Wiki Page, Math 13 - Perez Sandbox : Module 17 - Parameters vs. Statistics (2b of 2)

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2026-01-17T21:15:23-08:00

Before we continue our discussion of sampling variability, we introduce some vocabulary.

A parameter is a number that describes a population. A statistic is a number that we calculate from a sample.

Let’s use this new vocabulary to rephrase what we already know at this point:

When we do inference, the parameter is not known because it is impossible or impractical to gather data from everyone in the population. (Note: In each example on this page, we assumed we knew the parameter so that we could investigate how statistics relate to the parameter. This is the first step in creating a probability model. However, when we do inference, we use a statistic to draw a conclusion about an unknown parameter.)

We make an inference about the population parameter on the basis of a sample statistic.

Statistics from samples vary.

In this course, if the variable is categorical, the parameter and the statistic are both proportions. If the variable is quantitative, the parameter and statistic are both means.

From our first example:

Parameter: A population proportion. For this population of students at a small college, 0.60 are eligible for financial aid.

Statistics: Sample proportions that vary. In the example, 0.75, 0.625, and 0.375 are all statistics that describe the proportion eligible for financial aid in a sample of 8 students.

From our second example:

Parameter: A population mean. For this population of students at a small college, the mean amount of financial aid is $1,500.

Statistics: Sample means that vary. In the example, $2,087.50, $1,325.00, and $687.50 are all statistics that describe the mean amount of financial aid received by a sample of 8 students.

We use different notation for parameters and statistics:

Notation for parameters and statistics

| (Population) Parameter | (Sample) Statistic
Proportion | |
Mean | |
Standard Deviation | |

Sometimes we refer to the sample statistics as “p-hat” and “x-bar.”

Here we use this notation for the information from our examples.

For our first example:

For the population of college students, p = 0.60.

For the 3 random samples of 8 students, we have the following p-hats

, ,

For our second example:

For the population of college students, = $1,500.

For the 3 random samples of 8 students, we have the following x-bars

, ,

Important Comments about Notation

Many statistics packages and introductory statistics textbooks use the notation shown in the table. The notation for means and standard deviations is common in the field of statistics. However, you will occasionally see other notation for proportions. In some statistical material, the Greek letter represents the population proportion and p represents the sample proportion. This can be particularly confusing because p is used in some statistical material for the population proportion and in other statistical material for a sample proportion. Whenever you work with symbols, always be sure you understand what the symbol represents. You should be able to interpret the symbol from the context of the material.

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