Free tool
Z-Score Calculator
A z-score tells you how many standard deviations a value sits from the mean: z = (x − μ) / σ. Enter your value, mean and standard deviation below to get the z-score, its percentile, and the tail probabilities — no printed z-table needed. Free, and nothing is stored.
Find a z-score from a value
Enter the value, the mean, and the standard deviation.
Find probabilities from a z-score
Replaces looking a value up in a printed z-table.
How z-scores work
Standardising a value does two things at once. Subtracting the mean re-centres the distribution on zero, and dividing by the standard deviation rescales it so one unit equals one standard deviation. What comes out is a pure number with no units attached — which is precisely why z-scores let you compare things that otherwise cannot be compared.
That is the real power of the idea. A 82 on a biology test and a 640 on the SAT Math section live on completely different scales, but convert both to z-scores and you can say which performance was genuinely stronger relative to everyone else who sat them.
The percentile answers a different question from the z-score itself: it is the share of the distribution falling below your value. A z-score of 1.5 sits at roughly the 93rd percentile, meaning about 93% of values are lower. One caution — percentiles and probabilities assume the data is roughly normally distributed. The z-score formula works on any data, but if your distribution is badly skewed, treat those probability readings with suspicion.
Common z-scores and what they mean
| z | P(Z < z) | P(−z < Z < z) | Why it matters |
|---|---|---|---|
| 1.00 | 84.13% | 68.27% | The familiar 68% inside one σ |
| 1.28 | 90.00% | 79.99% | 90th percentile |
| 1.645 | 95.00% | 90.00% | One-tailed 5% critical value |
| 1.96 | 97.50% | 95.00% | The 95% confidence interval |
| 2.00 | 97.72% | 95.45% | The 95% inside two σ |
| 2.576 | 99.50% | 99.00% | The 99% confidence interval |
| 3.00 | 99.87% | 99.73% | The 99.7% inside three σ |
These are the standard normal values behind the 68-95-99.7 rule and the usual confidence-interval critical values.
Frequently asked questions
What is a z-score?+
A z-score tells you how many standard deviations a value sits from the mean. It is calculated as z = (x - mean) / standard deviation. A z-score of 0 means the value is exactly average, a positive z-score means above average, and a negative one means below average. Because it strips away the original units, it lets you compare scores from completely different tests or measurements.
How do you calculate a z-score by hand?+
Subtract the mean from your value, then divide by the standard deviation. For example, a score of 82 on a test with a mean of 70 and a standard deviation of 8 gives (82 - 70) / 8 = 1.5, so the score is 1.5 standard deviations above the mean.
What is a good or unusual z-score?+
There is no universally good z-score - it depends on context. As a rule of thumb, roughly 68% of values fall between z = -1 and z = 1, about 95% between -2 and 2, and about 99.7% between -3 and 3. Statisticians commonly treat values beyond z = 2 as unusual and beyond z = 3 as rare outliers.
Why is z = 1.96 used for 95% confidence?+
Because 95% of a standard normal distribution lies between z = -1.96 and z = 1.96, leaving 2.5% in each tail. That is what makes 1.96 the critical value for a two-tailed 95% confidence interval. For 99% confidence the equivalent value is 2.576.
Can I use a z-score if my data is not normally distributed?+
You can always compute one, since the formula only needs a mean and a standard deviation - but the percentile and probability readings assume a roughly normal distribution. For strongly skewed data those probabilities will be misleading, so check the shape of your distribution before interpreting them.
Is this z-score calculator free?+
Yes - it runs entirely in your browser, needs no signup, and stores nothing you enter.
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