Normal Distribution Calculator
Probability between bounds, with the full empirical rule.
Work out Normal Distribution. Probability between bounds, with the full empirical rule. Handles the case where the obvious answer is wrong.
Probability between the bounds
68.269%
z from -1 to 1
The empirical rule in full: 68.27% of a normal distribution lies within one standard deviation of the mean, 95.45% within two and 99.73% within three. The often-quoted "95% within two" is really 95.45%, and the exact 95% band is ±1.96σ — which is 70.6 to 129.4 here. All of this assumes normality, which is worth testing rather than presuming.
How the Normal Distribution Calculator works
The probability of landing between two values on a normal distribution, with the empirical rule stated precisely: 68.27% within one standard deviation, 95.45% within two, 99.73% within three. The often-quoted 95% is really ±1.96σ, not ±2σ.
Also known as: bell curve calculator · normal probability calculator · empirical rule calculator · gaussian distribution calculator
The empirical rule, precisely
68.27% of a normal distribution lies within one standard deviation of the mean, 95.45% within two, and 99.73% within three. The familiar 68-95-99.7 is a rounding of those.
The often-quoted "95% within two standard deviations" is really 95.45%. The exact 95% band is ±1.959964σ, which is where the 1.96 in every confidence interval comes from.
When it does not apply
The normal distribution describes symmetric, unbounded data with thin tails. A great deal of real data is none of those things.
Income is right-skewed, waiting times are bounded below at zero, and counts are discrete. Assuming normality on skewed data produces probabilities that are confident and wrong — usually badly underestimating how often extreme values occur.
The central limit theorem rescues averages rather than individual observations: means of large samples tend toward normal even when the underlying data does not. That distinction is what makes the normal distribution useful in statistics while being a poor model of most raw measurements.
Frequently asked questions
What is the empirical rule?
That 68.27% of a normal distribution lies within one standard deviation of the mean, 95.45% within two and 99.73% within three. The rounded 68-95-99.7 version is the usual shorthand.
Why is the 95% band 1.96 standard deviations?
Because exactly 95% lies within ±1.959964σ. Two standard deviations captures 95.45%, which is close but not the same — and the distinction matters in significance testing.
How do I find the probability between two values?
Convert both to z-scores and take the difference of their cumulative probabilities. This page does both steps.
When does the normal distribution not apply?
Whenever the data is skewed or bounded — income, waiting times and counts are all poorly described by it. Assuming normality on skewed data produces confident and wrong probabilities.
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