# Z-Score Calculator > Calculate the z-score (standard score) for a value. Enter the raw score, mean, and standard deviation to see how many SDs above or below the mean it sits. - **Canonical URL:** https://boringtoolskit.com/z-score-calculator/ - **Category:** Everyday Calculators - **Evidence level:** authored-methodology - **Reviewed:** July 26, 2026 - **Last verified:** 2026-10-12 - **Privacy:** Inputs and results stay in this browser unless the page clearly labels an external request. ## How it works z = (x - mu) / sigma, where x is the raw score, mu is the population mean, and sigma is the population standard deviation. The result z counts standard deviations between x and the mean: positive means above, negative means below. To convert back, x = mu + z * sigma. ## Worked example An exam score of 85 with a class mean of 70 and a standard deviation of 10: z = (85 - 70) / 10 = 1.5. The score sits 1.5 standard deviations above the class average, which corresponds to roughly the 93rd percentile under a normal curve, so about 93 percent of the class scored lower. ## Assumptions - Calculations use z = (x - mu) / sigma with the values you enter; nothing is estimated behind the scenes. - Percentile readings assume the underlying distribution is approximately normal. - Inputs are treated as exact population parameters, not sample estimates. - Results are for education and planning only; verify important numbers independently. ## Common questions ### What is a z-score in plain English? It is the number of standard deviations a value sits from the mean of its distribution. A z-score of 2 means two standard deviations above average, a z-score of -1.5 means one and a half below, and 0 means exactly average. ### What does a negative z-score mean? It means the value is below the mean. The negative sign is directional information, not an error. A z-score of -2 is exactly as far from the mean as a z-score of 2, just in the other direction. ### How do I convert a z-score back to a raw score? Multiply the z-score by the standard deviation and add the mean: x = mu + z * sigma. For example, with a mean of 70 and a standard deviation of 10, a z-score of 1.5 corresponds to a raw score of 70 + 1.5 * 10 = 85. ### What is a good z-score? There is no good or bad in the abstract; it depends on the question. If higher is better, positive z-scores beat negative ones. If you are quality checking a process, extreme z-scores in either direction are what you hunt for. The useful benchmark is rarity: |z| above 2 covers less than 5 percent of a normal distribution. ### What is the difference between a z-score and a percentile? The z-score measures distance from the mean in standard deviation units. The percentile measures what share of the distribution falls below the value. The two connect through the normal curve: a z-score of 1 corresponds to about the 84th percentile. Without a distributional assumption, a z-score alone tells you nothing about percentiles. ### When should I use a t-score instead of a z-score? Use a t-score when the population standard deviation is unknown and estimated from a small sample, roughly fewer than 30 observations. The t-distribution has heavier tails to reflect the extra uncertainty. As the sample grows, the t-distribution converges to the normal distribution and the distinction disappears. ### Can z-scores be used on non-normal data? Yes for the basic calculation: z = (x - mu) / sigma is just arithmetic and works on any data with a mean and standard deviation. What you lose is the percentile interpretation. For skewed data, standardized values still show relative distance from the mean, but do not read them against the normal-curve table. ### Why does my z-score look wrong for very small standard deviations? The formula divides by sigma, so as the standard deviation approaches zero the z-score blows up. If sigma is 0, all values equal the mean and a z-score is undefined. Check that you entered the standard deviation, not the variance; mixing up sigma and sigma squared is the classic input error and it changes every result. ## Related tools - /standard-deviation-calculator/ - /percentile-calculator/ - /confidence-interval-calculator/ - /statistics-calculator/ ## Sources - [NIST/SEMATECH e-Handbook of Statistical Methods, normal distribution](https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm) - [Calculation methodology and verification standard](/methodology/) --- *Local planning estimate. Verify current terms, rates, and policies before acting.*