Normal Distribution and Z-Score Calculator
Enter the mean and standard deviation, then choose a question: the z-score of a value, the chance of being below, above or between values, or the value that matches a probability.
Worked examples
Chance below a value
Mean 100, SD 15, below 120
z = 1.333, 90.88%
About 9 in 10 values are below 120.
Z-score of a test mark
Mark 85, mean 70, SD 10
z = 1.5, percentile 93.3
(85 - 70) ÷ 10.
About Normal Distribution and Z-Score Calculator
The normal distribution is the bell shaped curve that describes many real measurements, such as heights, test marks and measuring errors. It is centred on the mean, and the standard deviation sets how wide it is. The chance of landing in a range is the area under the curve over that range, and since the whole area is 1, the area below a value is the chance of being below it. About 68% of values are within one standard deviation of the mean, 95% within two and 99.7% within three.
A z-score says how many standard deviations a value is from the mean: the value minus the mean, divided by the standard deviation. A mark of 85 in a test with a mean of 70 and a standard deviation of 10 has a z-score of 1.5, which is higher than about 93.3% of results. Choose z-score, below, above, between or outside to answer the usual exam and quality control questions, or work backwards from a probability such as 0.95 to the value with that much of the distribution below it. With a mean of 100 and a standard deviation of 15, about 90.9% of values are below 120.
How to use Normal Distribution and Z-Score Calculator
- Enter the mean and standard deviation of the distribution.
- Choose what to find from the list, such as the chance of being below a value.
- Enter the value or values, or the z-score or probability, whichever the question needs.
- Read the result, and the z-scores and tail probabilities beneath.
Common uses
- Working out percentiles and grade boundaries
- Statistics homework with tables replaced by exact values
- Quality control limits for sizes and weights
- Comparing results measured on different scales
Frequently asked questions
What is a z-score?
It is the number of standard deviations a value lies above or below the mean. A z-score of 0 is exactly average, 1 is one standard deviation above and -2 is two below.
How do I find a percentile?
Choose the z-score option, or the chance of being below. The chance of being below a value, as a percentage, is its percentile.
How do I find the value for the top 10%?
Choose the last option and enter 0.9 as the chance below. The value returned has 90% of the distribution below it and 10% above.
Do I need a normal distribution table?
No. The page calculates the cumulative probability directly, accurate to many figures, so there is no need to round the z-score to two decimals and look it up.
When is it right to use the normal distribution?
When the data is roughly symmetric and bell shaped. It is also a good approximation for the binomial distribution when there are many trials.