Mathematical models in probability and statistics

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Statistics 1 (WST01/01) is a single 90-minute paper of 75 marks, and its first topic is the idea that sits under the whole unit: a statistical model.

The modelling point itself is worth only a mark or two when it is asked directly, but every later topic (a histogram, a probability, a regression line, the Normal distribution) is a model of something, and the closing parts of S1 questions often ask whether a model is suitable.

This section explains what a model is, why statisticians use one, and how a model is tested and improved.

How the marks work

M1 is a method mark: a correct method with this question's own numbers in it, where small slips are forgiven. dM1 can only be earned if the M mark before it was.

A1 is an accuracy mark for a correct answer, and it needs its M mark. A1* marks an answer printed in the question: every line must be shown and correct. cso means correct solution only.

B1 stands on its own: a value read from a diagram, or a statement with its reason. B1ft and A1ft (follow through) keep the mark when you use your own earlier wrong answer correctly.

awrt means "anything which rounds to" the printed value; oe means "or equivalent", so a fraction or a decimal both score. isw (ignore subsequent working): a correct answer is not undone by later work.

A misread that keeps the question's character costs two A or B marks, not the whole question.

S1 habits: inexact answers to 3 significant figures unless the stem says otherwise, every sum written out before it is evaluated, and every comment in context with figures.

Formulae for this unit

Learn it: mean $= \bar{x} = \dfrac{\sum x}{n}$ or $\dfrac{\sum fx}{\sum f}$ ; standard deviation $= \sqrt{\text{variance}}$ ; interquartile range $= \mathrm{IQR} = Q_3 - Q_1$ . These are formulae you are expected to know, and they are not printed in the booklet.

Learn it: the variance of data, $\dfrac{\sum (x - \bar{x})^{2}}{n} = \dfrac{\sum x^{2}}{n} - \bar{x}^{2}$ (or $\dfrac{\sum fx^{2}}{\sum f} - \bar{x}^{2}$ for a frequency table). The must-know formula list gives standard deviation $= \sqrt{\text{variance}}$ but does not list this formula itself; every recent S1 paper needed it, so learn it.

Learn it: frequency density $= \dfrac{\text{frequency}}{\text{class width}}$ , and the coding rules in the coding section of this unit. Neither is on the must-know formula list, but the papers expect both without prompting.

Given in the question: any rule for outliers and any measure of skewness. You do not have to remember an outlier rule: "Any rule to identify outliers will be specified in the question."

The idea: a simplified description of the real world

A mathematical model is a description of a real situation in mathematical terms: an equation, a probability distribution, a diagram with numbers on it.

A statistical model describes something that varies.

The heights of seedlings in a tray, the number of customers arriving in an hour and the score on a spinner all vary from one observation to the next, and a statistical model says how they vary: which values are likely, which are rare, and what the typical value and spread are.

Every model leaves things out.

"A fair spinner with five equal sectors" ignores the friction in the pin, the way it is flicked and the slight unevenness of the card.

That is not a weakness to apologise for; it is the point.

A model with every detail in it would be as complicated as the real thing and would teach you nothing.

A good model keeps what matters for the question being asked and drops the rest, so that you can reason with it.

So a model is judged by one test: do its predictions agree well enough with what actually happens? Not perfectly, because real data always vary; well enough for the purpose.

Why statisticians use models

These are the reasons a mark scheme accepts, in plain words:

A model simplifies a real-world situation, so that it can be analysed. This was the most common correct answer in a recent series.

A model improves understanding of a real situation: it picks out which features matter.

A model lets you describe and analyse a situation, including one that is hard, slow, dangerous or expensive to observe directly (the spread of a disease, the failure of a component).

A model lets you test ideas: change one assumption and see what the model predicts, without changing the real system.

A model can be used to make predictions, and can be refined when the predictions and the data disagree.

Two answers do not score.

A model is not "more accurate" than the real world: it is a simplification, so it is less detailed by design.

And if the question has already given some reasons (for example "models are quick and cheap and give predictions") and asks for one other, repeating those ideas in new words scores nothing; "easy to use" on its own has also been refused.

The modelling cycle

Building a model is not a single step.

It is a loop, shown in the diagram below.

A flow diagram of the statistical modelling cycle: six boxes joined by downward arrows. 1 Recognise a real-world problem; 2 Make assumptions and devise a model; 3 Use the model to make predictions; 4 Collect data from the real world; 5 Compare the data with the predictions; 6 Accept the model, or refine it. An arrow labelled refine runs from box 6 back up to box 2.

