Z score calculator with p-values, percentiles and a normal curve
A z score tells you how many standard deviations a value sits above or below the mean. This z score calculator computes it from x, μ and σ and immediately shows the probabilities that go with it: the left-tail area P(Z < z), the right-tail area, the two-tailed probability and the percentile, all drawn on a shaded standard normal curve. It also works backwards — from a z score to the raw value, or from a percentile to the z score that marks it — and it can standardise a whole column of raw data at once.
The probabilities are computed from the error function with a precision of about fourteen decimal places, so values like P(Z > 1.96) = 0.0250 and z₀.₉₉₅ = 2.5758 match statistical tables exactly. A reference table of critical z values for the common confidence levels is included for exam revision and confidence intervals.
Everything runs in your browser. For the descriptive statistics of a data set use the statistics calculator; for comparing means with small samples use the t test calculator.
How to use the Z Score Calculator
- In x → z, enter the value, the mean and the standard deviation; choose which probability to shade on the curve.
- In z → x, enter a z score with μ and σ to get the corresponding raw value, for example the score at z = −1 on a test with mean 100 and SD 15.
- In Percentile ↔ z, type a percentile such as 97.5 to get z = 1.96, or type a z to get its percentile.
- In Raw data, paste a list of numbers, pick population σ or sample s, and read the mean, standard deviation and the z score and percentile of every value. Copy the table if you need it elsewhere.
- Open Common z values for the critical values used in 90 %, 95 % and 99 % confidence intervals and hypothesis tests.
Z Score Calculator features
All four probabilities
Left tail, right tail, two-tailed and the central area between −z and z, each shown as a decimal and a percentage.
Accurate inverse normal
Percentile-to-z uses Acklam's rational approximation refined with Halley steps, giving z to better than 10⁻¹⁰.
Shaded normal curve
An SVG bell curve with the chosen region shaded and the z position marked, redrawn as you type.
Raw data mode
Standardise a whole data set with population or sample standard deviation, with counts of values beyond ±2 and ±3.
Critical value table
z for 80 % to 99.9 % confidence, one- and two-tailed, plus the 68–95–99.7 rule.
How to calculate a z score
The formula is z = (x − μ) ÷ σ: subtract the mean from the value and divide by the standard deviation. A score of 85 on a test with mean 70 and standard deviation 10 gives z = (85 − 70) ÷ 10 = 1.5, meaning the score is one and a half standard deviations above average. From the standard normal distribution, P(Z < 1.5) = 0.9332, so the score is at the 93rd percentile and only 6.7 % of scores are higher.
To go the other way, x = μ + z × σ. If an IQ test has mean 100 and standard deviation 15, the score one standard deviation below average is 100 + (−1)(15) = 85. For percentiles, invert the normal cumulative distribution: the 97.5th percentile corresponds to z = 1.960, which is why ±1.96 brackets the middle 95 % of a normal distribution.
| z | P(Z < z) | P(Z > z) | Two-tailed | Used for |
|---|---|---|---|---|
| 1.282 | 0.9000 | 0.1000 | 0.2000 | 80 % CI |
| 1.645 | 0.9500 | 0.0500 | 0.1000 | 90 % CI, one-tailed 5 % |
| 1.960 | 0.9750 | 0.0250 | 0.0500 | 95 % CI, two-tailed 5 % |
| 2.326 | 0.9900 | 0.0100 | 0.0200 | 98 % CI, one-tailed 1 % |
| 2.576 | 0.9950 | 0.0050 | 0.0100 | 99 % CI |
| 3.000 | 0.9987 | 0.0013 | 0.0027 | 3-sigma limits |
Which tail do I need?
A z score assumes the data are roughly normal. For skewed data the percentile from the raw-data table may be a better guide than the one from the curve, and for small samples with an unknown σ the t distribution replaces the normal; the t test calculator handles that case.
- Left tail P(Z < z) is the percentile: the share of the population below the value. Use it for "what fraction scored lower".
- Right tail P(Z > z) answers "what fraction is higher" and is the one-sided p-value for a test of "greater than".
- Two-tailed P(|Z| > |z|) doubles the smaller tail and is the usual p-value when the hypothesis is simply "different".
- Between −z and z is the coverage of a symmetric interval: 68.27 % for z = 1, 95.45 % for z = 2 and 99.73 % for z = 3.
Z Score Calculator FAQ
What is a good z score?
There is no good or bad in itself: z simply locates a value. Beyond ±2 is unusual (about 5 % of a normal population) and beyond ±3 is rare (0.3 %), which is why ±3 is a common outlier threshold.
How do I turn a z score into a p-value?
Look up the area in the tail. For z = 1.96 the right-tail area is 0.025 and the two-tailed p-value is 0.05. The z score calculator shows both plus the left-tail area and the percentile.
How do I find the z score for the 95th percentile?
Use the Percentile ↔ z mode: 95 gives z = 1.645. Note that the 97.5th percentile (1.96) is what you need for a two-sided 95 % interval.
Should I use population or sample standard deviation in raw-data mode?
Use population σ (divide by n) when the data set is the whole population you care about; use sample s (divide by n − 1) when the data are a sample and you are estimating the spread of a larger population.
Can a z score be negative?
Yes. A negative z means the value is below the mean; z = −1 is one standard deviation below, at roughly the 16th percentile.
Is the z score calculator accurate for extreme values?
Yes. The tail areas are computed from the complementary error function rather than a lookup table, so z = 5 correctly gives a right-tail probability of 2.87 × 10⁻⁷.