Basic Statistics > What is a Parameter in Statistics?
What is a Parameter in Statistics?
A parameter is a numerical characteristic of a population, and it is measured from the population. It is any summary number, such as mean or percentage, that describes the entire population. In comparison, a statistic is a numerical characteristic of a sample, and it is estimated from the sample.
In “regular” mathematics, a parameter refers to a value within an equation that is transferred throughout the equation. For example, m and b are parameters in the slope equation y = mx + b.
However, the parameter definition in statistics is quite different. Parameters in statistics are used to describe a population, not just one equation. For example, the mean and variance of a population are both parameters. The mean represents the average value within the population, while the variance tells us something about how spread out values are throughout the population.
| Feature | Parameter definition in Math | Parameter definition in Statistics |
|---|---|---|
| Definition | A value used to calculate the value of another variable | A numerical characteristic of a population |
| Symbol | Roman lowercase such as x, y, z | Greek letters such as μ, σ or Roman lowercase. |
| Use | Calculate the value of another variable | Describe a population |
Parameters are not always known. In many cases, they must be estimated from samples. However, parameters are still important because they provide a benchmark against which sample statistics can be compared.
| Measurement | Sample Statistic | Population Parameter |
| Proportion | p | P |
| Mean | x̄ | μ |
| Standard deviation | s | σ |
| Variance | s2 | σ2 |
| Number of items | n | N |
| Correlation coefficient | r | ρ |
A parameter never changes, because everyone (or everything) was surveyed to find the parameter. For example, you might be interested in the average age of everyone in your class. Maybe you asked everyone and found the average age was 25. That’s a parameter, because you asked everyone in the class.
Now let’s say you wanted to know the average age of everyone in your grade or year. If you use that information from your class to take a guess at the average age, then that information becomes a statistic. That’s because you can’t be sure your guess is correct (although it will probably be close!).
Statistics vary. You know the average age of your classmates is 25. You might guess that the average age of everyone in your year is 24, 25, or 26. You might guess the average age for other colleges in your area is the same. And you might even guess that’s the average age for college students in the U.S.. These may not be bad guesses, but they are statistics because you didn’t ask everyone.
Origin of the word Parameter
The parameter definition in statistics became commonplace in the early 20th century. The word parameter can be found in early texts such as E. Czuber’s 1914 work Wahrscheinlichkeitsrechnung, Vol. I (Probability Theory Vol.1). However, the term was first introduced in the field of statistics by Ronald Fisher in his 1922 book, On the Mathematical Foundations of Theoretical Statistics. Fisher, an English statistician, biologist, and geneticist, is widely regarded as the founder of modern statistics.
Several other statisticians contributed to the popularization of the parameter definition in statistics. Jerzy Neyman, a Polish-American statistician known for his work on hypothesis testing, utilized the term “parameter” in his 1937 paper, “Sufficient Statistics and Useful Tests of Statistical Hypotheses.”
Notation
Parameters are usually Greek letters (e.g. σ) or capital letters (e.g. P). Statistics are usually Roman letters (e.g. s). In most cases, if you see a lowercase letter (e.g. p), it’s a statistic. This table shows the different symbols. Some might look the same but look closely for small and capital letters.
| Measurement | Statistic (Roman or lowercase) | Parameter (Greek or uppercase) |
| Population Proportion | p | P |
| Data Elements | x | X |
| Population Mean | x̄ | μ |
| Standard deviation | s | σ |
| Variance | s2 | σ2 |
| Number of elements | n | N |
| Correlation Coefficient | r | ρ |
Tip: In statistics, the word parameter rarely pops up. That’s because ALL we deal with is statistics! You might see something like “population mean.” That makes it more obvious it’s about the whole parameter. When you see just “mean,” that’s usually a statistic.
Parameter Accuracy
Accuracy describes how close your statistic is to a particular population parameter. For example, you might be studying weights of pregnant women. If the sample median of your population is 150 pounds and your sample statistic is 149 pounds, then you can make a statement about the accuracy of your sample.
Statistics in general aren’t as accurate as we’d like, although they are the best tool we have right now for making predictions about populations. According to The Economist, scientific papers aren’t very reliable. John Loannisis, a Greek epidemiologist, thinks that as many as 50 percent of scientific papers turn out to be wrong.
Factors that contribute to false results include sample sizes that are too small, poorly designed studied and researcher bias caused by financial interests or personal agendas.
Q. In a census, how do the statistics that get computed compare to the population’s corresponding parameters?
What this question is really asking is, how accurate is the census?. The answer is (surprisingly) that the census is very accurate, give or take a tiny percentage. That tiny percentage is more likely to be minority groups, people of lower income, and people who live in rural areas.
At the time of writing, the most recent census was the 2010 census.
Historically, the census is getting more accurate: The 2010 census stated that the total population of the U.S. was 308,745,538 in 2010, a 9.7 percent increase from the 2000 census count. It over-counted the total U.S. population by 0.01 percent. That’s only 36,000 people — not bad when you consider the population of the U.S. is over 300 million.
Compare that to an over-count of 0.49 percent in 2000 (about one million people) and an under-count of 1.61 percent in 1990. Some key facts:
- Renters were under-counted.
- Homeowners were over-counted.
- 2.1 percent of black Americans were missed. According to the Denver Post, this was a huge improvement over the 1940 figures, when it was estimated the black under-count was 8.4 percent.
- 1.5 percent of Hispanics were missed.
- Non-Hispanic whites were over-counted.
Why are these groups not counted as accurately?
- Accessibility: People in rural areas can be hard or impossible to reach by mail. Some locations use “general delivery” addresses rather than fixed addresses.
- Language barrier and education: People who are learning the English language may have difficulty understanding the census form. People with lower education and literacy may also not understand the importance of responding to the census.
- Suspicion of the Government: Some people may think that the census will be used against them. Some people don’t want to be in a government database. For example: undocumented immigrants, people with warrants out on them, people owing debts or child support.
What is a Parameter in Statistics: Related article: How to tell the difference between a statistic and a parameter.