Z-Score Calculator
Turn a raw value into a z-score, percentile and p-value using the exact normal distribution, with the standard z-table and the arithmetic shown.
Z-Score Calculator: with the default inputs, z-score is 2.
The single observation you want to place on the curve.
The average of the population or distribution.
Must be greater than 0 — a distribution with no spread has no z-scores.
How many standard deviations x sits from the mean.
- Percentile (area to the left)
- 97.725%
- Area to the right
- 2.275%
- One-tailed p-value
- 0.02275P(Z ≥ |z|) — the chance of a result this extreme in one direction.
- Two-tailed p-value
- 0.0455P(|Z| ≥ |z|) — the chance of a result this extreme in either direction.
- How unusual is it?
- Unusual — between 2 and 3 SD above the mean (outside the middle 95%)
Assumptions
- Percentiles and p-values assume the underlying data is normally distributed; the z-score itself does not.
- μ and σ are treated as known population parameters, not sample estimates.
- Φ(z) is evaluated with Hart's double-precision rational approximation (absolute error < 1e-15), not a rounded table.
| z | Percentile (area left) | One-tailed p | Two-tailed p |
|---|---|---|---|
| 0 | 50% | 0.5 | 1 |
| 0.5 | 69.1462% | 0.308538 | 0.617075 |
| 1 | 84.1345% | 0.158655 | 0.317311 |
| 1.2816 | 90% | 0.1 | 0.2 |
| 1.6449 | 95% | 0.05 | 0.1 |
| 1.96 | 97.5% | 0.025 | 0.05 |
| 2 | 97.725% | 0.02275 | 0.0455 |
| 2.3263 | 99% | 0.01 | 0.02 |
| 2.5758 | 99.5% | 0.005 | 0.01 |
| 3 | 99.865% | 0.00135 | 0.0027 |
| 3.2905 | 99.95% | 0.0005 | 0.001 |
The rows at 1.644854, 1.959964 and 2.575829 are the critical values behind the familiar 90%, 95% and 99% confidence levels.
How this is worked out
The formula
z = (x − μ) ÷ σ x = the raw value μ = mean σ = standard deviation Percentile = Φ(z), the area under the standard normal curve to the left of z One-tailed p = 1 − Φ(|z|) Two-tailed p = 2 × (1 − Φ(|z|))
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Raw value (x)
- The single observation you want to place on the curve.defaults to 130
- Mean (μ)
- The average of the population or distribution.defaults to 100
- Standard deviation (σ)
- Must be greater than 0 — a distribution with no spread has no z-scores.0 or more · defaults to 15
What you get back
- Z-scoremain answer
- How many standard deviations x sits from the mean.
- Percentile (area to the left)
- Area to the right
- One-tailed p-value
- P(Z ≥ |z|) — the chance of a result this extreme in one direction.
- Two-tailed p-value
- P(|Z| ≥ |z|) — the chance of a result this extreme in either direction.
- How unusual is it?
What this assumes
- Percentiles and p-values assume the underlying data is normally distributed; the z-score itself does not.
- μ and σ are treated as known population parameters, not sample estimates.
- Φ(z) is evaluated with Hart's double-precision rational approximation (absolute error < 1e-15), not a rounded table.
About this calculator
A z-score rewrites a raw number as a distance. Instead of "you scored 130 on an IQ test", it says "you scored two standard deviations above the mean" — and that second sentence travels. It works for test scores, lab results, heights, manufacturing tolerances and A/B tests alike, because the units cancel out.
What it computes
Subtract the mean, divide by the standard deviation. That's the whole formula. The result is positive when the value is above the mean, negative when below, and zero when it lands exactly on it. A z of 1.5 means "one and a half standard deviations above average" no matter whether the original numbers were dollars, millimetres or reaction times.
How to read the results
- Percentile is the area under the normal curve to the left of your z. A z of 2 gives 97.72%, so about 97.7% of the population falls below that value.
- One-tailed p-value is the chance of landing at least this far from the mean in one direction; the two-tailed p-value is the chance of landing that far in either direction. The two-tailed value is simply twice the one-tailed one.
- The how unusual line is a plain-language reading of the same number. The rough landmarks worth memorising are the empirical rule: about 68% of values sit within ±1 SD, 95% within ±1.96 SD, and 99.7% within ±3 SD.
Where z-scores mislead
The percentile and p-value are only trustworthy if the underlying data is roughly normally distributed. The z-score itself is always defined — it is just arithmetic — but converting it to a percentile assumes a bell curve. Incomes, waiting times, insurance claims and web-page load times are all strongly right-skewed, and a z of 2 on data like that is nowhere near the 97.7th percentile.
Two more traps. First, if you calculated the mean and SD from the same small sample you are testing, the sampling error in σ matters and you should use a t-distribution instead. Second, a small p-value says the result is unlikely under the assumption that nothing is going on; it does not say how likely your hypothesis is, and it says nothing at all about whether the difference is large enough to care about.
Working backwards
Use Solve for to run the formula in reverse: fix the z-score you want and let the calculator find the raw value, the mean or the standard deviation that produces it. That is how cut scores ("what mark puts a student in the top 10%?") are set.
Frequently asked questions
▸What is a good z-score?
There is no universally good value — it depends on whether high or low is better. As a rule of thumb, |z| under 1 is ordinary, 2 is uncommon (about 1 value in 20 is that far out), and 3 is rare (about 1 in 370).
▸Can a z-score be negative?
Yes. A negative z simply means the value is below the mean. The size of |z| is what measures how unusual it is; the sign only tells you the direction.
▸How do I turn a z-score into a percentile?
Take the area under the standard normal curve to the left of z, written Φ(z). This calculator does it exactly; a printed z-table does the same to four decimals.
▸Should I use a z-score or a t-score?
Use z when you know the population standard deviation, or when your sample is large (roughly n ≥ 30). Use t when you are estimating the standard deviation from a small sample — the t-distribution has fatter tails to account for that extra uncertainty.
▸What p-value counts as significant?
Convention in most fields is 0.05 two-tailed (|z| ≥ 1.96), and 0.01 (|z| ≥ 2.576) where the cost of a false positive is high. The threshold is a convention, not a law of nature — report the p-value itself rather than just 'significant'.
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