What a z-score tells you
A z-score, also called a standard score, measures how far a value is from the mean in units of the standard deviation. It puts numbers from different scales on one common scale. A test score of 85 means little on its own, but if the class mean is 70 and the standard deviation is 10, the score is 1.5 standard deviations above average, and that is something you can compare with a z-score from any other test.
The sign shows the side: a positive z-score lies above the mean, a negative one below it, and zero is exactly the mean. The size shows the distance. A z-score of −2 is twice as far below the mean as −1.
Worked example
Say a student scores 85 on an exam where the mean is 70 and the standard deviation is 10. Subtract the mean, 85 − 70 = 15, and divide by the standard deviation, 15 ÷ 10 = 1.5. The z-score is 1.5. If the scores follow a normal distribution, the area under the bell curve to the left of z = 1.5 is about 0.93319, so the student did better than about 93% of the class and sits at about the 93rd percentile. A classmate with 62 has z = (62 − 70) ÷ 10 = −0.8, about the 21st percentile.
From z-score to probability and percentile
The standard normal distribution is the bell curve with a mean of 0 and a standard deviation of 1. Any normal distribution turns into it when you convert its values to z-scores, so one curve answers questions about all of them. The total area under the curve is 1, and the area over a stretch of z values is the probability that a value lands there:
- Left of z: P(Z < z), the share of values below z. Times 100 it is the percentile.
- Right of z: P(Z > z) = 1 − P(Z < z), the share above z. For a z-test this is the one-tailed p-value.
- Between two z-scores: P(z₁ < Z < z₂), the left area of z₂ minus the left area of z₁.
- Two tails: P(|Z| > |z|), the share at least as far from the mean as z on either side, the two-tailed p-value.
Here are some values from the calculator itself:
| z | Left, P(Z < z) | Right, P(Z > z) | Between −z and z |
|---|---|---|---|
| 0.5 | 0.6915 | 0.3085 | 38.29% |
| 1 | 0.8413 | 0.1587 | 68.27% |
| 1.5 | 0.93319 | 0.06681 | 86.64% |
| 2 | 0.97725 | 0.02275 | 95.450% |
| 2.5 | 0.993790 | 0.006210 | 98.758% |
| 3 | 0.998650 | 0.001350 | 99.7300% |
So about 68.27% of a normal distribution lies within one standard deviation of the mean, 95.45% within two and 99.73% within three, which is where the 68-95-99.7 rule of thumb comes from.
Finding z from an area
Sometimes you know the area and need the z-score: the cutoff for the top 10%, the score at the 90th percentile, or the critical value for a confidence interval. Choose "Area to z", type the percentage and pick where it sits. Add a mean and standard deviation to get the cutoff in your own units too. Common values:
| Area | z |
|---|---|
| 90% in the middle | ±1.6449 |
| 95% in the middle | ±1.9600 |
| 99% in the middle | ±2.5758 |
| Top 5% (right tail) | 1.6449 |
| Top 1% (right tail) | 2.3263 |
Mean and standard deviation from your data
If you have the raw numbers instead of the mean and standard deviation, choose "From data" and paste them. The calculator works out the mean and the standard deviation and lists the z-score and percentile of every value. Pick the population standard deviation, which divides by n, when your numbers are the whole group, and the sample standard deviation, which divides by n − 1, when they are a sample from a bigger group. The sample version is a little larger, so z-scores come out slightly closer to 0. The standard deviation calculator shows that part step by step.
When percentiles from z-scores do not fit
A z-score itself is just arithmetic and works for any data. Turning it into a percentile or a probability assumes the data follows a normal distribution. For skewed data, like incomes or waiting times, or data with two peaks, the real percentile can be far from what the bell curve says. With your own data at hand, compare: count how many of your values lie below x and see whether that share is close to the percentile shown.
How accurate the numbers are
Typed values are handled as exact decimals, so (0.3 − 0.1) ÷ 0.1 gives exactly 2. The bell curve areas use W. J. Cody's rational approximations of the error function, published in Mathematics of Computation in 1969 and available in the SPECFUN library on netlib. We checked the calculator against 60-digit arithmetic at more than 13,000 points: the areas agree to better than one part in ten billion, also far out in the tails. That matters there: P(Z > 9) is about 1.129 × 10⁻¹⁹, while computing it as 1 − P(Z < 9) in ordinary double precision gives 0.
How to use the z-score calculator
- 1Pick what to find: the z-score of a value, the value for a z-score, a probability for z-scores, or the z-score for an area.
- 2Type the value and the mean and standard deviation, or choose "From data" and paste your numbers to get the mean and standard deviation from them.
- 3For a probability or an area, choose left, right, between or two tails. The shaded bell curve shows which part of the curve you are looking at.
- 4Read the result with its percentile and the steps, then copy the result or a link that opens the calculator with your numbers.
Sources
- CALERF, SPECFUN library on netlib: W. J. Cody's rational approximations of the error function (Mathematics of Computation, 1969), used for every area on this page
Frequently asked questions
What is a z-score?
A z-score, also called a standard score, tells you how many standard deviations a value lies above or below the mean: z = (x − μ) ÷ σ. A z-score of 2 means two standard deviations above the mean, −0.5 means half a standard deviation below it, and 0 means the value equals the mean.
How do I turn a z-score into a percentile?
Find the area under the standard normal curve to the left of z, P(Z < z), and multiply by 100. For z = 1.5 that area is about 0.93319, so the value sits at about the 93rd percentile: roughly 93% of a normal distribution lies below it. The calculator shows this area for every z-score, with the shaded curve.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. A z-score of −1 is one standard deviation below the mean and sits at about the 16th percentile of a normal distribution. The sign only tells you the side; the size tells you the distance.
What is a good z-score?
That depends on what you measure. For a test score, a higher z-score is better; for a race time, a lower one is. In a normal distribution about 68% of values have a z-score between −1 and 1 and about 95% lie between −1.96 and 1.96, so a z-score beyond ±2 marks a fairly unusual value either way.
Should I use the sample or the population standard deviation?
Use the population standard deviation (dividing by n) when your numbers are the whole group you care about, like every student in one class. Use the sample standard deviation (dividing by n − 1) when your numbers are a sample from a larger group you want to describe. If you paste data, the calculator lets you pick either and shows both.
What z-score gives a 95% confidence level?
For 95% in the middle of a normal distribution, z is about 1.96: 2.5% of the area lies above 1.96 and 2.5% below −1.96. For 90% it is about 1.645 and for 99% about 2.576. Choose "Area to z" and "Middle" to get these values for any percentage.