1The idea in one paragraph

Statistics answers questions about a group, the population, by measuring some of its members, the sample. That trade is only worth making if the sample behaves like the population, and it only does that if chance, not a person's convenience, chose it. So before any mean or graph, ask four things: who the data are about, how the members were chosen, what kind of values were recorded, and whether anything in the list is missing, impossible or extreme. Random methods (simple random, systematic, stratified) protect you against bias; non-random ones (quota, convenience) invite it. A value far from the rest, an outlier, is tested with a precise rule, then judged in context: it may be the most interesting fact in the data, or a typing slip.

2Populations and samples

The population is the whole set of people or things you want to draw a conclusion about. The question defines it, not convenience: if a head teacher asks how long her students sleep, the population is every student in the school, not the students in one tutor group.

Measuring every member is a census. Sometimes it is possible, but usually it is too slow, too expensive, or impossible because measuring destroys the item: a factory that tests every match by striking it has nothing left to sell. So we measure a sample, a subset of the population, and use what we find to estimate the same quantity for the whole population. Figure 1 shows the trade.

A random sample is one chosen by a process in which chance alone decides who is included, so that every member of the population has the same chance of being chosen. Randomness matters because any human choice lets preferences leak in: a teacher asked to "pick a few typical students" will pick the ones she thinks of first, and those are not typical. A random sample does not guarantee a perfect match with the population; an unlucky draw can happen. What it guarantees is that the errors come from chance rather than from a built-in lean, and chance errors shrink as the sample grows.

One exam convention to know now: at SL, the data set in a question is treated as the population unless the question says otherwise. This matters in 4.3, when your GDC offers two standard deviations and you must pick the right one.

3Discrete and continuous data

Discrete data can take only particular, separate values, usually because they are counted: the number of siblings, goals in a match, cars in a car park. Shoe sizes (7, 7½, 8) are discrete too, even though they are not whole numbers, because nothing lies between size 7 and size 7½.

Continuous data are measured, and can take any value within a range: time, mass, length, temperature. The value you record is rounded to whatever your instrument can show, so a sprint time written as 13.47 s really means "somewhere between 13.465 s and 13.475 s". Figure 2 puts the two side by side.

The quick test: between any two possible values, is there always another possible value? If yes, the data are continuous. Two cases need a sentence of judgement. Age is continuous (you grow older every second), but "age in completed years" is recorded like discrete data. Money is strictly discrete, since it moves in cents, but large amounts such as salaries are usually treated as continuous. If a question hinges on it, say which you are assuming and why.

The distinction decides how the data are presented. Continuous data are grouped into class intervals written as inequalities with no gaps, such as 20 ≤ t < 25, so every value has exactly one home. That is 4.2.

4Reliability of data sources, and bias in sampling

A data source is reliable when you have good reason to believe its figures describe what they claim to. Figure 3 gives the six questions to ask of any data set, in an exam or in a newspaper.

Bias is a systematic tendency for the sample to differ from the population in one direction. It is not bad luck, and that is why a bigger sample does not cure it: a biased method applied to 10,000 people gives a very precise wrong answer. The common sources are these.

• Selection bias. The list the sample is drawn from, the sampling frame, leaves some groups out. An online survey cannot reach people without internet access.
• Non-response bias. The people who reply differ from those who do not. Busy people rarely fill in long questionnaires.
• Self-selection. Volunteers choose themselves, so only people with strong views take part in a phone-in poll.
• Response bias. The question or the setting pushes the answer: "Do you agree the canteen is overpriced?" invites yes, and people overstate how much exercise they take.

The guide points to a famous real case. Before the 1936 US presidential election, the magazine The Literary Digest mailed about ten million straw-poll ballots to names taken largely from telephone directories, car registrations and its own subscriber list. More than two million came back, and they pointed to a clear win for Alf Landon. Franklin Roosevelt won in a landslide. In 1936 telephones and cars were far more common in wealthier households, which leaned towards Landon, and the people who bothered to return a ballot were not typical either. George Gallup, with a far smaller but better-chosen sample, called the winner correctly. Two million responses could not rescue a biased frame.

A modern version: the city of Boston tried a smartphone app, Street Bump, that logged jolts as drivers went over potholes. Reports could only come from people who owned smartphones and installed the app, so there was a real risk of repair crews being sent disproportionately to better-off districts. The data were plentiful, and still not representative.

5Missing data and errors in recording

Real data arrive with gaps and mistakes. Two rules handle almost everything.

