The normal distribution
Drop balls through a bean machine and watch a bell curve appear. Then use μ, σ and z-scores to find what share of values lies below a point.
Standards: HSS-ID.A.4HSS-ID.A.2
1Wonder
Heights of people, errors in measurements, scores on a big test: plot enough of them and the same shape keeps appearing, a bell. Why does chance make such a neat pattern?
2Explore
This activity needs a bigger screen
Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.
- Press Drop 1 a few times. Can you predict where a single ball will land?
- Press Add 1000 at once. What shape do the bins make? Does the red curve fit?
- Change the chance of bouncing right to 0.7. What happens to the mean?
- Switch to Bell curve. Make σ small, then large. What changes, and what stays the same?
- Drag x to one standard deviation above the mean. What percent of values are below it?
3Explain
Many small random pushes
At every peg a ball goes left or right by chance. Its final bin is the result of many small random pushes added together.
A bell-shaped, symmetric distribution described by its mean μ and standard deviation σ. About 68% of values lie within 1σ of μ, 95% within 2σ and 99.7% within 3σ.
4Practice
Heights are normal with mean 160 cm and SD 8 cm. About what percent of people are between 152 and 168 cm?
5Recap
- Adding up many small random effects produces a bell curve.
- The normal distribution is symmetric around its mean μ; its spread is the standard deviation σ.
- 68–95–99.7: the share of values within 1, 2 and 3 σ of the mean.
- A z-score z = (x − μ)/σ says how unusual a value is.