How to draw a bell curve from the mean and standard deviation
To draw a bell curve you need two numbers. The mean sets where the peak sits, and the standard deviation sets how wide the bell is. Put the mean in the middle, step out in standard deviations on each side, and the curve drops to about 61% of its height at 1 SD, 14% at 2 SD and 1% at 3 SD.
Step by step
- Draw a horizontal axis and mark the mean (70) at the centre.
- Mark one SD (10) steps either side, so 40, 50, 60, 70, 80, 90, 100.
- Work out the height at each mark (table below).
- Plot the points and join them with a smooth, symmetric curve. It should be flat at the top and inflect (change from bending down to bending up) exactly at ±1 SD.
The heights
f(x) = e−z²/2 ÷ (σ√(2π)), where z = (x − μ) ÷ σ
| z | Score x | Height f(x) | % of peak |
|---|---|---|---|
| −3 | 40 | 0.0004 | 1.1% |
| −2 | 50 | 0.0054 | 13.5% |
| −1 | 60 | 0.0242 | 60.7% |
| 0 | 70 | 0.0399 | 100.0% |
| 1 | 80 | 0.0242 | 60.7% |
| 2 | 90 | 0.0054 | 13.5% |
| 3 | 100 | 0.0004 | 1.1% |
If you only need the shape, use the last column. Whatever the mean and SD, the curve is 100% of its peak at the centre, about 60.7% at ±1 SD, 13.5% at ±2 and 1.1% at ±3.
What changes the shape
- A bigger SD makes the bell wider and lower. The area underneath always stays 1.
- A smaller SD makes it narrower and taller.
- A different mean slides the curve sideways and nothing else.
The area under the curve is fixed by the 68–95–99.7 rule. About 68% of the area lies within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD. That is why scores of 50 to 90 cover about 95% of students in this example.
Skip the drawing
The bell curve generator plots any mean and SD and shades the area you choose. For what a low and a high SD look like on real data, see the standard deviation graph page.
Related calculators
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Bell curve generator
Type a mean and SD and get the curve, shaded.
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Standard deviation graph
What a low and high SD look like on a chart.
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Empirical rule calculator
The 68–95–99.7 areas for any mean and SD.
Common questions
How do you draw a normal distribution with a given mean and standard deviation?
Mark the mean in the centre, mark ±1, ±2 and ±3 standard deviations along the axis, plot the heights from the density formula (or 100%, 60.7%, 13.5% and 1.1% of the peak), and join the points with a smooth bell.
What does a larger standard deviation do to the curve?
It makes the bell wider and lower. The area under the curve is always 1, so spreading it out lowers the peak. A smaller SD gives a narrow, tall curve.
What does changing the mean do?
It slides the whole curve left or right without changing its shape.
Do I need a normal distribution to use this?
Yes. The bell curve only describes data that is roughly normal. For skewed data, draw a histogram instead and read the SD from the data.
