1The idea in one paragraph

Weigh 500 loaves from one bakery line and draw the histogram: most are close to the target, a few are a little heavy or light, very few are far off, and the bars make a symmetric bell. The normal distribution is the smooth curve that models that bell. It is fixed by two numbers, the mean μ, which says where the centre is, and the standard deviation σ, which says how spread out it is, and we write X ~ N(μ, σ²). For a quantity like mass, which can take any value in a range, probability is area under the curve: the chance a loaf weighs between 790 g and 820 g is the area above that interval. Your GDC finds those areas, and it also works backwards, from an area to the boundary that produces it. Almost every mark on this page comes from setting the question up correctly, and a quick sketch is how you do that.

2A curve for measured quantities

Figure 1 shows a histogram of 500 loaf masses with a normal curve over it. The data are real-looking and a little ragged; the curve is the tidy model. The bars are drawn as frequency densities, so the area of each bar is its number of loaves. Scale everything down so the total area is 1 instead of 500, and area becomes proportion, which is probability.

This is the key change from 4.8. A binomial variable is discrete: it can be 3 or 4, never 3.6, and each value has its own probability, drawn as a bar. The normal distribution models a continuous variable, one that can take any value in an interval: a mass, a time, a length. For a continuous variable:

• probability is area under the curve, and the whole area is 1;
• the probability of one exact value is 0, because a line has no width. P(X = 800) = 0, which says that no loaf weighs exactly 800.000… g;
• so P(X < 780) and P(X ≤ 780) are the same number. Including or excluding the end point changes nothing, which is the opposite of the binomial, where it changes everything.

Where the normal distribution occurs. Many measured quantities come out roughly normal: the masses of items filled or baked by a machine set to a target, the heights of adults of one sex in one population, the lengths of leaves from one species of tree, repeated measurements of the same quantity in a laboratory, where the random errors scatter symmetrically around the true value. What these have in common is that each value is the result of many small, independent influences, some pushing up and some pushing down. Most of the time they roughly cancel, which is why values bunch near the middle; now and then most push the same way, which makes the thin tails.

Many quantities are not normal. Incomes and house prices have a long tail to the right: a few very large values and none below zero. Waiting times are also skewed. Section 8 shows how to tell when a normal model is wrong.

The curve has a history worth knowing. In the 1730s Abraham de Moivre found it as an approximation to binomial probabilities when the number of trials is large; you can already see the bell forming in the binomial bar charts of 4.8 when p = 0.5. A century later the Belgian statistician Adolphe Quetelet fitted it to human measurements and built from it the idea of l'homme moyen, the "average man". His enthusiasm was also a warning: treating the average as the ideal, and every departure from it as an error, is a use of the model that the mathematics does not justify.

3The properties of the normal curve

Figure 2 labels the shape. For X ~ N(μ, σ²):

• It is bell-shaped, with a single peak at x = μ.
• It is symmetric about the vertical line x = μ, so the left half is the mirror image of the right. Exactly half the area lies on each side: P(X < μ) = P(X > μ) = 0.5.
• Because of the symmetry, the mean, the median and the mode are all equal to μ.
• The total area under the curve is 1.
• The tails get ever closer to the horizontal axis but never touch it, so in the model any value is possible, even though values far from μ are very unlikely.
• The curve changes from bending downwards to bending upwards (its points of inflection) at x = μ − σ and x = μ + σ. That is a good guide for sketching: the "shoulders" of the bell are one standard deviation from the centre.

Notation. In N(μ, σ²) the second number is the variance, σ². So X ~ N(800, 144) and X ~ N(800, 12²) are the same statement, and the standard deviation is 12. The calculator wants σ, not σ².

What μ and σ do. Figure 3 changes one at a time. Changing μ slides the whole curve left or right without changing its shape. Changing σ changes the spread: a larger σ makes the curve wider and, because the area must stay 1, lower. A small σ makes it tall and narrow.

4The 68–95–99.7 rule

For every normal distribution, whatever μ and σ are, the same proportions lie within the same number of standard deviations of the mean. You need to know these three:

About 68% of values lie within μ ± σ, about 95% within μ ± 2σ, and about 99.7% within μ ± 3σ.

Figure 4 draws them for the loaves, X ~ N(800, 12²). About 68% weigh between 788 g and 812 g, about 95% between 776 g and 824 g, and almost all, 99.7%, between 764 g and 836 g.

Combine the rule with symmetry and you can split the areas further, as the figure does. The central 68% splits into 34% each side of μ. The band between one and two standard deviations holds (95 − 68) ÷ 2 = 13.5% on each side. Beyond two standard deviations, (100 − 95) ÷ 2 = 2.5% on each side.

