The normal distribution
The bell-shaped normal curve describes heights, measurement errors, test scores, and, most importantly, the behaviour of sample averages.
The normal curve
A normal distribution is symmetric and bell-shaped, centred at its mean \(\mu\), with spread set by its standard deviation \(\sigma\). The inflection points of the curve are one standard deviation from the mean. Every normal distribution is the same shape after rescaling, which is what makes it so convenient.
The 68-95-99.7 rule
In any normal distribution, about 68% of values lie within \(1\sigma\) of the mean, 95% within \(2\sigma\), and 99.7% within \(3\sigma\). If adult heights have mean 170 cm and standard deviation 8 cm, about 95% of adults are between 154 and 186 cm.
68\%,\ 95\%,\ 99.7\%Standardising
Convert any normal value to a z-score, \(z = \frac{x - \mu}{\sigma}\), to use the standard normal distribution, which has mean 0 and standard deviation 1. Tables or software then give areas. For example, \(P(Z < 1.96) \approx 0.975\), and \(P(Z < 0) = 0.5\).
Z = \frac{X - \mu}{\sigma} \sim N(0, 1)Worked example
IQ scores are approximately normal with mean 100 and standard deviation 15. What proportion exceed 130? \(z = \frac{130 - 100}{15} = 2\). About 2.5% of values lie above \(z = 2\) (half of the 5% outside \(\pm 2\sigma\)), so about 2.5% of people have IQ above 130.
Working backwards
To find a cutoff, start with the area and find \(z\), then convert: \(x = \mu + z\sigma\). For the top 10% of IQ scores, \(z \approx 1.28\), so the cutoff is \(100 + 1.28(15) \approx 119\).
Checking normality
Not all data are normal. Check with a histogram (roughly symmetric and bell-shaped?) or a normal probability plot, where normal data fall close to a straight line. Skewed data, such as incomes, should not be analysed with normal calculations directly.
Why the normal matters so much
The central limit theorem, covered in the sampling unit, says that averages of many independent observations are approximately normal, whatever the shape of the original data. This is why normal-based methods appear throughout statistics.
Common mistakes
Applying the 68-95-99.7 rule to skewed data. Forgetting to standardise before using a standard normal table. Confusing 'above' and 'below' areas. Using the variance in place of the standard deviation in the z formula.
Key takeaways
Normal curves are symmetric bells defined by \(\mu\) and \(\sigma\). Use the 68-95-99.7 rule for quick estimates and z-scores for exact areas. Work backwards with \(x = \mu + z\sigma\). Check that data are roughly normal before relying on these methods.
Flashcard
Symmetric bell-shaped curve
Check your understanding
Heights are normal with mean 170 cm and SD 8 cm. About what percent are between 162 and 178 cm?