1. Recognise a real-world problem. For example: are the five colours on a fairground spinner equally likely?

2. Make assumptions and devise a model. Assume each sector is equally likely, so each has probability $\frac{1}{5}$ .

3. Use the model to make predictions. In 200 spins, the model predicts each colour about $200 \times \frac{1}{5} = 40$ times.

4. Collect data from the real world. Spin it 200 times and record the colours.

5. Compare the data with the predictions. Are the observed frequencies close to 40?

6. Accept the model, or refine it. If the agreement is good, use the model. If not, change the assumptions (perhaps the sectors are not equal) and go round again.

The comparison in step 5 is a judgement in S1, not a calculation with a rule.

In the data topics "Significance tests will not be expected", so you compare by eye and say whether the differences look like ordinary random variation or like a pattern.

In S2 and S3 you meet hypothesis tests and goodness-of-fit tests that turn this judgement into a calculation.

Why it works: predictions vary even when the model is right

Even a perfect model does not reproduce data exactly.

Spin a genuinely fair five-sector spinner 200 times and the counts will not all be 40: something like 38, 44, 41, 35, 42 is typical.

This is random variation, and it is why a model is judged on whether the differences are small and patternless.

A difference that is large, or that always goes the same way (one colour well above 40 every time you repeat the experiment), points to a wrong assumption.

Later in the unit the standard deviation gives a way to say how big "ordinary" variation is.

Assumptions and limitations

Every model rests on assumptions, and its limitations come from them.

When asked for a limitation, name an assumption and say how reality might differ from it, in context:

The model "each customer arrives independently" fails if customers come in groups.

A model that predicts heights from a curve that never ends allows heights below zero or above any limit, which cannot happen; the model is only useful over the range where it fits.

A model built on last year's data may not hold this year if conditions change.

Connections

Every later topic in S1 is a model of this kind.

A histogram or box plot summarises data so that you can choose a model; probability builds models from sample spaces; a regression line models how one variable depends on another; a discrete random variable and the Normal distribution are probability models whose predictions you compare with data.

The same cycle runs through mechanics (M1 models a car as a particle and the road as smooth), economics and data science, where a model is trained on data, tested on new data and refined.

Key terms

Mathematical model: a simplified description of a real-world situation in mathematical terms, used to analyse it and make predictions.

Statistical model: a model for a quantity that varies, saying which values are likely and how they are spread.

Assumption: a simplifying statement a model takes as true, such as "each sector is equally likely".

Refining a model: changing its assumptions when its predictions do not agree well enough with observed data.

Random variation: the natural difference between observed values and a model's predictions, even when the model is correct.

Worked example 1: comparing observed frequencies with a model

A spinner has five equal sectors coloured red, blue, green, yellow and white.

A model assumes that each colour is equally likely.

The spinner is spun 200 times.

The results are

Colour

red

blue

green

yellow

white

Frequency

(a) Find the frequency of each colour predicted by the model. (1)

(b) Comment on the suitability of the model, giving a reason. (1)

1. (a) Each colour: $200 \times \frac{1}{5} = 40$ B1

2. (b) The observed frequencies are all close to 40 (from 35 to 44) with no colour far above or below, so the model is suitable B1

Answer: 40 each; the model is suitable.

Differences of up to 5 either side of 40, in no particular pattern, are the size of random variation you expect in 200 spins.

Worked example 2: refining a model

A second five-sector spinner is spun 200 times, giving frequencies 21, 30, 47, 52 and 50 for red, blue, green, yellow and white.

(a) State, giving a reason, why the model of equally likely colours is not suitable for this spinner. (1)

(b) Find a refined model, giving the probability of each colour. (2)

1. (a) Red (21) is about half the predicted 40, while yellow and white (52 and 50) are well above it; the differences are large and not random-looking, so the colours are not equally likely B1

2. (b) Use the relative frequencies as probabilities: P(red) $= \frac{21}{200}$ , P(blue) $= \frac{30}{200}$ , P(green) $= \frac{47}{200}$ M1

3. P(red) $= 0.105$ , P(blue) $= 0.15$ , P(green) $= 0.235$ , P(yellow) $= 0.26$ , P(white) $= 0.25$ A1

Answer: the refined model gives the probabilities above, which add to 1.

The next step of the cycle is to test the refined model on a new set of spins, not on the same 200, because those data were used to build it.

Worked example 3: assumptions and limitations in context

A bus is timetabled to arrive at a stop every 12 minutes.

A model for the waiting time of a passenger who arrives at a random time assumes that the buses run exactly to the timetable, so every waiting time from 0 to 12 minutes is equally likely.

