z-score represents
A z-score expresses how many standard deviations a value sits above or below the mean — a z-score of 0 is exactly at the mean, +1 is one standard deviation above, and −2 is two standard deviations below. Because the standard normal distribution has a well-known, consistent shape, any z-score can be converted into a cumulative probability (the area under the curve up to that point), which is what the second and third tools on this page compute.
Frequently asked questions
Why does this only work well for normally distributed data? Z-scores and the normal distribution's probability relationships assume the data follows a bell-shaped (Gaussian) distribution — for skewed or otherwise non-normal data, the probability interpretation of a z-score is not reliable.
What does a z-score of 1.96 have to do with 95% confidence? About 95% of a normal distribution's area falls between z = −1.96 and z = +1.96 — this specific value is why 1.96 shows up so often in statistics as the standard z-score for a 95% confidence interval.
How is "probability between two z-scores" calculated? It subtracts the cumulative probability at the lower z-score from the cumulative probability at the higher one, isolating just the area of the curve that falls between those two points.