Z-Score Calculator
Convert a raw score to a z-score and percentile, find P(Z < z), P(Z > z) or the area between two values, or get the z for a given probability, on a shaded curve.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Z-Score Calculator?
A z-score says how many standard deviations a value sits above or below the mean. Enter a raw score with the mean and standard deviation and this calculator returns the z-score, the percentile, the left-tail and right-tail probabilities, the share within ±z of the mean and the two-tailed p-value, and shades the matching area on a normal curve labelled in your own units.
It also works the other way round and in between. Type a z-score to look up its probabilities without a printed table, type a probability to get the z (and raw score) that cuts it off on the left, on the right, in the middle or in both tails, or give two values to find the chance of landing between them. Every result updates as you type.
- Raw score to z, z to probability, probability to z and between-two-values modes
- Left, right, central and two-tailed areas with percentiles
- Shaded normal curve with an axis in your own units
- Accurate far into the tails (full precision at z = 8)
- Shows the formula and every intermediate value
- Copy and share link
How to use the Z-Score Calculator
- Choose what you want: Raw score to z, z to probability, Probability to z, or Between two values.
- Enter the raw score (or z, or probability, or both bounds) and the mean and standard deviation of the distribution.
- Read the z-score and percentile, then the left-tail, right-tail, central and two-tailed probabilities.
- For Probability to z, say whether the area is left of z, right of z, between -z and z, or split across both tails.
- Check the shaded curve, then copy the result or share the link.
Worked examples
A test score of 85 when the mean is 70 and SD is 10
Input
x = 85, μ = 70, σ = 10
Output
z = 1.5, P(Z < 1.5) = 0.933193, 93.3193rd percentile, P(Z > 1.5) = 0.066807
The score is one and a half standard deviations above average, which beats about 93% of test takers if scores are normally distributed.
A low reading
Input
z = -2.1
Output
P(Z < -2.1) = 0.017864, two-tailed p = 0.035729
Fewer than 2 in 100 values fall this far below the mean. As a two-sided test statistic it would be significant at the 5% level but not at 1%.
Cut-off for the top 10%
Input
Probability to z: 90% to the left, μ = 70, σ = 10
Output
z = 1.281552, raw score 82.815516
A score of about 82.8 marks the 90th percentile. Use "Right of z" with 10% to get the same z.
Between two values
Input
a = 60, b = 80, μ = 70, σ = 10
Output
P(60 < X < 80) = 0.682689
This is the one-standard-deviation band of the empirical rule, 68.27%.
Background and insight
The standardisation formula is z = (x - μ) / σ. Once a value is in z units, the standard normal distribution gives the probabilities: Φ(z) is the area to the left of z. Φ has no closed form, so printed tables give it to four decimal places for z in steps of 0.01. This calculator computes it directly with George Marsaglia's 2004 series for |z| below 3 and a continued fraction in the tails, which keeps about 15 significant digits even at z = 8, where the upper tail is 6.22 x 10⁻¹⁶. Many online tools subtract from 1 at that point and report 0.
The reverse lookup, Φ⁻¹(p), uses Peter Acklam's rational approximation followed by one Halley correction step against the accurate Φ, which gives the familiar critical values to full precision: 1.644854 for 95% one-sided, 1.959964 for 95% two-sided and 2.575829 for 99% two-sided. The regression tests compare results with these published constants and with Python's statistics.NormalDist.
The empirical rule is a useful sanity check: about 68.27% of a normal distribution lies within one standard deviation of the mean, 95.45% within two and 99.73% within three. Those percentages only hold when the data really are close to normal; z-scores of skewed data still measure distance from the mean, but their percentiles will be off.
Common use cases
Grading and test scores
Turn an exam or SAT-style score into a percentile, or set a grade boundary at a chosen percentile.
Hypothesis tests
Get the one- or two-tailed p-value for a z test statistic without a printed table.
Quality control
Find the share of parts expected outside tolerance limits from the process mean and standard deviation.
Growth and health charts
Read what a reported z-score (such as a child's height-for-age) means as a percentile.
Tips and limitations
- A z-score is only as good as the mean and standard deviation you feed it. Use the population values if you have them.
- Percentiles from z assume a normal distribution. For skewed data, rank the values instead.
- Two-tailed p-values double the smaller tail. Use one tail only when the direction was fixed before you saw the data.
- Set the mean to 0 and the standard deviation to 1 to work purely in z units.
- For small samples where σ is estimated from the data, a t distribution is more appropriate; the Confidence Interval Calculator uses it.
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x - μ) / σ. A score of 85 with mean 70 and standard deviation 10 has z = (85 - 70) / 10 = 1.5.
How do I convert a z-score to a percentile?
Find the area to the left of z under the standard normal curve, Φ(z), and multiply by 100. For z = 1.5 that is 0.9332, so the value sits at about the 93rd percentile.
What does a negative z-score mean?
The value is below the mean. A z of -2 means two standard deviations below average, which in a normal distribution is around the 2nd percentile.
What z-score is statistically significant?
For a two-tailed test at the 5% level, |z| above 1.96 is significant; at 1%, |z| above 2.576. For a one-tailed test at 5% the cut-off is 1.645.
What is the probability between two z-scores?
Subtract the left-tail areas: P(a < Z < b) = Φ(b) - Φ(a). Between -1 and 1 that is 0.6827, between -2 and 2 it is 0.9545.
Is a z-score of 3 unusual?
Yes, for normal data. Only 0.135% of values lie above z = 3, and 0.27% lie more than three standard deviations from the mean in either direction.
Is there an in-depth guide to the Z-Score Calculator?
Yes. “Mean, Median, Standard Deviation and Z-Scores Explained” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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