Z-score calculator
Enter a value with the mean and standard deviation of its distribution to get the z-score, or enter a z-score to get the matching value. The left-tail, right-tail and two-tailed probabilities come from the standard normal distribution.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose whether to find z from a value, or a value from z.
- Enter the mean and standard deviation of the distribution.
- Enter x (or z) and read the probabilities below the headline result.
How it works
z = (x − μ) / σ, and the reverse is x = μ + z × σ. The left-tail probability is the standard normal cumulative distribution Φ(z) = ½ × erfc(−z/√2). The right tail is computed directly rather than as 1 − Φ(z), so far-tail values such as z = 6 (about 9.9 × 10⁻¹⁰) keep their precision. The two-tailed p-value is twice the tail beyond |z|.
Worked example
An exam has a mean of 70 and a standard deviation of 10, and you scored 85.
- z = (85 − 70) / 10 = 1.5
- Left tail: 0.933193, so you scored higher than about 93.3% of test takers (the 93rd percentile).
- Right tail: 0.0668072
- Two-tailed p-value: 0.133614
Tips
- About 68%, 95% and 99.7% of a normal distribution lies within 1, 2 and 3 standard deviations of the mean.
- Use the sample or population standard deviation consistently; this tool takes the value you give it.
- Critical values to remember: 1.645 (one-tailed 5%), 1.96 (two-tailed 5%), 2.576 (two-tailed 1%).
Frequently asked questions
What is a z-score?
A z-score says how many standard deviations a value lies from the mean, z = (x − μ) / σ. A z of 1.5 means 1.5 standard deviations above the mean, and −2 means two below it. It lets you compare values from different scales.
How do I get a p-value from a z-score?
Use the standard normal curve. The left-tail value is the area up to z, the right-tail value is the area beyond it, and the two-tailed p-value is twice the smaller tail. For z = 1.96 the left tail is 0.975002 and the two-tailed p is about 0.05.
When should I use one tail and when two?
Use one tail when you test a direction, such as whether a value is higher than the mean. Use two tails when any difference, high or low, counts. The two-tailed p-value is double the one-tailed value for the same |z|.
How accurate are the probabilities?
They come from the error function evaluated with a series for small arguments and a continued fraction for the far tail, giving results accurate to about 10 or more digits, for example Φ(1.96) = 0.9750021. Output is rounded to 6 significant digits.
Does a normal distribution need to be assumed?
The z-score itself is just arithmetic and works for any data. The probabilities, however, assume the values follow a normal (bell-shaped) distribution. For skewed data they can mislead.