How this z-score calculator works
A z-score (also called a standard score) rescales a raw value into the number of standard deviations it lies from the mean. This makes values from different distributions directly comparable. The formula is z = (x − μ) / σ, where x is your value, μ is the population mean and σ is the standard deviation.
Once the z-score is known, the calculator converts it to a percentile using the standard normal cumulative distribution function P(Z < z). We evaluate it with an accurate Abramowitz–Stegun approximation of the error function (erf), so the percentile and p-values match standard z-tables to several decimal places. The standard deviation must be greater than zero, otherwise the z-score is undefined.
Reference values: z = 0 → 50%, z = 1 → 84.1%, z = 1.96 → 97.5%, z = 2 → 97.7%. These assume the underlying data is approximately normally distributed.
Frequently asked questions
- What is a z-score?
- A z-score, or standard score, tells you how many standard deviations a value lies from the mean of its distribution. A z-score of 0 is exactly at the mean, a positive z-score is above the mean, and a negative z-score is below it.
- How do I calculate a z-score?
- Subtract the mean from your value and divide by the standard deviation: z = (x − μ) / σ. For example, with x = 85, μ = 70 and σ = 10, z = (85 − 70) / 10 = 1.5. The standard deviation must be greater than zero.
- What does a negative z-score mean?
- A negative z-score means the value is below the mean. For instance, z = −1 sits one standard deviation below the mean, which corresponds to roughly the 15.9th percentile of a normal distribution.
- How do I convert a z-score to a percentile?
- The percentile is the cumulative probability P(Z < z) of the standard normal distribution, expressed as a percentage. For example z = 0 gives the 50th percentile, z = 1 gives about the 84.1st percentile, and z = 1.96 gives the 97.5th percentile.
- What is a good z-score?
- There is no universally good z-score — it depends on context. In statistical testing, |z| above 1.96 is often treated as significant at the 5% level (two-tailed). When higher is better, a larger positive z-score means a value well above average.
- What is the 68-95-99.7 rule?
- Also called the empirical rule, it states that for a normal distribution about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3 standard deviations.