Statistics From Zero · Lesson 5 of 22
The Bell Curve, Explained at the Bar
In one sentence: When lots of small, random nudges add up, the results pile up in a bell shape around the average, and then the standard deviation tells you exactly how rare anything is.
The idea
Measure lots of things that are each pushed around by many small, random influences, and the results tend to make the same shape: most values bunch up near the average, fewer and fewer appear as you move away from it, and the two sides look like mirror images. Draw it as a chart and you get a hill with a rounded top and long, thin slopes on each side. It looks like a bell, so everybody calls it the bell curve. Your lecturer calls it the normal distribution.
The bell needs only two numbers to describe it completely: the average (where the middle of the bell is, Lesson 2) and the standard deviation (how wide the bell is, Lesson 3). Know those two, and you know how common or how rare any value is.
A barman pours a pint
Uncle Gary's local, a busy Friday. The barman pours pint after pint. A US pint is 16 ounces, but no pint is ever exactly 16. A little more foam here, a moment of distraction there, a slightly tilted glass, a customer shouting an order mid-pour. Each nudge is tiny and random, some push the pint up, some push it down.
Most of the time the nudges roughly cancel out and the pint lands close to 16 ounces. Occasionally several nudges happen to push the same way, and a pint comes out noticeably short or generously full. Very rarely, almost all of them push the same way.
Let's say (this is an example, not a measurement) the barman's pints average 16 ounces with a standard deviation of 0.3 ounces. Line up a thousand Friday pints by size and they form a bell.
Gary, naturally, is convinced that every one of his pints is short.
This is why the bell curve turns up everywhere: it's what you get whenever a result is the sum of many small, independent pushes. Height (many genes, plus food, plus sleep), measurement errors, exam scores on a big well-written test and the weight of bags of chips all come out roughly bell-shaped.
The 68-95-99.7 rule
Here's the payoff. For any bell curve, no matter what it measures:
- About 68% of values land within 1 standard deviation of the average.
- About 95% land within 2 standard deviations.
- About 99.7% land within 3 standard deviations.
Back at the bar. A pint of 15.7 ounces is 1 standard deviation short. About 1 pint in 6 is at least that short, so Gary getting one is completely normal. A pint of 15.4 ounces is 2 standard deviations short: about 1 in 44. On a busy night, a few of those will happen to somebody. A pint of 15.1 ounces is 3 standard deviations short: about 1 in 740. If Gary gets one of those, it might be bad luck. If he gets one every Friday, Gary has a point, and possibly a case.
The same rule works on people. In the US, adult men average roughly 5 feet 9 inches, with a standard deviation of roughly 3 inches. So about two thirds of men are between 5 feet 6 and 6 feet 0, and about 95% are between 5 feet 3 and 6 feet 3. A man of 6 feet 6 is 3 standard deviations above average: taller than all but about 1 in 740. That's why basketball scouts know most of them by name.
A thousand Friday pints
The z-score: how unusual is this?
Counting "how many standard deviations away from the average" is so useful it has its own name: the z-score.
z-score = (your value minus the average) ÷ the standard deviation
A z-score of 0 is perfectly average. 1 is one standard deviation above. A z-score of 2 below average is written as "minus 2" and means unusually low. The magic is that z-scores let you compare things measured in completely different units: pints, inches, exam points, anything. "How many standard deviations away" means the same thing everywhere.
Where the bell curve lies to you
Not everything is a bell. The bell only appears when lots of small pushes add up and none of them dominates. When that's not true, it can fail badly:
- Lopsided data, like incomes or house prices, has a long tail on one side (the billionaire from Lesson 2). There's no mirror image, and the 68-95-99.7 rule doesn't apply.
- Fat tails: in financial markets, very large daily moves turn up far more often than a bell curve says they should. If you use a bell to estimate how rare a crash is, you'll badly underestimate it. That's one reason we always look at the actual history of results instead of trusting a neat formula.
The bell curve is a model: an extremely useful one, and wrong whenever its assumptions are. A good statistician checks the shape of the data before reaching for the rule.
Try it yourself
Cousin Kevin comes home with two exam results. In statistics he got 72; the class average was 60 with a standard deviation of 6. In history he got 80; the class average was 75 with a standard deviation of 10. Both sets of marks were roughly bell-shaped.
Kevin says history went better, because 80 is more than 72. Is he right?
Show the answer
Work out the z-scores.
Statistics: (72 minus 60) ÷ 6 = 12 ÷ 6 = 2. Two standard deviations above the class average. That beats about 98% of the class.
History: (80 minus 75) ÷ 10 = 5 ÷ 10 = 0.5. Half a standard deviation above average. That beats about 69% of the class.
The statistics result is much more impressive, even though the number is lower. The raw mark only tells you the score; the z-score tells you how it compares. (Kevin has asked us to point out that he is now "statistically gifted". We have agreed to say this once.)
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Normal distribution (Gaussian distribution) | The bell curve |
| μ (mu) and σ (sigma) | The average and the standard deviation of the whole population |
| Symmetric | Both sides are mirror images, so the mean, median and mode are all in the middle |
| Empirical rule | The 68-95-99.7 rule |
| z-score (standardised score) | How many standard deviations a value is from the average |
| Standard normal distribution | A bell curve with average 0 and standard deviation 1: the scale z-scores live on |
| Z-table | A lookup table that turns a z-score into "what share of values are below this" |
| Skewed | Lopsided, with a long tail on one side, so not a bell |
| Fat tails (heavy tails) | Extreme values happen more often than the bell curve predicts |
Three things to remember
- Many small, random pushes added together make a bell curve. That's why it turns up so often.
- 68-95-99.7: within 1, 2 and 3 standard deviations of the average. A z-score tells you how many standard deviations away something is.
- Check the shape first. Incomes are lopsided and markets have fat tails, and there the bell curve badly underestimates extremes.
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.