About Probability Between Two Z-Scores
This calculator finds the probability that a normal variable falls between two values. It subtracts the area to the left of the lower value from the area to the left of the upper value, and shades the region. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The calculator opens with -1.96 and 1.96, which enclose 95% of a standard normal distribution.
How to use Probability Between Two Z-Scores
- Choose what you want to find: z from x, x from z, a probability, a percentile to z, or a z-table lookup.
- Enter the values. For a raw score you need the mean and the standard deviation of the distribution.
- For a probability, pick Below, Above, Between or Outside and enter one or two limits.
- Read the headline answer, the percentile and the other areas. The shaded bell curve shows which area is meant.
- Open the working to see the formula with your numbers filled in.
Features
- Z from x, x from z, and percentile to z-score in one page.
- Probabilities below, above, between and outside, with the shaded area on a bell curve.
- Percentile and both tail areas for every z-score.
- Step-by-step working in written-out formulas.
- Full standard normal z-tables, for negative and positive z, on the z-table page.
- Accurate to many decimals, not rounded to the four digits of a printed table.
- Copy the results as text. Works on a phone.
Area between two z-scores
Subtract the smaller cumulative probability from the larger one.
P(a < Z < b) = Phi(b) - Phi(a)Worked examples
Between -1 and 1 gives 0.682689
About 68%.
Between -2 and 2 gives 0.9545
About 95%.
Between -3 and 3 gives 0.9973
About 99.7%.
Privacy: how it works
- Everything is calculated in your browser tab. The values you enter are not sent to Ulkaa or to any other server.
Tips
- Percentile and probability below are the same number, shown once as a percent and once as a decimal.
- A z-score beyond 3 in either direction is rare for normal data, about 0.27% in total.
- To find a critical value for a confidence level, use percentile to z with 100 minus half the tail: 97.5 for 95%.
- The standard deviation must be greater than 0.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your values are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your values are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is the 68-95-99.7 rule?
About 68% of a normal distribution lies within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
Does the order of the two values matter?
No. The calculator uses the smaller as the lower limit.
More Z-Score Calculator pages
- Z-Score CalculatorTurn a raw score into a z-score and a percentile, or find the probability under the normal curve.
- Z-Table (Standard Normal Table)Look up the area to the left of any z-score, in the calculator or in the full tables below.
- Z-Score to Percentile CalculatorEnter a score, the mean and the standard deviation to get the z-score and its percentile.
- Percentile to Z-Score CalculatorEnter a percentile to find the z-score with that share of the distribution below it.
- Z-Score Probability CalculatorChoose a region, enter a value and read the probability under the normal curve.
- Find X From a Z-ScoreEnter a z-score, the mean and the standard deviation to get the raw score.