Z-Score Calculator
Calculate the z-score (standard score) for any raw value, or flip it around: enter a z-score or a probability and work backward. All four modes return left-tail, right-tail, and two-tailed probabilities from the standard normal distribution.
Formula
Worked example
x = 85, mu = 70, sigma = 10: z = (85 - 70) / 10 = 1.5. Left-tail P = 93.32%, two-tailed P = 13.36%. Reverse: z = 1.5, mu = 70, sigma = 10 gives x = 85.
How the Z-Score Calculator Works
The calculator covers four modes. The default mode takes a raw value (x), a mean, and a standard deviation and returns the z-score using the formula z = (x - mu) / sigma. The reverse mode takes a z-score plus a mean and standard deviation and returns the raw value using x = mu + z * sigma. The "probability from z" mode accepts only a z-score and returns all four probability regions of the standard normal curve: left-tail, right-tail, two-tailed, and the central (middle) area. The "between two z-scores" mode returns the probability that a random observation falls between any two z-values.
Understanding the Probability Outputs
Every z-score maps to four distinct probability regions under the standard normal curve. The left-tail probability, P(X less than or equal to x), is the cumulative probability, equivalent to the percentile. The right-tail probability, P(X greater than x), is 1 minus the left tail. The two-tailed probability, P(|X| greater than |z|), is the probability of a value at least this far from the mean in either direction, and equals twice the smaller tail. The middle area, P(-|z| less than X less than |z|), is 1 minus the two-tailed probability and represents the proportion of the distribution within that many standard deviations of the mean. For z = 1.96, the two-tailed probability is about 5%, which is why z = 1.96 is the standard 95% confidence interval critical value.
When to Use Population vs. Sample Standard Deviation
The z-score formula uses the population standard deviation (sigma) when you know the true spread of the entire population, such as IQ scores or standardized test norms that are defined over millions of observations. Use the sample standard deviation (s) when you have measured a subset and are estimating the population parameters. The formula is identical either way, but the interpretation differs: with sample statistics, the resulting percentile carries additional uncertainty, especially for small samples where a t-distribution would be more appropriate. The calculator accepts whichever SD you provide.
Practical Uses of Z-Scores
Z-scores appear across many fields. In clinical medicine, growth charts and bone density (DEXA) reports express results as z-scores relative to an age-matched reference population: a z-score below -2 typically signals clinical significance. In finance and quality control (Six Sigma), a process running at six sigma produces fewer than 3.4 defects per million opportunities, corresponding to a z-score of about 4.5 under the industry convention. In education, standardized tests like the SAT and ACT convert raw scores to scaled scores by z-scoring relative to the reference population, then applying a linear shift. In data science and machine learning, z-score normalization (also called standardization) rescales features to have mean 0 and standard deviation 1 before fitting distance-based models.
Limitations to Keep in Mind
Z-scores and the percentile conversions here assume the underlying data follow a normal (Gaussian) distribution. When distributions are skewed, heavy-tailed, or multimodal, the percentile derived from the standard normal CDF will be inaccurate. The calculator uses population-level formulas; if you only have sample statistics, the resulting percentile carries estimation uncertainty. For small samples, a t-distribution is statistically more appropriate than the standard normal for inference. Finally, a z-score standardizes relative to a chosen reference: selecting the wrong mean or standard deviation produces a misleading score even if the arithmetic is correct.
Common z-score critical values
| Confidence level | Two-tailed z | One-tailed z | Left-tail % | Right-tail % |
|---|---|---|---|---|
| 80% | 1.2816 | 0.8416 | 90.00% | 10.00% |
| 90% | 1.6449 | 1.2816 | 95.00% | 5.00% |
| 95% | 1.9600 | 1.6449 | 97.50% | 2.50% |
| 99% | 2.5758 | 2.3263 | 99.50% | 0.50% |
| 99.7% | 3.0000 | 2.7478 | 99.87% | 0.13% |
Standard reference z-scores used in hypothesis testing and confidence intervals.
Frequently asked questions
What does a z-score of 1.96 mean?
A z-score of 1.96 means the value is 1.96 standard deviations above the mean. Critically, z = 1.96 is the standard two-tailed critical value for a 95% confidence interval: exactly 2.5% of the standard normal distribution lies above it and 2.5% lies below -1.96, leaving 95% in the middle. It is one of the most commonly cited z-scores in statistics.
What is the difference between a z-score and a percentile?
A z-score is a raw standardized distance from the mean in units of standard deviations. A percentile converts that distance into a proportion: it states what percentage of the reference population scores at or below that value. The two are directly linked via the cumulative distribution function (CDF) of the normal distribution. For example, z = 0 corresponds to the 50th percentile, z = 1 to roughly the 84th percentile, and z = 2 to roughly the 97.7th percentile.
How do I find the raw score from a z-score?
Use the reverse formula: x = mean + z * standard deviation. For example, a z-score of 1.5 with a mean of 70 and SD of 10 gives x = 70 + 1.5 * 10 = 85. This calculator's "raw value from z-score" mode does exactly this and also returns all associated probabilities.
What is a good or bad z-score?
There is no universally good or bad z-score; context determines significance. In quality control (Six Sigma), values beyond plus or minus 3 are flagged as outliers because fewer than 0.3% of a normal distribution falls outside that range. In clinical medicine, such as bone density or pediatric growth, a z-score below -2 is typically considered clinically significant. In hypothesis testing, |z| greater than 1.96 corresponds to statistical significance at the 5% level (two-tailed).
What is the difference between a one-tailed and two-tailed z-test?
A one-tailed test checks whether a value is significantly above (right tail) or significantly below (left tail) the mean, using only one end of the distribution. A two-tailed test checks whether a value is significantly different in either direction, so it splits the significance level across both tails. For a 5% significance level, one-tailed uses z = 1.645, while two-tailed uses z = plus or minus 1.96.
Can z-scores be used with non-normal data?
You can compute a z-score for any numeric value regardless of the underlying distribution, but the percentile conversion assumes normality. If your data are skewed or otherwise non-normal, the calculated percentile will not accurately reflect the true proportion of the population below that value. In those cases, non-parametric methods or distribution-specific transformations provide more reliable percentile estimates.