How to use the z-score and normal distribution calculator
- Choose Z-score of a value to turn a measurement into standard deviations from the mean, Probability from z if you already have a z-score, or Value from percentile to work backward.
- Enter the value, the mean and the standard deviation. Optionally add a second value to get the probability between the two.
- Read the z-score, the probability below and above, the two-tailed probability and the percentile.
- In percentile mode, enter a percentile such as 97.5 to get the z-score and the value that sits at that point.
Formula
z = (x - mean) / standard deviation
x = mean + z x standard deviation
Probability below = area under the standard normal curve to the left of z
Probability above = 1 - probability below
Two-tailed probability = 2 x probability above |z|
The normal curve probabilities and the percentile to z step are computed numerically, with accuracy well beyond the six decimals that are shown.
Worked example
A test has a mean of 100 and a standard deviation of 15. A score of 130 has z = (130 - 100) / 15 = 2.
The probability below is 0.97725, so the score is at the 97.725th percentile, the probability above is 0.02275 and the two-tailed probability is 0.0455.
The probability of a score between 115 and 130 is 0.135905. A z-score of 1.96 has 0.975002 below it, and the value at the 97.5th percentile is 100 + 1.96 x 15 = 129.3995.
What a z-score tells you
A z-score says how many standard deviations a value is from the mean. A z of 0 is exactly average, a z of 1 is one standard deviation above, and a z of -2 is two below. Because it is unit free, it lets you compare things measured in different units, such as a test score and a height.
Reading the probabilities
If data follow a normal curve, the area to the left of z is the share of values below it, which is its percentile. Roughly 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. The two-tailed probability counts how unusual a value is in either direction, which is the p value used in many tests.
When a z-score is misleading
A z-score assumes a roughly bell shaped distribution. For skewed data, such as incomes, or small samples, percentiles from the normal curve can be far off. For help with the mean and spread of your own numbers use the descriptive statistics calculator or the standard deviation calculator. For planning a survey, see the sample size calculator.
Assumptions and limits
- The data are modeled as a normal distribution with the mean and standard deviation you enter.
- The standard deviation is the one for the whole population, or your best estimate of it.
- Values more than 8 standard deviations from the mean are not reported.
- Probabilities tinier than 0.000001 are shown in scientific notation.
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from the value and divide by the standard deviation. A score of 130 with a mean of 100 and a standard deviation of 15 gives a z of 2.
What is the z-score for the 95th percentile?
About 1.6449. For the 97.5th percentile it is about 1.96, which is the number behind the common 95% confidence interval.
What is a two-tailed probability?
It is the chance of a value at least as far from the mean as yours, in either direction. For z = 2 it is 0.0455, which is twice the one sided 0.02275.
Can a z-score be negative?
Yes. A negative z means the value is below the mean. The probability below it is then less than 0.5.