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.
Parameters and Statistics
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.
Examples of Parameters and Statistics
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.