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.
|Mean||μ or Mu||x̄ or x-bar|
|Standard deviation||σ or Sigma||s|
|Correlation||ρ or rho||r|
|Proportion||P||p̂ or p-hat|
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.
|Mean weight of all German Shepherd dogs.||Mean weight of a random sample of 200 German Shepherds.|
|Median income of a county.||Median income of a random sample of 50 from that county.|
|Standard deviation of all transaction times in a particular bank.||Standard deviation of a random sample of 500 transaction times at that bank.|
|Proportion of all people who prefer Coke over Pepsi.||Proportion of a random sample of 100 people who prefer Coke over Pepsi.|
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.