About Z-Score Calculator
This z-score calculator is free, needs no sign-up and runs in your browser, so nothing you type is uploaded. Enter a raw score, the mean and the standard deviation and it returns the z-score, the percentile and the probabilities below and above that score, with a bell curve that shades the area.
It works the other way too: give it a z-score and it returns the raw score, give it a percentile and it returns the z-score, or ask for the probability that a value falls below, above, between or outside two limits. Leave the mean at 0 and the standard deviation at 1 to work straight in z-scores.
The pages for the z-table show the full standard normal table, one row for each tenth of z and one column for each hundredth, with the area to the left of z. The same numbers drive the calculator, so a table lookup and a calculated answer always agree.
How to use Z-Score Calculator
- Choose what you want to find: z from x, x from z, a probability, a percentile to z, or a z-table lookup.
- Enter the values. For a raw score you need the mean and the standard deviation of the distribution.
- For a probability, pick Below, Above, Between or Outside and enter one or two limits.
- Read the headline answer, the percentile and the other areas. The shaded bell curve shows which area is meant.
- Open the working to see the formula with your numbers filled in.
Features
- Z from x, x from z, and percentile to z-score in one page.
- Probabilities below, above, between and outside, with the shaded area on a bell curve.
- Percentile and both tail areas for every z-score.
- Step-by-step working in written-out formulas.
- Full standard normal z-tables, for negative and positive z, on the z-table page.
- Accurate to many decimals, not rounded to the four digits of a printed table.
- Copy the results as text. Works on a phone.
Z-score formulas
A z-score says how many standard deviations a value is above or below the mean. The area to the left of z on the standard normal curve is the percentile of the value, written Phi(z).
z = (x - mean) / sd
x = mean + z * sd
percentile = Phi(z) x 100
P(a < Z < b) = Phi(b) - Phi(a)
P(Z > a) = 1 - Phi(a)Worked examples
x = 85, mean 70, standard deviation 10 gives z = 1.5
(85 - 70) / 10 = 1.5. The area to the left is 0.933193, about the 93rd percentile.
P(Z < 1.96) gives 0.975002
This is the value that makes 1.96 the two-sided 95% critical value: 2.5% lies above it and 2.5% lies below -1.96.
P(-1 < Z < 1) gives 0.682689
About 68% of a normal distribution lies within one standard deviation of the mean.
IQ with mean 100 and sd 15, z = -2 gives x = 70
100 + (-2) x 15 = 70.
97.5th percentile gives z = 1.96
The 95% critical value is 1.96.
Good to know
When a z-score is the right tool
The probabilities here assume the values follow a normal distribution, or that you are working with a sample mean from a reasonably large sample. A z-score by itself, measured against a mean and standard deviation, is useful for any data, but turning it into a percentile is only right for roughly bell-shaped data.
If you estimated the standard deviation from a small sample, a t-score with the right degrees of freedom is more accurate than a z-score.
Reading a z-table
A standard z-table lists the area to the left of z. Find the row for the first decimal of z and the column for the second decimal, and read where they cross. For a negative z use the negative table, or use the symmetry Phi(-z) = 1 - Phi(z). For the area to the right, subtract the table value from 1.
Privacy: how it works
- Everything is calculated in your browser tab. The values you enter are not sent to Ulkaa or to any other server.
Tips
- Percentile and probability below are the same number, shown once as a percent and once as a decimal.
- A z-score beyond 3 in either direction is rare for normal data, about 0.27% in total.
- To find a critical value for a confidence level, use percentile to z with 100 minus half the tail: 97.5 for 95%.
- The standard deviation must be greater than 0.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your values are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your values are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I calculate a z-score?
Subtract the mean from the value and divide by the standard deviation. For 85 with mean 70 and standard deviation 10 that is (85 - 70) / 10 = 1.5.
What does a z-score of 1.5 mean?
The value is 1.5 standard deviations above the mean. On a normal curve that is about the 93rd percentile.
How do I convert a z-score to a percentile?
Find the area to the left of z under the standard normal curve and multiply by 100. The calculator does this directly, and the z-table page lists the same values.
How do I find the probability above a z-score?
Subtract the area to the left from 1. The calculator shows both in its Above and Below choices.
What is a z-score of 1.96?
It leaves 2.5% in the upper tail, so 95% of a normal distribution lies between -1.96 and 1.96. It is the usual critical value for a 95% confidence interval.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean.
Why does my answer differ slightly from a printed z-table?
Printed tables round to four decimals and often round z to two. The calculator keeps full precision, so the last digit can differ.
Popular Z-Score Calculator pages
- Z-Table (Standard Normal Table)Look up the area to the left of any z-score, in the calculator or in the full tables below.
- Z-Score to Percentile CalculatorEnter a score, the mean and the standard deviation to get the z-score and its percentile.
- Percentile to Z-Score CalculatorEnter a percentile to find the z-score with that share of the distribution below it.
- Z-Score Probability CalculatorChoose a region, enter a value and read the probability under the normal curve.
- Find X From a Z-ScoreEnter a z-score, the mean and the standard deviation to get the raw score.
- Probability Between Two Z-ScoresEnter a lower and an upper value to get the area between them under the normal curve.