A missing value is left out, never written as 0. An impossible or clearly mistyped value is corrected from the original record if you can, removed if you cannot, and you say what you did.

A PE teacher records twelve students' 400 m times in seconds (invented data): 72.4, 68.1, 75.0, 81.3, 0, 70.6, 7.25, 77.9, 69.8, 74.2, 73.5, 79.5. The 0 is a student who was absent. The 7.25 is impossible for 400 m (a world-class time is over 40 s), and looks like 72.5 with the decimal point in the wrong place.

The raw mean of 62.5 s describes no student in the class. Either cleaned figure is honest, provided you state that n changed and why. Never "fix" a value by guessing without evidence: if the original sheet is lost, remove the 7.25 rather than invent a replacement.

6Outliers: the rule, then the judgement

An outlier is a data item more than 1.5 × IQR from the nearest quartile, where the interquartile range IQR = Q₃ − Q₁ is the width of the middle half of the data.

x is an outlier if x < Q₁ − 1.5 × IQR or x > Q₃ + 1.5 × IQR.

The two values Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR are called the fences. Everything between them is ordinary.

Worked example. Fifteen employees of a small firm record their commuting times in minutes (invented data), already in order:

12, 14, 15, 17, 18, 19, 20, 21, 22, 22, 24, 25, 27, 30, 58

With n = 15 the median is the 8th value, 21. The lower half is the seven values below it, and its median is Q₁; the upper half is the seven values above it, and its median is Q₃. This is the method most GDCs use.

Figure 4 draws the fences. Notice that they are measured from the quartiles, not from the median.

The rule tells you a value is unusual. It does not tell you what to do with it. That needs context.

• A valid outlier is a genuine member of the population. The 58-minute commuter probably lives in the next town. Removing that person would make the firm's commuting look shorter than it is, so keep the value and mention it.
• An error is a value that could not be true, or that has a clear recording explanation: 580 minutes, a height of 17.2 m, a mass typed in grams among masses in kilograms. Correct it from the source, or remove it and say so.

Outliers matter because they pull some measures much harder than others. With the 58 the mean commute is 22.9 minutes; without it, 20.4 minutes. The median moves only from 21 to 20.5. You will use this in 4.3 when choosing which average to report, and in 4.2, where a box-and-whisker diagram marks each outlier with a cross.

7Five sampling techniques

Take one invented setting for all five. A tennis club has 480 members: 144 juniors, 264 adults and 72 seniors. The committee wants the views of 40 members on new opening hours.

Simple random sampling. Number the members 1 to 480, then let a random number generator choose 40 different numbers, ignoring repeats. Every possible group of 40 is equally likely to be chosen. It needs a complete list of the population, and by bad luck it can under-represent a small group such as the seniors.

Systematic sampling. Take the list in some order and choose every kth member, where k = population size ÷ sample size.

It is quick and spreads the sample through the whole list. Its weakness is a list with a repeating pattern: if the club's list happened to run in blocks of 12 by court booking slot, every chosen member could come from the same slot. Only the start is random, so it is not a simple random sample: members 5 and 6 can never both be chosen.

Stratified sampling. Split the population into strata, non-overlapping groups that might answer differently (here, the age groups), then take a random sample from each stratum in proportion to its size.

Within each stratum the members are chosen at random, for example by simple random sampling. When the proportions do not come out whole, round to the nearest whole number and check the total. A college with 680 first-year and 560 second-year students wanting a sample of 50 gets 27.4 and 22.6, so 27 and 23, which total 50. Stratified sampling is usually the most representative method, because every group appears in its true proportion. The cost is that you must know which stratum each member is in and how big each stratum is.

Quota sampling. The collector is given a quota for each group, 12 juniors, 22 adults and 6 seniors, and fills it with whoever is available until each quota is met. It looks like stratified sampling, and the difference is the whole point: in a quota sample, the people within each group are not chosen at random. The adults found at the club at 10 a.m. on a Tuesday are not typical adults. Quota sampling is quick and needs no list, but the collector's choices bring bias.

Convenience sampling. Ask whoever is easiest to reach, such as the first 40 members through the door on Saturday. It is fast and cheap, and it is almost always biased, because the people who are easy to reach share habits. Figures 5 and 6 show all five methods on the same group of 40.

How effective is each? A sampling method is effective when it is likely to give a representative sample at a cost you can bear.

When a question asks you to "comment on the effectiveness" or "suggest an improvement", name the flaw, say which way it could push the result, and name a random method that removes it.