Normal Distribution Calculator
A normal distribution is fixed by two numbers: its mean and its standard deviation. Give it those, pick a region, and this tool returns the probability of landing there, the area below, above, between or outside a value. Flip it around and it works backwards from a probability to the cutoff value that marks it. Everything runs in your browser.
P(X < 130) = 0.9772
How this is computed
pnorm() and qnorm().
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The four questions a normal distribution answers
Every mode above maps to one R function. Areas use pnorm(); working back from a probability uses qnorm().
| You want | The region | In R |
|---|---|---|
| Area below a value | P(X < x) | pnorm(x, mean, sd) |
| Area above a value | P(X > x) | pnorm(x, mean, sd, lower.tail = FALSE) |
| Area between a and b | P(a < X < b) | pnorm(b, mean, sd) - pnorm(a, mean, sd) |
| The value for a probability | find x | qnorm(p, mean, sd) |
The 68-95-99.7 rule
These three figures fall straight out of the Between mode. Set the bounds to the mean plus and minus one, two or three standard deviations.
| Within | Region | Area |
|---|---|---|
| 1 SD | μ ± 1σ | 68.27% |
| 2 SD | μ ± 2σ | 95.45% |
| 3 SD | μ ± 3σ | 99.73% |
The familiar round "95%" comes from ±1.96 SD, not exactly ±2; the Inverse mode with a 95% central area returns that 1.96.
Frequently asked questions
How do I find a probability from a normal distribution?
Pick the region you want, below, above, between or outside a value, then enter the mean, the standard deviation and the cutoffs. The probability is the shaded area under the curve. It is computed with the normal CDF, exactly what R's pnorm() returns.
What is the inverse normal distribution?
The inverse works backwards from a probability to the value that marks it. Ask for the value with 10% of the area to its right and you get the 90th-percentile cutoff. In R that is qnorm(). Use the Inverse mode and choose whether the probability sits to the left, to the right, or as a central band.
What is the difference between this and a z-score?
A z-score standardises one value to the mean-0, SD-1 scale. This calculator works directly on your own mean and SD, so you never have to convert first. The z-score is still shown alongside each answer, and there is a dedicated z-score calculator if that is what you need.
How do I get the 68-95-99.7 percentages?
Use the Between mode with bounds at the mean plus and minus one, two and three standard deviations. The exact areas are 68.27%, 95.45% and 99.73%. The round 95% figure everyone quotes actually corresponds to ±1.96 SD.
Does this calculator match R?
Yes. Every probability and cutoff uses the same algorithm as R's pnorm() and qnorm() and agrees to nine decimal places, including deep tails past four standard deviations where naive formulas lose precision.