Parameters and Statistics

In statistics, understanding the distinction between key concepts is essential for analyzing data effectively. Two foundational terms, parameter and statistic, play a crucial role in summarizing and interpreting information.

What is a Parameter?

A parameter is a numerical measurement describing some characteristic of a population.

An example of a parameter is the average height of all people living in Nashville, TN, and its surrounding suburbs. While it’s theoretically possible to measure the height of every individual in the area to determine this parameter, doing so would require a complete census, which is impractical. Many people might refuse to participate, and certain groups, such as homeless individuals or undocumented immigrants, can be difficult to locate and include. As a result, although this exact number exists, the inability to conduct a full census means it will likely remain unknown.

This is why we take samples. While we can’t calculate this number for the entire population, we can measure the heights of a subset of individuals who are willing to participate. The average height calculated from this subset is called a statistic, which is used to estimate the unknown parameter.

What is a Statistic?

A statistic is a numerical measurement describing some characteristic of a sample. It is calculated from a sample drawn from some population that is intended to be studied.

Returning to our Nashville example, we could take a random sample of 500 individuals and calculate their average height. If we get an average height of 5 feet 7 inches, then this value is a statistic since it is a value derived from a sample.

Note: To ensure accuracy, the sample should be randomly selected from different neighborhoods and include a diverse mix of ages, genders, ethnicities, and socio-economic backgrounds. This approach can provide a reasonably accurate estimate of the average height of people living in the Nashville area, provided it is free from bias or sampling errors, which we will discuss shortly.

Telling the Difference Between a Parameter and a Statistic

These two terms can be confusing so here is a helpful memory device for sorting out which term goes with what value:

  • Parameter refers to a Population (both start with a P)
  • Statistic refers to a Sample (both start with a S).
Try the example below to determine if you can identify numerical values as parameters or statistics.

Example : Statistic vs Parameter

For each scenario below, tell if each of the bolded values are a parameter or a statistic. If the value is a statistic, tell what could be changed about the scenario to make the bolded word become a parameter.

  1. A study of all 2223 passengers aboard the Titanic found that 706 survived when it sank.
  2. In a large sample of households, the median annual income per household for high school graduates is $19,856 (based on data from the U.S. Census Bureau).
  3. Among the Senators in the current Congress, 45% are Democrats.
  4. The author measured the voltage supplied to his home on 40 different days, and the average (mean) value is 123.7 volts.

Solution: Statistic vs Parameter

Here are the solutions to the above example problem:

  1. The number 706 is a parameter
  2. The value $19,856 is a statistic. It would be a parameter by removing "sample of households" and adding "all the households in Tennessee" or "every households in the county": putting in wording that indicates it is a complete collection of households.
  3. The percentage 45% is a parameter
  4. The value 123.7 is a statistic. This would be a parameter if the "40 different days" was change to "all 365 days of the year" or "all 30 days of this month" to indicate that the voltage measurements are a complete collection.

\[ \tag*{\(\blacksquare\)} \]

Conclusion

Understanding the distinction between parameters and statistics is fundamental in statistical analysis. While parameters describe entire populations, they are often theoretical and difficult to calculate due to practical constraints. Statistics, on the other hand, are calculated from samples and provide a way to estimate these unknown parameters. By carefully designing samples to be representative and minimizing bias, we can make reliable conclusions about populations from data we collect from samples. Using statistics to estimate parameters forms the foundation for much of the work in statistics, allowing us to draw meaningful conclusions from data and apply them to real-world contexts.

Now that we have introduced several of the key terms used in this course, we can explain a little more in-depth what a statistics course entails. This course will cover how to organize and summarize data, a process known as descriptive statistics. Data can be summarized through graphs or by numerical measures which will be discussed in future readings. After some time in the course we will introduce formal methods for drawing conclusions about a population from reliable sample data. This process is known as inferential statistics. Effective data interpretation relies on sound data collection methods and careful analysis. While the course includes numerous mathematical formulas, the focus is primarily on understanding the data rather than performing extensive calculations, which can be handled by calculators or computers. A solid grasp of statistical fundamentals fosters confidence in decision-making.