The z score
Subtract the mean and divide by the standard deviation. That turns any normal curve into the standard one, where the tables and this page agree.
A value of 115 with a mean of 100 and a standard deviation of 15 has a z score of 1.
Worked example
With a mean of 100 and a standard deviation of 15, the chance of a value below 115 is 0.841345, so the chance above it is 0.158655.
The chance of falling between 115 and 130 is 0.135905, and 115 sits at the 84.1345th percentile.
Reading the numbers
About 68% of a normal distribution lies within one standard deviation of the mean, 95% within two, and 99.7% within three.
Those figures are why a z score beyond 2 is unusual and beyond 3 is very unusual.
When a normal model fits
Heights, measurement errors and averages of many independent readings are close to normal.
Skewed data such as incomes, or counts of rare events, are not, and a normal probability will mislead there.
Inputs for the normal distribution calculator
Mean and standard deviation
The centre and spread of the curve. The standard deviation must be above zero.
Value of interest
The point whose probability you want.
Upper value
Optional. Set it above the first value to get the chance of falling between the two.
CALCULATOR QUESTIONS
Normal distribution calculator questions
What is the chance of a value below 115 when the mean is 100 and the standard deviation is 15?
0.841345, or about 84.1%. The z score is 1.
What does a z score of 1 mean?
The value is one standard deviation above the mean. About 84% of the distribution falls below it.
How do I find the chance between two values?
Put the higher value in the upper box. The page subtracts the lower probability from the higher one.
What is a percentile?
The percentage of the distribution that falls below a value. A z score of 1 is the 84.1345th percentile.