Normal Distribution Calculator
Find the z-score and the probabilities P(X<x) and P(X>x) for a value in a normal distribution.
z-score
1.3333
How many standard deviations x sits from the mean: z = (x − μ) / σ.
P(X < x)
0.908789
Area to the left of x.
P(X > x)
0.091211
Area to the right of x.
Probabilities use an erf-based approximation of the normal CDF, accurate to about four decimals. Percentages describe where x falls within the distribution.
How it works
Lots of real-world measurements — heights, test scores, measurement errors — cluster around an average and thin out symmetrically on both sides, which is the shape of a normal distribution. To place a single value x within that curve, this tool needs the distribution's mean and standard deviation.
First it converts x into a z-score with z = (x − μ) / σ, which counts how many standard deviations x sits above or below the mean. Then it feeds that z into an erf-based approximation of the normal cumulative distribution function to find P(X<x), the area of the curve to the left of x. Subtracting from 1 gives P(X>x), the area to the right.
For example, an IQ of 120 with a mean of 100 and a standard deviation of 15 has a z-score of about 1.33, meaning roughly 91% of scores fall below it and about 9% above. The erf approximation is accurate to about four decimal places, which is plenty for coursework and quick checks.
Frequently asked questions
What does the z-score tell me?
The z-score is how many standard deviations your value is from the mean. A z of 0 sits right at the mean, positive z-scores are above it, and negative ones are below. It lets you compare values from different normal distributions on the same scale.
How accurate are the probabilities?
They use a standard erf-based approximation of the normal CDF that's accurate to roughly four decimal places across the usual range of z-scores. That's more than enough for homework, statistics courses, and everyday estimates.
How do I find the probability between two values?
Calculate P(X<x) for the larger value and P(X<x) for the smaller value, then subtract the smaller result from the larger. The difference is the probability that a value lands between the two.