How do I find the area under a bell curve?

Enter the mean and standard deviation, then choose a region to shade (for example X < 85). The tool computes the area — which is the probability that a value falls in that region — and shows it as a decimal and a percent, along with the z-score of the boundary.

A z-score is how many standard deviations a value sits from the mean: z = (x − mean) / SD. A z-score of −1 means one SD below the mean. Z-scores let you compare values from different normal distributions and look probabilities up on a standard z-table.

What do the mean and standard deviation change?

The mean slides the whole bell left or right — it is the center and the peak. The standard deviation sets the spread: a small SD gives a tall, narrow bell; a large SD gives a short, wide one. The area under the curve is always 1 regardless of either.

Bell Curve Generator

Free bell curve generator: enter a mean and standard deviation to draw a normal distribution, shade any region, and read off the probability and z-score. Download as PNG, SVG, PDF, or CSV.

Mean Standard deviation Shade region Chart title X-axis label

Gridlines

Enter a mean and standard deviation, then shade a region to see its probability and z-score.

What a bell curve is

A bell curve is the graph of a normal distribution — a symmetric, bell-shaped curve defined entirely by two numbers: the mean, which sets where the peak sits, and the standard deviation (SD), which sets how wide the bell spreads. Most of the area lies near the mean and thins out toward the tails. Because the total area under the curve is exactly 1, the area over any range of x-values is the probability that a value falls in that range.

Reading a probability off the curve

Shading a region turns the picture into a probability. Shade everything below a value a and you get P(X < a); shade above it for P(X > a); shade between two values for P(a < X < b). This generator reports that area as both a decimal and a percent, and shows the z-score of each boundary — the number of standard deviations it sits from the mean, z = (x − mean) / SD. A z-score lets you compare values from different normal distributions on one scale.

Worked example

Say test scores are normal with a mean of 100 and an SD of 15, and you want the share of scores below 85. The z-score is (85 − 100) / 15 = −1.00, so 85 is one standard deviation below the mean. The area to the left of z = −1 is about 0.1587, so roughly 15.87% of scores fall below 85 — which is exactly what this tool shades and reports when you enter mean 100, SD 15, and the region X < 85.

The 68–95–99.7 rule

The 68–95–99.7 rule is a fast sanity check: about 68% of the area lies within one SD of the mean, about 95% within two, and about 99.7% within three. So for mean 100 and SD 15, about 95% of values fall between 70 and 130. If a shaded region you expected to be large comes back tiny (or vice versa), re-check whether you entered the SD and the boundary correctly.

How to use it

Enter the mean and the standard deviation of your distribution.
Pick a region to shade — less than, greater than, between, or outside two values — and type the boundary value(s).
Read the probability (as a decimal and a percent) and the z-score of each boundary below the curve.
Download the curve as PNG, SVG, or PDF, or export the sampled curve as CSV.

FAQ

How do I find the area under a bell curve?: Enter the mean and standard deviation, then choose a region to shade (for example X < 85). The tool computes the area — which is the probability that a value falls in that region — and shows it as a decimal and a percent, along with the z-score of the boundary.
What is a z-score?: A z-score is how many standard deviations a value sits from the mean: z = (x − mean) / SD. A z-score of −1 means one SD below the mean. Z-scores let you compare values from different normal distributions and look probabilities up on a standard z-table.
What do the mean and standard deviation change?: The mean slides the whole bell left or right — it is the center and the peak. The standard deviation sets the spread: a small SD gives a tall, narrow bell; a large SD gives a short, wide one. The area under the curve is always 1 regardless of either.