Before the method: the idea behind it
The shape that keeps appearing
Measure the heights of a thousand people, the weights of a thousand loaves, or the errors in a thousand instrument readings, and the same bell shape emerges each time. It is not a coincidence, and the reason it happens explains why this one distribution dominates statistics, and why the rest of your course depends on it.
Part one
Why so many things are normal
A person’s height is not decided by one thing. It is the accumulated result of many genes, nutrition, sleep, illness and chance, each nudging the total up or down a little.
Whenever a quantity is the sum of many small independent influences, the extremes require an unlikely run of nudges in the same direction, while middling values can happen countless ways. The result is a distribution that is dense in the middle and thins out symmetrically towards both tails: the normal distribution.
That is why it describes naturally occurring variation so well. It is also why it does not fit everything. Incomes are not normal, because a few people earn enormously more than the median and the distribution has a long right tail. Nor are quantities with a hard floor near the mean, like the time taken to run 100 m. Recognising when the model is inappropriate is examined just as often as using it.
| Feature of a normal distribution | What it means |
|---|---|
| Symmetric about μ | Mean, median and mode are all equal |
| Bell-shaped | Dense in the middle, thinning towards both tails |
| Points of inflection at μ ± σ | The standard deviation is visible on the graph |
| Total area 1, never touching the axis | It is a density function with infinite domain |
Part two
Two numbers control everything
The mean μ says where the curve sits; the standard deviation σ says how wide it is. Nothing else varies.
Drag the two sliders. Notice that the inflection points, where the curve stops bending one way and starts bending the other, always sit exactly one standard deviation either side of the mean. That is what makes σ something you can literally see on the graph.
Changing μ slides the curve. Changing σ makes it wider and flatter, or narrower and taller, but the area underneath is always exactly 1.
The area staying fixed at 1 is why a wider curve must also be shorter. A large σ spreads the same total probability over a broader range, so no single region holds much of it. That is precisely what “more variable” means.
Part three
The 68, 95, 99.7 rule
Because every normal curve has the same shape, the proportion within a given number of standard deviations is always the same.
About 68% of all values lie within one standard deviation of the mean.
These three numbers are worth memorising, because they let you answer many questions without a calculator, and they give you a sanity check on every calculator answer. If a question asks for the probability within two standard deviations and your CAS returns 0.75, something is wrong.
The symmetry also does a lot of work. Since the halves match, the probability below the mean is always exactly 0.5, and any tail beyond μ + kσ has the same size as the tail below μ − kσ.
Part four
One curve to rule them all
Every normal distribution can be converted into a single standard one, with mean 0 and standard deviation 1.
Subtracting the mean shifts the curve to be centred on zero; dividing by the standard deviation rescales it to have width 1. The result is called the standard normal distribution, written Z ~ N(0, 1), and the converted value is called a z-score.
A z-score answers a single question: how many standard deviations from the mean is this value? A z-score of 2 means two standard deviations above the mean, whatever the original units. That makes it a universal measure of how unusual a value is, so you can compare a height with an exam mark.
Before calculators, standardising was essential: a single table of standard normal probabilities could serve every normal distribution. Your ClassPad works with any μ and σ directly, so you often will not need it. But z-scores remain essential for two things: comparing values from different distributions, and finding an unknown μ or σ when a probability is given.
Where this shows up in real life
Quality control and tolerances
Manufacturers specify tolerances in standard deviations. The phrase “six sigma” means a process so tightly controlled that defects lie six standard deviations from the mean, which corresponds to about two failures per billion.
Standardised testing and growth charts
ATAR scores, IQ scores and paediatric growth percentiles all report where you sit on a normal curve rather than a raw score. A child on the 95th percentile for height is about 1.64 standard deviations above the mean, computed exactly as in this lesson.
Before you start
What to reach for
| The question asks for… | Use |
|---|---|
| a probability, given μ and σ | Normal CD with the correct bounds |
| a value with a given probability below it | Inverse Normal |
| a probability at 1, 2 or 3 standard deviations | The 68, 95, 99.7 rule, no calculator needed |
| how unusual a value is, or a comparison across contexts | z-score |
| an unknown μ or σ, given a probability | Inverse Normal for the z-score, then solve z = (x − μ)/σ |
| whether the normal model is suitable | Check continuity, symmetry, and the 68, 95, 99.7 proportions |
Next: the working method, twenty practice questions, ten harder ones, and the ClassPad guide.