The normal distribution
| English | Chinese | Pinyin |
|---|---|---|
| normal distribution | 正态分布 | zhèng tài fēn bù |
| z-score | 标准分 | biāo zhǔn fēn |
The shape that keeps appearing
- Heights, exam scores, measurement errors, the total of many small independent effects — all pile up into the same symmetric hump.
- The normal distribution 正态分布 is that shape, and it is described entirely by two numbers.
- Those two are the mean and the standard deviation you met in the last unit.
How many numbers are needed to describe a normal distribution completely?
The mean fixes where it sits and the standard deviation fixes how wide it is.
The empirical rule
- About 68% of values lie within one standard deviation of the mean.
- About 95% lie within two, and about 99.7% within three.
- Those three numbers answer most normal-distribution questions in this module without a table.
What percentage of values lie within two standard deviations of the mean?
68 within one, 95 within two, 99.7 within three. These three answer most questions here.
The z-score
- A z-score 标准分 says how many standard deviations a value sits from the mean.
- It is what makes two different scales comparable: a $z$ of 1.5 means the same thing on a maths test and on a height chart.
- Negative $z$ is below the mean; that sign carries meaning and dropping it changes the answer.
A test has mean 62 and standard deviation 8. What is the z-score of 78?
(78 − 62) ÷ 8 = 2, so the score is two standard deviations above the mean.
A test has mean 62 and standard deviation 8. What proportion score above 78?
Two standard deviations above the mean. By the empirical rule 95% lie within two either side, so 5% lie outside — and by symmetry half of that, 2.5%, is above.
About one student in forty. Halving the 5% is the step students skip, and it doubles the answer.
What percentage score above that value? Give the percentage.
5% lies outside two standard deviations, and symmetry puts half of it above: 2.5%.
Put a normal-distribution question in the order that avoids a tail error.
The sketch is what tells you whether the answer should be small or large before you trust the arithmetic.
Sketch the curve and shade what you want, before calculating. The sketch tells you whether the answer should be near 2%, near 50% or near 98%, and it catches a tail-direction error that no amount of careful arithmetic will.
Not everything is normal. Income is not, and neither is anything strictly positive with a long right tail. Applying the empirical rule to skewed data gives confident nonsense, and questions in this module sometimes supply skewed data precisely to see whether you notice.
The empirical rule can be applied to any data set.
It assumes a normal shape. Income and other skewed data break it, and the answer looks confident and is wrong.