Parameters and Statistics

Two Kinds of Numerical Summaries

In Census Versus Sample, you distinguished a population from a sample. That distinction also tells us how to describe a numerical summary. A summary calculated for an entire population is a parameter. A summary calculated from a sample is a statistic.

The same type of summary can be either one. A mean calculated from every member of a population is a parameter; a mean calculated from a sample is a statistic. To classify a number, you need to know which group its data summarize—not just the number itself or the word used to describe it.

A population is the complete group the statistical question concerns. A sample is the subset from which data are collected. For example, if the question concerns all students enrolled at a school, the mean for all those students is a parameter. The mean for 30 students selected from the school is a statistic.

A parameter is fixed for a defined population and a defined variable, even when its value is not known. A statistic is calculated from the particular sample collected. If a different sample is selected, its statistic may have a different value. This does not mean the population parameter changed; it means the sample-based summary changed.

A census that successfully collects data from every member of a defined population can provide the population value for the variable measured. A sample, by contrast, provides a statistic describing the sampled units. In either case, the group and variable must be clear before the number can be classified.

Common Notation for Parameters and Statistics

Statistics uses different symbols for several population summaries and their sample counterparts. These symbols help you identify what a reported number represents. You have already used many of these summaries in earlier tutorials; the key here is to notice whether the summary comes from a population or a sample.

The population mean is written \(\mu\), while the sample mean is written \(\bar{x}\). The population proportion is \(p\), while the sample proportion is \(\hat{p}\). The population standard deviation is \(\sigma\), while the sample standard deviation is \(s\). These pairings are useful, but the group described remains the deciding factor.

Not every statistic needs one of these specific symbols. A sample median, sample range, or sample IQR is also a statistic because it summarizes sample data. The population could have a median, range, or IQR as well; those would be population parameters if calculated for the full population. Earlier tutorials on quartiles and the IQR explain those summaries.

A Reliable Way to Classify a Number

When a problem gives you a number and asks whether it is a parameter or a statistic, do not guess from its size or from its symbol alone. First identify the population of interest, then find out which units supplied the data used for the number. Finally, decide whether those units are the whole population or only a subset.

This method also prevents a common mix-up: the word “average” does not tell you whether a number is a parameter or a statistic. “Average” describes the kind of calculation; it does not tell you which group was used.

Worked Example: Screen Time at a School

Worked Example: Responses About an Online Portal

Worked Example: A Sample Median and a Population Median

What Counts as a Summary?

A parameter or statistic is a numerical summary of a group, not simply any number appearing in a data set. One visitor’s wait of 8 minutes is an individual observation. It is not, by itself, a summary of the sample or of the population. The sample mean of 7 minutes, however, summarizes the six recorded waits.

A count can also be part of a summary, depending on what the number represents. In the portal example, 68 is the count of sample members who answered yes; it is based on sample data. The proportion \(68/100=0.68\) is the sample statistic summarizing the share of surveyed patients who answered yes. Always describe what a number counts or summarizes before assigning a label.

A group’s size alone is not the same as a measure such as its mean or proportion. If a problem reports that a school has 240 students, that identifies the population size in the screen-time example. It does not tell you the students’ mean screen time. Pay attention to the variable being summarized as well as the units included.

Common Mistakes and AP Exam Tips

• Classifying by the symbol alone. Symbols such as \(\mu\) and \(\bar{x}\) are helpful, but your explanation should still identify whether the value summarizes the population or a sample.
• Assuming a large sample makes a statistic a parameter. A sample remains a subset whether it contains 30 people or 30,000. A summary calculated from that subset is a statistic.
• Thinking a parameter must be unknown. A parameter is defined by the population it describes, not by whether someone knows its value. A census may make a population parameter available.
• Calling an individual observation a statistic. A single response or measurement is an observation. A statistic is a numerical summary calculated from sample data.
• Forgetting to name the variable. A statement such as “the answer is a parameter” is incomplete if the group and quantity are unclear. Say what population and variable the number describes.
• Treating a sample result as an exact population value. A sample proportion describes the sampled units. Do not label it as the population proportion unless the data cover the whole defined population.

For a clear AP response, state the population, identify which units supplied the data, and connect those facts to the classification. For example: “The 0.68 is a statistic because it is the proportion of the 100 surveyed patients who reported using the portal; the population consists of all 800 patients.”