Population Parameters and Sample Statistics

In a study, the terms population, sample, parameter, and statistic answer different questions. The population is the full group of interest. The sample is the smaller group from which data were collected. A parameter is a number that summarizes the population, while a statistic summarizes the sample. A sample statistic can help estimate a population parameter, but the two are not the same value by definition.

Population and sample describe groups

A population is the complete set of people, objects, or measurements the question is about. It might be all current students at a school, all transactions in a year, or all registered voters in a defined region. The population is not always a huge national group; it is whichever full group the study intends to draw conclusions about.

A sample is the portion of the population actually observed or surveyed. If researchers ask 200 students at a university about study habits, those respondents are the sample. If the intended question concerns every student at that university, the full student body is the population. Define the target population from the study question, not from the fact that the data set is available.

Parameter and statistic describe numbers

A parameter is a numerical characteristic of the whole population. Examples include the population mean μ, population proportion p, or population standard deviation σ. A parameter may be unknown because collecting data from every member of the population is impractical. It is still a fixed value for the specified population and time, even when researchers do not know it.

A statistic is a numerical summary calculated from sample data. Common notation includes sample mean x̄, sample proportion p̂, and sample standard deviation s. A statistic can be computed from the observations in hand. Another sample drawn from the same population may produce a different statistic, because the included observations differ.

A worked example

Suppose a school wants the mean time all of its current students spend reading each week. The population is every current student at that school. The population mean reading time is the parameter of interest. Staff survey 80 students selected from the school and calculate an average of 6.5 hours. Those 80 students form the sample; 6.5 hours is a sample statistic. It estimates the unknown population parameter but does not automatically equal it.

If the school instead records reading time for every student, the calculated mean describes the full population and is a parameter for that defined group. The same arithmetic formula can calculate a mean from either a sample or a population. Whether it is called a statistic or parameter depends on which group the numbers summarize, not on which formula is used.

Why sample statistics vary

Different samples can yield different sample means, proportions, or ranges. This variation is called sampling variability. If one random sample of 80 students averages 6.5 hours and a second averages 6.1, neither sample statistic changes the underlying parameter; each is a different estimate based on different observations. Larger, well-designed samples generally provide more information, but size alone cannot correct a systematically biased selection method.

The distinction matters when interpreting claims. 'The average among surveyed respondents was 6.5 hours' reports a statistic. 'All students at the school average 6.5 hours' is a claim about a population parameter and requires evidence that the sample supports that generalization. A sample summary should not silently be presented as an exact population fact.

Sampling method affects what a statistic can tell you

A sample statistic is useful for estimating a population parameter only when the sample and method fit the question. A convenience sample taken from students in one advanced reading class may overrepresent students who read more. A random sample drawn from an appropriate list gives members of the defined population a known chance of selection, although nonresponse and measurement problems can still introduce bias.

Even an unbiased statistic has uncertainty because it is based on a sample rather than a census. A confidence interval, margin of error, or other inferential method may describe that uncertainty. The key conceptual relationship remains: a statistic is computed from sample data and is used to estimate a parameter for the population.

Identify the terms in a problem

Read a study description in this order. First, identify the group the researchers want to describe. That is the population. Next, identify the units from which measurements were actually collected; those are the sample. Then distinguish the target numerical quantity from the number calculated using the sample. The population quantity is the parameter; the sample calculation is the statistic.

For example, in a poll of 500 residents about a city proposal, the target population might be all eligible city voters. The 500 respondents are the sample. The true fraction of all eligible voters who support the proposal is a population parameter. The fraction of the 500 respondents who support it is a sample statistic. Do not call the poll's observed fraction a parameter just because it is written as a percent.

Common mix-ups

One mistake is to use 'population' for the people who answered the survey. Those are usually the sample; the population is the wider group the study intends to describe. Another is to call any percentage a parameter. A percentage calculated from a sample is a statistic, while the population percentage is a parameter. A third mistake is to assume that a sample statistic must match the parameter exactly. Sampling variation makes that unlikely in many real studies.

• Population: full group the study aims to describe.
• Sample: subset actually observed or measured.
• Parameter: numerical summary of the population, often unknown.
• Statistic: numerical summary calculated from sample data.
• A statistic estimates a parameter; it is not automatically equal to it.
• A biased sample may produce a misleading estimate regardless of its size.

Exam takeaway

The group answers 'population or sample?' The number answers 'parameter or statistic?' Full group plus its numerical summary means population and parameter; observed subset plus a value calculated from it means sample and statistic. Statistics estimate population parameters, with possible sampling variation and bias.