Normal distribution calculator
Pick a mean and a standard deviation, set the bounds, and see the probability as a shaded area under the bell curve. Tails, intervals, z-scores and percentiles are all one row in the calculator, and the bounds can be sliders.
How to use it
- 1In the calculator, type
normaldist(100, 15): the mean first, then the standard deviation. The bell curve appears; press Zoom to fit under the row to frame it. - 2Press Find a probability under the row and type the bounds. Use -inf or inf for a tail, so P(X ≥ 130) is from 130 to inf.
- 3The bounds can be expressions or sliders, like
mu - sigma. For a percentile, typeinvnorm(0.9, 100, 15).
The 68–95–99.7 rule
Every normal distribution has the same shape, so the share of values within k standard deviations of the mean is the same for all of them:
| Within | Probability | Outside |
|---|---|---|
| μ ± 1σ | 0.68269 (68.27%) | 31.73% |
| μ ± 2σ | 0.9545 (95.45%) | 4.55% |
| μ ± 3σ | 0.9973 (99.73%) | 0.27% |
So for IQ scores with mean 100 and standard deviation 15, about 68% of people score between 85 and 115, and about 95% between 70 and 130.
z-scores: one table for every normal
A z-score says how many standard deviations a value is from the mean:
An IQ of 130 has z = (130 − 100)/15 = 2. The probability of scoring at most 130 is the same as P(Z ≤ 2) for the standard normal: 0.9772, so a score of 130 is higher than about 97.7% of the population. The calculator does the standardising for you, but z-scores are still the best way to compare values from different normal distributions.
Percentiles: working backwards
A probability question asks “how likely is a value in this range?”. A percentile question runs the other way: “which value has 90% of the distribution below it?”. That is the inverse of the cumulative distribution function, invnorm(p, mu, sigma).
For IQ scores, invnorm(0.9, 100, 15) = 119.22: the 90th percentile.
The functions, TI-84 style
| Function | Gives |
|---|---|
normalpdf(x, mu, sigma) | The height of the bell curve at x (a density, not a probability) |
normalcdf(a, b, mu, sigma) | P(a ≤ X ≤ b) |
normalcdf(x) | P(Z ≤ x) for the standard normal |
invnorm(p, mu, sigma) | The value with probability p below it |
Examples to try
normaldist(0, 1)normaldist(0, 1)normaldist(0, 2)normaldist(0, 3)Frequently asked questions
How do I find a probability from a normal distribution?
How do I find P(X > a)?
What is a z-score?
How do I find a percentile?
Why is the probability of an exact value zero?
Related tools and guides
Binomial probabilities shown as bars, with mean and SD.
P-values from data or statistics, shaded on the distribution.
Box and whisker plots, dot plots and histograms from a list.
Linear, polynomial, exponential or any custom model, with error bars.
Make the mean and standard deviation sliders and watch the shaded probability change as you drag.