AP Statistics

What a statistical study does

A statistical study collects data from a sample in order to answer an investigative question about a larger population. A study is needed when the population is too large, or too hard to reach, for every member to be measured. A school cannot ask every adult in a city how they travel to work, so it asks some of them and uses what it finds to say something about all of them.

A single piece of information about one item or individual is a datum, and a collection of them is a data set. The population is all the items or individuals of interest, and its size is written $N$. A sample is the subset of the population from which data are actually obtained, and its size is written $n$. The word "in context" means that every component of a study and every calculation is tied back to the real situation it came from: a mean is not just 14.2, it is the mean number of minutes late for the buses in this depot.

Parts of an investigative question

An investigative question has a defined purpose, and it is fixed before the data are analysed. Changing the question after seeing which result looks interesting would make the analysis meaningless, because an effect found only by searching for one is easy to find by chance. The question must also be posed so that the data it needs can actually be collected and analysed.

A well-formed question has three parts.

1. A part that guides data collection, phrased in terms of the population and the variable to be measured.
2. A part that guides the choice of analysis. For a hypothesis test it makes clear the parameter and the direction of the alternative (not equal, greater than, less than, an association, or not independent). For a confidence interval it makes clear the parameter and that the aim is to estimate it within a range of plausible values.
3. A part that says what kind of conclusion can be drawn: the population to which conclusions apply and, in an experiment that uses random assignment, whether a cause-and-effect conclusion is possible.

For example, "Is the proportion of students at this college who walk to campus greater than 0.30?" names a parameter (a population proportion), a direction (greater than), a population (students at this college) and a variable (whether a student walks).

Observational units and variables

An observational unit is the item or individual from which a datum is collected. The observational units might be people, animals, schools, trees or transactions. A variable is a characteristic that may change from one observational unit to another. If every unit had the same value there would be nothing to study.

Data on variables are not only numbers in a spreadsheet. Photographs, sounds, videos and text, once a characteristic has been recorded from them, are data too.

Parameters and statistics

A parameter is a number that describes the variable of interest for a whole population, such as its mean or the proportion with some property. It is a single, fixed value, usually unknown. A statistic is the same kind of number worked out from a sample. A statistic is known once the sample is taken, but it varies from one sample to another, and its value is usually not exactly the parameter. It is still the basis for making inferences about the parameter.

Suppose 400 of a company's 5,000 employees are surveyed, and 232 say they would use a new canteen. The population is all 5,000 employees, the sample is the 400, the statistic is $\hat{p} = 232/400 = 0.58$, and the parameter is the proportion of all 5,000 employees who would use it, which is unknown.

Types of variable

A categorical variable, also called a qualitative variable, takes values that are category names or group labels: blood type, favourite subject, whether a patient recovered. A quantitative variable, also called a numerical variable, takes numbers as values, recording something measured or counted, and usually has units: height in centimetres, number of siblings, time in minutes.

A numerical code does not make a variable quantitative. Postcodes and shirt numbers are numbers used as labels, and adding or averaging them means nothing, so they are categorical. The test is whether arithmetic on the values is meaningful.

Quantitative variables are divided further.

Counts are discrete. Measurements are continuous in principle, even though a recorded value is rounded to a whole number of seconds or to the nearest 0.1 kg. A variable does not become discrete because the instrument rounds.

Identifying components

Given a description of a study, the working steps are to name the observational unit, then the variable or variables measured on it, then the type of each variable, then whether a number described is a parameter or a statistic.

A researcher measures the resting heart rate, in beats per minute, of 60 randomly chosen members of a running club with 900 members, and reports a mean of 58.4. The observational unit is a club member, the variable is resting heart rate (quantitative, continuous), the population is the 900 members, the sample has $n = 60$, and 58.4 is a statistic, $\bar{x}$, estimating the unknown population mean $\mu$.