MISSION · Data and probability
The Normal Distribution
The shape that turns up whenever many small effects add together
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The idea in one sentence
Measure enough of almost anything shaped by many small independent causes and the values pile up in the same symmetric bell, described entirely by its mean and standard deviation.
The shape and what fixes it
The normal curve is symmetric about the mean, highest there, thinning out in both directions and never quite reaching zero. Two numbers pin it down completely:
- The mean \(\mu\) sets where the peak sits.
- The standard deviation \(\sigma\) sets how wide it spreads.
Change the mean and the whole curve slides. Change \(\sigma\) and it gets taller and narrower, or flatter and wider — the area underneath stays at 1, because it accounts for all the data.
The 68–95–99.7 rule
For any normal distribution:
| Within | Contains |
|---|---|
| \(\mu \pm 1\sigma\) | about 68% |
| \(\mu \pm 2\sigma\) | about 95% |
| \(\mu \pm 3\sigma\) | about 99.7% |
This makes “unusual” quantifiable. Adult male heights in the UK are roughly normal with a mean of 175 cm and \(\sigma\) of 7 cm. So about 68% of men are between 168 and 182 cm, and someone at 196 cm is three standard deviations up — in the top 0.15%, about one man in 700.
Only 0.3% of values fall more than \(3\sigma\) from the mean, which is why manufacturing and medicine both treat that as the boundary of routine variation.
The z-score
To compare values from different distributions, convert each to the number of standard deviations it sits from its own mean:
\[z = \frac{x - \mu}{\sigma}\]A test score of 78 where the mean is 70 and \(\sigma\) is 4 gives \(z = 2\). Another test scoring 90 with mean 85 and \(\sigma\) of 10 gives \(z = 0.5\). The 78 is the better performance, despite the smaller number, because it’s further out in its own distribution.
Standardising this way is what lets one table of probabilities serve every normal distribution there is.
Why it appears so often
Because of the central limit theorem: add up many small independent random effects and their total tends to a normal distribution, regardless of how the individual effects were distributed.
Height is influenced by many genes and many environmental factors, each contributing a little. Measurement error is the sum of many tiny disturbances. Averages of samples are literally sums. That’s why the bell turns up in fields that have nothing else in common.
Where it doesn’t apply
Plenty of real data isn’t normal, and assuming it is causes real damage. Incomes are strongly right-skewed — a few enormous values and a long tail. Waiting times can’t be negative, so they’re bounded on one side. Anything with a hard floor or ceiling, or driven by one dominant cause rather than many small ones, will not be bell-shaped.
Look at the data before reaching for the rule.
Common mix-ups
- Applying 68–95–99.7 to skewed data, where it’s simply wrong.
- Confusing the z-score with the probability. A z of 2 isn’t a 2% chance; it’s about the top 2.3%.
- Forgetting the curve is two-tailed. Within \(2\sigma\) is 95%, so each tail beyond is 2.5%, not 5%.
- Treating a normal model as exact. It’s an approximation that’s often excellent and occasionally terrible.
Check yourself
Exam marks are normal with mean 60 and \(\sigma = 8\). What range holds the middle 95%, and what is the z-score of a mark of 76?
The middle 95% is \(60 \pm 16\), so 44 to 76. A mark of 76 has \(z = \frac{76 - 60}{8} = 2\), putting it at the top edge of that range.
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