Free maths tool · Level: Ages 16 to 18
Normal distribution
Drag the two bounds across the curve and read off how much sits between them. The z-scores follow automatically.
For a normal distribution, the probability that a value falls between two bounds is the area under the bell curve between them. The mean μ sets where the curve sits, the standard deviation σ sets how wide it is. The total area is always 1, so a wider curve is automatically flatter.
What μ and σ each do
Move μ and the whole curve slides across without changing shape. Move σ and the shape does change: a bigger σ makes it wider and flatter, a smaller σ makes it narrow and tall.
Mixing those two up is the most common mistake. The mean says where the centre is, the standard deviation says how far from that centre the values typically sit.
The 68 - 95 - 99.7 rule
Put the bounds exactly one σ either side of the mean and you get about 68 per cent. At two σ about 95 per cent, at three σ about 99.7 per cent.
The tool spots those positions and says so. Those three numbers are worth knowing by heart, because they let you estimate without a table or a calculator.
What the z-score is for
A z-score says how many standard deviations a value sits from the mean: z = (x - μ) / σ. A z of 2 means two σ above the mean, whatever unit you are measuring in.
That is why one table covers every normal distribution. You convert to a z-score first, then look it up. Drag a bound and watch the z change alongside the raw value.
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Frequently asked questions
- What is a normal distribution?
- A symmetric bell-shaped distribution around the mean. Values near the mean are the most common, and they get rarer the further out you go.
- How do you calculate a probability with a normal distribution?
- You find the area under the curve between your two bounds. In practice you convert the bounds into z-scores and look those up in a table or on a calculator. This tool does both steps at once.
- What does the standard deviation actually mean?
- How far the values sit from the mean on average. A small σ means nearly everything is close to the centre, a large σ means the values are spread out.
- Why is the area always 1?
- Because every possible outcome falls somewhere under the curve, and the probability of "anything at all" is 1. So making the curve wider forces it to get flatter.