(a) Find the mean waiting time and the proportion of passengers waiting more than 9 minutes predicted by the model. (2)

(b) In a survey of 80 passengers, the mean wait was 7.4 minutes and 31 passengers waited more than 9 minutes. Comment on the model in the light of these data, and state which assumption should be refined. (2)

1. (a) Waiting times are spread evenly from 0 to 12, so the mean is $\frac{0 + 12}{2} = 6$ minutes B1

2. Proportion over 9 minutes $= \frac{12 - 9}{12} = 0.25$ , so about $0.25 \times 80 = 20$ of 80 passengers B1

3. (b) Observed: mean 7.4 > 6, and $\frac{31}{80} = 0.3875$ > 0.25; passengers wait longer than predicted, so the model does not fit well B1

4. The assumption that buses run exactly to the timetable should be refined (buses are often late or bunch together, making some gaps longer than 12 minutes) B1

Answer: the model predicts a mean of 6 minutes and 25% waiting over 9 minutes; the data show longer waits, so refine the timetable assumption.

Notice the answer names the assumption and says how reality differs from it, in the bus context.

Exam tip

When a part asks for a reason why models are used, give a different idea from any the stem has already listed, and use the scheme's own kind of phrase: a model "simplifies a real-world situation" or "improves understanding of a real-world problem".

One clear reason earns the mark; a list containing a wrong one can lose it.

And never expect a perfect fit: a model is not wrong just because the counts are not exactly as predicted.

Ask whether the differences are small and patternless.

Common mistakes

Reasons the scheme refuses. In a recent report, "models give more accurate or precise results" and "quick", "cheap", "easy" or "to make predictions" were among the unacceptable answers (the stem had already said models are cheap, quick and predictive). The tempting idea is that a model beats reality; it cannot, because it simplifies. Write about simplifying or understanding the real world instead.

Check your understanding

1. (a) Give one reason for using statistical models. (1)

(b) Explain why "a statistical model is more accurate than the real situation" is not a valid reason. (1)

Total: [2]

2. A model for a six-sided die assumes it is fair. The die is rolled 300 times, with these results.

Score

Frequency

(a) Find the frequency of each score predicted by the model. (1)

(b) Comment on whether the model is suitable. (1)

Total: [2]

3. Two coins are spun together 400 times. A model assumes each coin is fair and the coins land independently.

(a) Show that the model predicts two heads on 100 occasions. (2)

The results were two heads 131 times, one head 190 times and no heads 79 times.

(b) Comment on the model, and state which assumption may be wrong. (2)

Total: [4]

4. The stages of the modelling cycle are listed in a jumbled order: P compare predictions with data; Q make assumptions and devise a model; R collect data; S recognise a real-world problem; T use the model to make predictions. Write down the stages in the correct order. (2)

Total: [2]

5. A student says: "The spinner model in Worked example 1 cannot be right, because none of the five frequencies was exactly 40." Explain whether or not the student is correct. (2)

Total: [2]

6. A café owner models the number of cups of coffee sold each morning, assuming that every customer buys exactly one cup and that customers arrive independently of each other. State one limitation of this model in context. (1)

Total: [1]

Answers

1. (a) Any one of: to simplify a real-world situation; to improve understanding of a real-world problem; to describe or analyse a situation; to test ideas or the effect of changing an assumption B1 (b) A model is a simplification of the real situation, so it leaves out detail and cannot be more accurate than reality B1

2. (a) $300 \times \frac{1}{6} = 50$ each B1 (b) The frequencies run from 44 to 61; most are close to 50 and the differences show no clear pattern, so the model is suitable (a comment that 61 for a four is rather high, but within ordinary variation, is also fine) B1

3. (a) P(two heads) $= \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ M1; $400 \times \frac{1}{4} = 100$ A1* (b) The model predicts 100, 200, 100 for two, one and no heads; 131 two heads and 79 no heads are far from 100, so the model is not suitable B1; the coins may be biased towards heads (or may not land independently) B1

4. S, Q, T, R, P: B1 for S, Q, T in that order; B1 for R then P to finish. (After P comes "accept or refine", which returns to Q.)

5. Not correct. Even when a model is right, observed frequencies show random variation, so exact agreement is not expected B1; the frequencies 35 to 44 are all close to 40 with no pattern, so the data support the model B1

6. For example: some customers buy more than one cup, or none; or customers often arrive in groups (friends, office workers), so arrivals are not independent B1 (any limitation tied to an assumption, in the café context)

Mathematical models in probability and statistics

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