CCEA GCSE Statistics

The stages of an investigation

A statistical investigation follows a cycle, and each stage feeds the next.

1. Planning: decide what question to answer, what data would answer it, where the data will come from and how much is needed.
2. Collecting data: gather the data by a method that suits the question, and record it in a form that can be processed.
3. Processing, representing and analysing the data: clean it, tabulate it, draw diagrams and calculate summary values such as averages and measures of spread.
4. Discussing and interpreting the results: say what the diagrams and calculations show, in the context of the original question, and how far the conclusion can be trusted.
5. Communicating: present the findings in a form the reader can use, which may be written sentences, tables or diagrams, usually a mixture.

The stages are not always followed in a straight line. Looking at the first results often shows that the question was too vague, or that a measurement was taken badly, and the investigation goes back to planning with an improved method. A good conclusion also ends with suggestions for what could be done better if the investigation were repeated.

Questions and hypotheses

An investigation starts from a question ("Do pupils who travel further to school arrive later?") or from a hypothesis, which is a statement that can be tested with data. A hypothesis is most useful when it names the things being compared and says which way the difference is expected to go, so that the data can support it or count against it.

• "Pupils in Year 12 sleep for fewer hours on a school night than pupils in Year 8."
• "Households with more bedrooms spend more on heating."
• "The mean reaction time of the right hand is shorter than the mean reaction time of the left hand."

"Sleep matters to students" is not a hypothesis, because no data could show it to be right or wrong. A hypothesis that says there is a relationship but not which variable goes where ("there is a link between age and sleep") can be tested, but it is weaker than one that states the direction.

Choosing suitable data

Once the question is fixed, the data that would settle it has to be identified. For the sleep hypothesis above the data needed is the number of hours slept on a school night, recorded for pupils in each of the two year groups, plus the year group itself. Data that does not help answer the question is not collected, because it costs time and gets in the way.

The data must also be something that can actually be obtained. Asking people how much they earn is a poor way to investigate income if many will refuse, and a question about last year's spending depends on people remembering accurately.

Population, sampling frame and sample

The population is the whole group the investigation is about. It is defined by the question and need not be people: it could be all the bottles filled by a machine in a day, all the secondary schools in Northern Ireland, or all the lessons in a term. The population for the sleep hypothesis is all pupils in Year 8 and Year 12, at the schools being considered.

A sampling frame is a list of every member of the population from which a sample can be chosen, such as a school register, an electoral register or a list of customer accounts. A good frame is complete, has no repeats and is up to date. For some populations no frame exists, for example everyone who visited a town centre last Saturday, and then random sampling from a list is impossible and another method has to be used.

A sample is the part of the population that is actually studied. Studying a sample is quicker and cheaper than studying everyone, and it is the only option when testing destroys the item (a batch of light bulbs tested until they fail). The price is that the sample may not represent the population, which is why the way it is chosen matters.

A census collects data from the whole population, so there is no sampling error, but it is slow and costly for a large population.

Variables

A variable is a characteristic that can take different values from one member of the population to the next, such as height, shoe size, favourite subject or journey time. The data is the collection of values recorded. An investigation into whether height affects jump distance has two variables: height and jump distance. In the language of data types, each is recorded for every member of the sample, so the data is bivariate; the types of data are set out in a separate note.

Worked example: a population and a frame

A school wants to find out how its Year 11 pupils get to school.

• The population is all Year 11 pupils at the school.
• A suitable sampling frame is the Year 11 register, which lists every pupil once. If there are 180 pupils, the register can be numbered 1 to 180.
• A sample of 30 is chosen from the frame, and the variable recorded for each pupil is the method of travel.

If instead the school wanted to know how all the town's residents travel to work, there would be no complete list of workers to sample from, and the school would need a method that does not depend on a frame.