Normal Distribution Calculator: Z-Score and Probability Between Two Values
Find the z-score and the chance of a value below, above or between two numbers on a bell curve, or the value for a chosen chance.
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How this calculator works
Enter the mean and standard deviation of your data, then the value x1. The calculator gives its z-score (how many standard deviations it sits from the mean) and the chance of a result below it and above it. Enter a second value x2 to get the chance of a result between the two.
To go the other way, enter a chance such as 0.975 and it gives the value that this share of results falls below. That is how a design value or a percentile is found: for example the value that is exceeded only 2.5 percent of the time.
The bell curve is drawn with your two values marked. The cumulative chance is calculated from the error function to full precision, not read from a printed table, so it stays accurate far out in the tails.
Formulas used
- z = \frac{x - \mu}{\sigma}
- P(X < x) = \Phi(z) = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)\right]
- x_p = \mu + z_p\,\sigma
Worked example
With the default numbers in the calculator above:
- Standard score (z-score) of x1: 1.00
- Chance of a value below x1: 0.84134 (84.134 %)
- Chance of a value above x1: 0.15866 (15.866 %)
- Chance of a value between x1 and x2: 0.13591 (13.591 %)
- z-score for the chance you chose: 1.96
- Value that this share of results falls below: 129.40
- Range holding about 95 % of the values: 70.60 to 129.40
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
What does a z-score of 2 mean?
The value is 2 standard deviations above the mean. About 97.7 percent of results are below it.
How do I know my data are normal?
Plot a histogram. A symmetric bell shape suggests it. Strongly skewed data, such as failure lives, are not normal.
Should I use the sample or population standard deviation?
Use the sample standard deviation for a set of readings from a larger population. The statistics calculator gives both.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Montgomery and Runger, Applied Statistics and Probability for Engineers (normal distribution); NIST/SEMATECH e-Handbook of Statistical Methods.
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