Statistics
Bell curve generator
Enter a mean and a standard deviation to draw the curve, with each sigma band shaded and the exact coverage labelled.
The 68–95–99.7 rule is a rounding of the exact coverages shown above. It holds only for data that is roughly normal.
Show the working, step by step
Generate data with this mean and standard deviation
Makes a set of numbers whose sample mean and sample standard deviation match what you enter — useful for examples, test data and classroom exercises. The values are drawn from a normal distribution and then rescaled, so the match is exact until the values are rounded.
Reading a bell curve
The height of the curve at any point is not a probability — it is a density. Probability is area, which is why questions like "what fraction scores above 120?" are answered by shading a region rather than reading a height. The normal distribution calculator does that shading for any range.
The features worth knowing how to spot:
- The peak sits at the mean, which is also the median and the mode.
- The inflection points — where the curve changes from concave to convex — are at exactly μ ± σ. This is the standard deviation made visible.
- The tails approach the axis but never touch it, so no value is strictly impossible.
- Total area is always 1, whatever the mean and standard deviation.
What each parameter does
| Change | Effect on the curve |
|---|---|
| Increase the mean | Slides right. Shape unchanged. |
| Decrease the mean | Slides left. Shape unchanged. |
| Increase σ | Wider and flatter — more spread, lower peak. |
| Decrease σ | Narrower and taller — tightly clustered data. |
The peak height is 1 / (σ√(2π)), so it depends only on σ. Doubling the standard
deviation halves the peak, because the area underneath has to stay at 1.
The equation
f(x) = (1 / (σ√(2π))) · e^(−(x − μ)² / (2σ²))
The (x − μ)² in the exponent is the same squared deviation that appears in the
standard deviation formula. Dividing it by
2σ² is what makes the curve's width scale with the standard deviation, and the
1 / (σ√(2π)) in front is exactly the constant needed to make the area come to 1.
A caution about "bell curves"
Plenty of real data is not bell-shaped, and drawing a smooth curve over it does not make it so. Incomes, waiting times and city sizes are all strongly right-skewed. Before relying on a normal model, plot the actual data — the box plot will show a badly off-centre median if the shape is wrong, and the standard deviation calculator reports what fraction of your values really do fall within one σ so you can compare it against the expected 68%. To draw the curve over your real values instead of from two parameters, plot your own data with SD bands.
Related calculators
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Normal distribution
Probabilities and areas for any range on this curve.
-
Empirical rule
What fraction each band holds.
-
Standard deviation
Plot your own data against a fitted curve.
-
Z-score calculator
Locate a single value on the curve.
Common questions
How do I make a bell curve graph?
Enter a mean and a standard deviation above. The curve is drawn immediately with each sigma band shaded and labelled. To graph your own data instead, put the numbers into the standard deviation calculator, which plots them against a fitted curve.
What determines the shape of the curve?
Only two numbers. The mean slides it left or right; the standard deviation sets how wide and flat it is. Every normal curve is the same shape stretched and shifted.
Where is the standard deviation on a bell curve?
At the inflection points — where the curve stops bending downwards and starts bending outwards. Those sit exactly one standard deviation either side of the mean, which makes σ something you can literally see on the graph.
Can I make a bell curve in Excel?
Yes: create a column of x values spanning about μ ± 4σ, then use
=NORM.DIST(x, mean, sd, FALSE) for the height at each, and chart the pair as
a scatter with smooth lines. Setting the last argument to TRUE gives the cumulative
S-curve instead. More Excel formulas.