What is the type of sample statistic that is used to make inferences about a given type of population parameter?

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Population vs sample
What kinds of numbers are parameters and statistics?
Statistical notation
Telling the difference between a parameter and a statistic
Quiz: Statistic or parameter?
Estimating parameters from statistics
Frequently asked questions about parameters and statistics

It is often of interest to learn about the characteristics of a large group of elements such as individuals, households, buildings, products, parts, customers, and so on. All the elements of interest in a particular study form the population. Because of time, cost, and other considerations, data often cannot be collected from every element of the population. In such cases, a subset of the population, called a sample, is used to provide the data. Data from the sample are then used to develop estimates of the characteristics of the larger population. The process of using a sample to make inferences about a population is called statistical inference.

Characteristics such as the population mean, the population variance, and the population proportion are called parameters of the population. Characteristics of the sample such as the sample mean, the sample variance, and the sample proportion are called sample statistics. There are two types of estimates: point and interval. A point estimate is a value of a sample statistic that is used as a single estimate of a population parameter. No statements are made about the quality or precision of a point estimate. Statisticians prefer interval estimates because interval estimates are accompanied by a statement concerning the degree of confidence that the interval contains the population parameter being estimated. Interval estimates of population parameters are called confidence intervals.

Although sample survey methods will be discussed in more detail below in the section Sample survey methods, it should be noted here that the methods of statistical inference, and estimation in particular, are based on the notion that a probability sample has been taken. The key characteristic of a probability sample is that each element in the population has a known probability of being included in the sample. The most fundamental type is a simple random sample.

For a population of size N, a simple random sample is a sample selected such that each possible sample of size n has the same probability of being selected. Choosing the elements from the population one at a time so that each element has the same probability of being selected will provide a simple random sample. Tables of random numbers, or computer-generated random numbers, can be used to guarantee that each element has the same probability of being selected.

A sampling distribution is a probability distribution for a sample statistic. Knowledge of the sampling distribution is necessary for the construction of an interval estimate for a population parameter. This is why a probability sample is needed; without a probability sample, the sampling distribution cannot be determined and an interval estimate of a parameter cannot be constructed.

A parameter is a number describing a whole population (e.g., population mean), while a statistic is a number describing a sample (e.g., sample mean).

The goal of quantitative research is to understand characteristics of populations by finding parameters. In practice, it’s often too difficult, time-consuming or unfeasible to collect data from every member of a population. Instead, data is collected from samples.

With inferential statistics, we can use sample statistics to make educated guesses about population parameters.

Population vs sample

In research, a population is the entire group that you’re interested in studying. This may be a group of people (e.g. all adults in the US or all employees of a company), but it can also mean a group containing other kinds of elements: objects, events, organizations, countries, species, organisms, etc.

A sample is a smaller group taken from the population. The sample is the group of elements that you will actually collect data from.

Population vs sampleYou want to identify the level of support for the death penalty among US residents. Since the population you’re interested in is all US residents, it’s not practical to collect data from the whole population. Instead, you use random sampling to survey a sample of 2000 participants.

What kinds of numbers are parameters and statistics?

Statistics and parameters are numbers that summarize any measurable characteristic of a sample or a population.

For categorical variables (e.g., political affiliation), the most common statistic or parameter is a proportion.

For numerical variables (e.g., height), the mean or standard deviation are commonly reported statistics or parameters.

Examples of statistics vs parameters

Sample statistic	Population parameter
Proportion of 2000 randomly sampled participants that support the death penalty.	Proportion of all US residents that support the death penalty.
Median income of 850 college students in Boston and Wellesley.	Median income of all college students in Massachusetts.
Standard deviation of weights of avocados from one farm.	Standard deviation of weights of all avocados in the region.
Mean screen time of 3000 high school students in India.	Mean screen time of all high school students in India.

Statistical notation

Different symbols are used for statistics versus parameters to show whether a sample or a population is being referred to.

Greek letters and capital letters usually refer to populations, whereas Latin letters and lower-case letters refer to samples.

Symbols for statistics vs parameters

	Sample statistic	Population parameter
Proportion	p̂ (called “p-hat”)	P
Mean	x̄ (called “x-bar”)	μ (Greek letter “mu”)
Standard deviation	s (Latin letter “s”)	σ (Greek letter “sigma”)
Variance	s2	σ2

Telling the difference between a parameter and a statistic

In news and research reports, it’s not always clear whether a number is a parameter or statistic. To figure out which type of number you’re dealing with, ask yourself the following:

Does the number describe a whole, complete population where every member can be reached for data collection?
Is it possible to collect data on this characteristic from every member of the population in a reasonable time frame?

If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters.

If the answer is no to either of the questions, then the number is more likely to be a statistic. Sampling is used to collect data from large populations and generalize the statistics to the broader population in an externally valid way.

Quiz: Statistic or parameter?

Estimating parameters from statistics

Using inferential statistics, you can estimate population parameters from sample statistics. To make unbiased estimates, your sample should ideally be representative of your population and/or randomly selected.

There are two important types of estimates you can make about the population parameter: point estimates and interval estimates.

A point estimate is a single value estimate of a parameter based on a statistic. For instance, a sample mean is a point estimate of a population mean.
An interval estimate gives you a range of values where the parameter is expected to lie. A confidence interval is the most common type of interval estimate.

Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.

Estimating a population parameter from a sample statisticIn your study on support for the death penalty among US residents, you find that 61% of participants in your sample support the death penalty. To estimate the population parameter, you calculate a point estimate and an interval estimate from your sample statistic.

Your point estimate is your sample statistic – you estimate that 61% of all US residents support the death penalty.

To find the interval estimate, you construct a 95% confidence interval that tells you where the population parameter is expected to lie most of the time. With random sampling, there is a 0.95 probability that the true population parameter for support for the death penalty among US residents lies between 57% and 65%.

Frequently asked questions about parameters and statistics

How do you know whether a number is a parameter or a statistic?

To figure out whether a given number is a parameter or a statistic, ask yourself the following:

Does the number describe a whole, complete population where every member can be reached for data collection?
Is it possible to collect data for this number from every member of the population in a reasonable time frame?

If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters.

If the answer is no to either of the questions, then the number is more likely to be a statistic.