Normal Distribution Calculator
Find normal distribution probabilities — P(X < x), P(X > x), or the probability between two values — for any mean and standard deviation, using the normal CDF.
P(X < 115)
0.841345
= 84.134474% · z = 1
Probabilities are areas under the normal curve, found by standardizing to a z-score and applying the standard normal CDF (Abramowitz-Stegun error function). Recall the 68-95-99.7 rule for one, two, and three standard deviations.
Frequently Asked Questions
How do I find a normal distribution probability?
Convert your value to a z-score with z = (x − μ) / σ, then look up the area under the standard normal curve. P(X < x) is the area to the left, P(X > x) is the area to the right, and the probability between two values is the difference of their left areas.
What is the 68-95-99.7 rule?
In a normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This empirical rule is a quick sanity check on probabilities from the full normal CDF.
What is the difference between the PDF and the CDF?
The probability density function (PDF) gives the height of the bell curve at a point — it is not a probability by itself. The cumulative distribution function (CDF) gives the area under the curve up to a point, which is the probability that a value is less than that point. This calculator uses the CDF.
How do I find the probability between two values?
Compute P(X < upper) − P(X < lower). Each term is the normal CDF at that value, so the difference is the area of the bell curve between them.
Does the calculator need the mean and standard deviation?
Yes. Enter the mean (μ) and standard deviation (σ) of your distribution along with the value(s) of interest. The calculator standardizes internally to a z-score and uses the standard normal CDF.