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Master: Distributions and the normal curve

Statistics and data · step 4 of 5

A question to start

Measure the heights of a thousand adults, sort them into groups (150 to 155 cm, 155 to 160 cm, and so on) and draw a bar for each group. You get a hill: low at both ends, high in the middle, and nearly the same on both sides. Do the same with the weights of a thousand apples, or with the errors in a thousand careful measurements, and the same hill appears again. Why does this one shape keep turning up, and what can you do with it?

The idea

Distributions and histograms

The distribution of a set of data tells you which values occur and how often. For numbers that can take many values, such as heights, you draw it with a histogram: split the number line into equal intervals (called bins or classes), count how many values fall in each, and draw touching bars. Unlike the bar chart in Play, the bars touch, because the number line has no gaps.

Shapes of distributions

  • Symmetric: the left and right sides are mirror images. The mean and median are about the same.
  • Skewed right (positively skewed): a long tail stretches out to the right. Incomes are the classic example: most people earn a moderate amount, and a few earn enormous amounts. The few huge values pull the mean above the median.
  • Skewed left: a long tail to the left, for example scores on a very easy test, where most people do well and a few do badly. The mean is below the median.
  • Bimodal: two humps. This often means two different groups have been mixed together.

So the shape tells you which average to trust, which connects back to the outliers of Explore.

The normal curve

Many measurements that come from adding up lots of small, independent effects (a person's height depends on many genes and on many years of meals) have a distribution close to one perfect, smooth, symmetric shape: the normal distribution, also called the bell curve.

A normal distribution is completely described by two numbers: its mean \(\mu\) (the Greek letter mu), which sets the centre, and its standard deviation \(\sigma\), which sets how wide it is. The curve's equation is

\[f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}}\]

You will not need to use this formula here. What matters is that the area under the curve between two values equals the fraction of the data between those values. The total area is exactly 1, which is all of the data.

The 68–95–99.7 rule

For every normal distribution, whatever its mean and standard deviation:

  • about 68% of the values lie within 1 standard deviation of the mean (between \(\mu - \sigma\) and \(\mu + \sigma\));
  • about 95% lie within 2 standard deviations;
  • about 99.7% lie within 3 standard deviations.
68%13.5%13.5% μ−3σμ−2σμ−σ μμ+σμ+2σμ+3σ

Because the curve is symmetric, you can split these numbers in half. Between \(\mu\) and \(\mu + \sigma\) is 34%. Between \(\mu + \sigma\) and \(\mu + 2\sigma\) is \((95 - 68) \div 2 = 13.5\%\). Above \(\mu + 2\sigma\) is \((100 - 95) \div 2 = 2.5\%\).

z-scores: how many standard deviations away

The z-score of a value says how many standard deviations it is above (positive) or below (negative) the mean:

\[z = \frac{x - \mu}{\sigma}\]

A z-score of 2 means "2 standard deviations above the mean", wherever the data came from. That lets you compare things measured on completely different scales. For values that are not whole numbers of standard deviations, a z-table (or a calculator) gives the fraction below any z. A few useful values, to one decimal place:

\(z\) −2 −1 0 0.5 1 1.5 2 3
% of values below \(z\) 2.3 15.9 50 69.1 84.1 93.3 97.7 99.9

The percent of values below a given value is its percentile. A z-score of 1 is about the 84th percentile.

Worked examples

Suppose the heights of the adults in a town are normally distributed with mean \(\mu = 170\) cm and standard deviation \(\sigma = 8\) cm. (These are made-up but realistic numbers.)

1. What fraction are between 162 cm and 178 cm? \(162 = 170 - 8\) and \(178 = 170 + 8\), so this is within 1 standard deviation of the mean: about 68%.

2. What fraction are taller than 186 cm? \(z = \dfrac{186 - 170}{8} = 2\). By the 68–95–99.7 rule about 2.5% are above \(z = 2\); the table gives the more exact \(100 - 97.7 = 2.3\%\).

3. How tall must you be to be in the tallest 16%? 16% above means 84% below, which is \(z = 1\) in the table. So \(x = \mu + 1 \times \sigma = 170 + 8 = 178\), that is 178 cm. In general, turning a z-score back into a value uses \(x = \mu + z\sigma\).

4. Comparing two tests. Amir scored 80 on a history test where the class mean was 70 and the standard deviation was 5. Bea scored 85 on a science test where the mean was 75 and the standard deviation was 10. Who did better compared with their class?

  • Amir: \(z = \dfrac{80 - 70}{5} = 2\).
  • Bea: \(z = \dfrac{85 - 75}{10} = 1\).

Both scored 10 points above their class mean, but Amir's 10 points were 2 standard deviations, far rarer than Bea's 1. Relative to their classes, Amir did better: about the 98th percentile against about the 84th.

Try it

A test is scaled so that scores are normally distributed with mean 100 and standard deviation 15.

  1. About what percent of people score between 85 and 115? About what percent score above 130?
  2. What is the z-score of a score of 112?
  3. About what percent score between 100 and 145?
  4. On a different test the standard deviation is 6, and a score of 64 has a z-score of −2. What is the mean?
  5. House prices in a city are strongly skewed right. Which is likely to be larger, the mean price or the median price? Which would you quote as the "typical" price?
Answers
  1. 85 to 115 is \(\mu - \sigma\) to \(\mu + \sigma\), so about 68%. 130 is \(\mu + 2\sigma\), so about 2.5% score above it (2.3% from the table).
  2. \(z = \dfrac{112 - 100}{15} = \dfrac{12}{15} = 0.8\).
  3. 145 is \(\mu + 3\sigma\). Within 3 standard deviations on both sides is 99.7%, so from the mean up to \(\mu + 3\sigma\) is half of that: about 49.85%, or roughly 50%.
  4. \(x = \mu + z\sigma\) gives \(64 = \mu + (-2)(6) = \mu - 12\), so \(\mu = 76\).
  5. The mean is larger, because a few very expensive houses pull it up. The median is the better "typical" price: half of the houses cost less and half cost more.

Common mistake

Using the 68–95–99.7 rule on data that is not normal. The rule is a property of the bell curve. For skewed data, such as incomes or the time people wait in a queue, the percentages can be badly wrong, and the mean plus or minus two standard deviations might even include impossible negative values. Draw a histogram first and look at its shape before you reach for the normal curve.

Where it leads