What is the relationship between a statistic and a parameter?

In several disciplines, the goal is to study a large group of individuals. These groups could be as varied as a species of bird, college freshmen in the U.S. or cars driven around the world. Statistics are used in all of these studies when it is infeasible or even impossible to study each and every member of the group of interest. Rather than measuring the wingspan of every bird of a species, asking survey questions to every college freshman, or measuring the fuel economy of every car in the world, we instead study and measure a subset of the group.

The collection of everyone or everything that is to be analyzed in a study is called a population. As we have seen in the examples above, the population could be enormous in size. There could be millions or even billions of individuals in the population. But we must not think that the population has to be large. If our group being studied is fourth graders in a particular school, then the population consists only of these students. Depending on the school size, this could be less than a hundred students in our population.

To make our study less expensive in terms of time and resources, we only study a subset of the population. This subset is called a sample. Samples can be quite large or quite small. In theory, one individual from a population constitutes a sample. Many applications of statistics require that a sample has at least 30 individuals.

What we are typically after in a study is the parameter. A parameter is a numerical value that states something about the entire population being studied. For example, we may want to know the mean wingspan of the American bald eagle. This is a parameter because it is describing all of the population.

Parameters are difficult if not impossible to obtain exactly. On the other hand, each parameter has a corresponding statistic that can be measured exactly. A statistic is a numerical value that states something about a sample. To extend the example above, we could catch 100 bald eagles and then measure the wingspan of each of these. The mean wingspan of the 100 eagles that we caught is a statistic.

The value of a parameter is a fixed number. In contrast to this, since a statistic depends upon a sample, the value of a statistic can vary from sample to sample. Suppose our population parameter has a value, unknown to us, of 10. One sample of size 50 has the corresponding statistic with value 9.5. Another sample of size 50 from the same population has the corresponding statistic with value 11.1.

The ultimate goal of the field of statistics is to estimate a population parameter by use of sample statistics.

There is a simple and straightforward way to remember what a parameter and statistic are measuring. All that we must do is look at the first letter of each word. A parameter measures something in a population, and a statistic measures something in a sample.

Below are some more example of parameters and statistics:

• Suppose we study the population of dogs in Kansas City. A parameter of this population would be the mean height of all dogs in the city. A statistic would be the mean height of 50 of these dogs.
• We will consider a study of high school seniors in the United States. A parameter of this population is the standard deviation of grade point averages of all high school seniors. A statistic is the standard deviation of the grade point averages of a sample of 1000 high school seniors.
• We consider all of the likely voters for an upcoming election. There will be a ballot initiative to change the state constitution. We wish to determine the level of support for this ballot initiative. A parameter, in this case, is the proportion of the population of likely voters that support the ballot initiative. A related statistic is the corresponding proportion of a sample of likely voters.

Parameters are numbers that describe the properties of entire populations. Statistics are numbers that describe the properties of samples.

For example, the average income for the United States is a population parameter. Conversely, the average income for a sample drawn from the U.S. is a sample statistic. Both values represent the mean income, but one is a parameter vs a statistic.

Remembering parameters vs statistics is easy! Both are summary values that describe a group, and there’s a handy mnemonic device for remembering which group each describes. Just focus on their first letters:

• Parameter = Population
• Statistic = Sample

A population is the entire group of people, objects, animals, transactions, etc., that you are studying. A sample is a portion of the population.

Types of Parameters and Statistics

Parameters and statistics use numbers to summarize the properties of a population or sample. There is a range of possible attributes that you can evaluate, which gives rise to various types of parameters and statistics. For example, are you measuring the length of a part (continuous) or whether it passes or fails an inspection (categorical)?

When you measure a characteristic using a continuous scale, you can calculate various summary values for statistics and parameters, such as means, medians, standard deviations, and correlations.

When the characteristic is categorical, the parameter or statistic will often be a proportion, such as the proportion of people who agree with a particular law.

Related post: Discrete vs Continuous Data

Statistic vs Parameter Symbols

While parameters and statistics have the same types of summary values, statisticians denote them differently. Typically, we use Greek and upper-case Latin letters to signify parameters and lower-case Latin letters to represent statistics.

Parameter vs Statistic Examples

In the examples below, notice how the same subject and summary value can be either a parameter or a statistic. The difference depends on whether the value summarizes a population or a sample.

Identifying a Parameter vs Statistic

If you’re listening to the news, reading a report, or taking a statistics test, how do you tell whether a summary value is a parameter or a statistic?

Real-world studies almost always work with statistics because populations tend to be too large to measure completely. Remember, to find a parameter value exactly, you must be able to measure the entire population.

However, researchers define the populations for their studies and can specify a very narrowly defined one. For example, a researcher could define the population as a specific neighborhood, U.S Senators (n=100), or a particular sports team. It’s entirely possible to survey the entirety of those populations!

The trick is to determine whether the summary value applies to an entire population or a sample of a population. Carefully read the narrative and make the determination. Consider the following points:

• A description that specifies the use of a sample indicates that the summary value is a statistic.
• If the population is very large or impossible to measure completely, the summary value is a statistic.
• However, if the researchers define the population as a relatively small group that is reasonably accessible, the researchers could potentially measure the entire group. The summary value might be a parameter.

Researchers and Parameters vs Statistics

Researchers are usually more interested in understanding population parameters. After all, understanding the properties of a relatively small sample isn’t valuable by itself. For example, scientists don’t care about a new medicine’s mean effect on just a few people, which is a sample statistic. Instead, they want to understand its mean effect in the entire population, a parameter.

Unfortunately, measuring an entire population to calculate its parameter exactly is usually impossible because they’re too large. So, we’re stuck using samples and their statistics. Fortunately, with inferential statistics, analysts can use sample statistics to estimate population parameters, which helps science progress.

Using a sample statistic to estimate a population parameter is a process that starts by using a sampling method that tends to produce representative samples—a sample with similar attributes as the population. Scientists frequently use random sampling. Then analysts can use various statistical analyses that account for sampling error to estimate the population parameter. This process is known as statistical inference.

Learn more about Descriptive vs Inferential Statistics and Statistical Inferences.