Worked example 1 (no calculator). For X ~ N(800, 12²), estimate:

These are approximations; the GDC gives 0.159, 0.0228 and 0.819. The rule is for estimates, for checking a calculator answer is sensible, and for Paper 1. It also gives a quick test of whether data look normal: if far fewer or far more than about two-thirds of a data set lie within one standard deviation of its mean, the normal model is doubtful.

5Symmetry: the Paper 1 method

On Paper 1 you have no calculator, so a normal question will hand you one probability and ask for another. Symmetry is the whole method, and a sketch makes it safe.

Worked example 2. X ~ N(800, 12²) and P(X < 785) = 0.106. Find P(785 < X < 815).

785 is 15 g below the mean and 815 is 15 g above it, so by symmetry the two tails are equal, as Figure 5 shows.

Two more facts from symmetry come up constantly. P(800 < X < 815) is half of the middle piece, 0.394. And P(X < 815) = 1 − 0.106 = 0.894. Before you use symmetry, check that the two values really are the same distance from μ; a question that gives P(X < 785) and asks about 820 cannot be done this way.

6Probabilities with technology

On Paper 2 the GDC finds any area. Every model has a normal cdf function that takes a lower bound, an upper bound, μ and σ, and returns the area between them. On a TI-84 Plus it is normalcdf(lower, upper, μ, σ); on a Casio it is Normal CD. For a tail that goes on for ever, use a very large number as the missing bound, such as 10⁹⁹ (typed 1E99 on a TI) or −10⁹⁹.

The routine that earns full marks is always the same: sketch, shade, write, calculate.

Worked example 3. Loaf masses X ~ N(800, 12²). Find the probability that a loaf chosen at random weighs (a) less than 780 g, (b) between 790 g and 820 g, (c) more than 825 g.

Figure 6 is the sketch for (b). It catches the two commonest errors: shading the wrong side, and an answer that could not possibly be right. An area that is clearly more than half the curve cannot be 0.25.

From a probability to a number of items. If 600 loaves are baked, the expected number under 780 g is 600 × P(X < 780) = 600 × 0.047790… = 28.7. As in 4.5, an expected number need not be a whole number, so the answer is 28.7, not 29.

7Inverse normal: from an area back to a value

Sometimes you know the area and want the boundary: "find the mass exceeded by only the heaviest 10% of loaves". That is an inverse normal calculation. The GDC's inverse normal function (invNorm on a TI-84, Inverse Normal on a Casio) takes an area, μ and σ, and returns the value of x with that much area to its left. At SL, μ and σ will always be given in an inverse normal question; finding an unknown μ or σ needs z-values and is 4.12.

The one step that needs thought is turning the question's area into an area to the left, and Figure 7 shows the two common cases.

Worked example 4. X ~ N(800, 12²).

(a) The heaviest 10% of loaves weigh more than k grams. Find k.

(b) Find the interval, symmetric about the mean, that contains the middle 90% of masses.

Check each answer against the sketch. The top 10% boundary must be above the mean, and it is. The two ends of the middle interval must be the same distance from 800, and they are: 19.7 g each side.

Joining the normal to the binomial. A Section B question often goes one step further. Suppose loaves are sold in packs of 6, chosen at random. What is the probability that at least one loaf in a pack weighs less than 780 g?

Carry the full calculator value of p into the second stage. Rounding it first to 0.05 would give 0.265, and the third figure would be wrong.

8When to trust the model

A normal model always gives an answer, so the thinking has to come from you. Three checks catch most misuse.

1. Is the variable continuous and measured? Masses, lengths and times, yes. Counts of people or goals, not in their own right.
2. Is the data roughly symmetric and bell-shaped? Look at a histogram, and compare the mean and median: if they are far apart, the data are skewed and the normal model is poor.
3. Does the model give impossible values much probability? The normal curve never ends, so it always gives some probability to values below zero. When μ is many standard deviations above zero, that probability is negligible. When it is not, the model is wrong.

Figure 8 shows the third check failing. A survey of daily screen time found a mean of 3.1 hours and a standard deviation of 2.4 hours. A normal model N(3.1, 2.4²) gives P(X < 0) = 0.0982: nearly one person in ten would spend a negative number of hours on screens. The real data are skewed to the right, so a normal model does not suit them. A quick rule: if μ − 2σ is below a value that is impossible (here, 3.1 − 4.8 = −1.7 hours), be suspicious.

Be most careful in the tails. A model fitted to the middle of a data set says least about the extremes, yet extremes are often what matters, such as the strongest flood a river defence must hold. Far out in the tails there are few data, and real data often have heavier tails than the normal curve allows.