Measures of Central Tendency

SAT Math Overview. Topic: Measures of Central Tendency

Overview

You should expect to see questions on the SAT that rely on an understanding of Measures of Central Tendency. What are Measures of Central Tendency? They are an important way of understanding how data sets cluster. In this Topic Overview, we will be going over the following concepts:

  • Mean
  • Median
  • Mode 
  • Range

Mean

Suppose you have a data set made up of n terms and suppose the data are arranged in order from least to greatest.:

 

A data set consisting of x1, x2, etc., up to x-n.

 

What is the “average” of this data set? What does it even mean to talk about the average?

 

Suppose the data are scattered from 0 to 100. Here’s one possible way the data points might cluster:

A data graph with data points ranging from zero to one hundred.

There are many other ways the data might cluster. The “average” would be the behavior of the data around the middle of the cluster of data.

The term “average” is a general term used to describe this clustering, but there are several mathematical terms that more precisely define this average. The first such term is called the mean.

Using the data set shown earlier (x1…xn), here is the formula for calculating the mean of a data set.

 

The formula for calculating the mean, which is the sum of the numbers in the data set divided by the number of numbers.

 

To see some worked-out examples of calculating the mean, click on this link to see a slide show. It includes a video, definitions, formulas, and examples.

The data set for calculating the mean can include negative numbers. To see some worked-out examples of calculating the mean when there are negative numbers in the data set, click on this link to see a slide show.

Another case to consider is what is called a “weighted average.” Click on this link to watch a slide show that explains weighted average in more detail. This includes a video tutorial on weighted averages. 

Median

The mean is a very effective, precise way of finding the central tendencies of data sets. But what about when the data sets are huge? For example, if you’ve listened to news reports you’ll often hear about “median household income.” Why not the “mean household income”?

There are millions of people in the data set of “household income.” Going back to our original data set, imagine this data set consisting of millions of data points.

A data set consisting of x1, x2, etc., up to x-n.

Imagine calculating the mean with this formula with millions of data points:

The formula for calculating the mean, which is the sum of the numbers in the data set divided by the number of numbers.

Do you see the problem? Inputting millions of data points is beyond what a spreadsheet can handle. The number of calculations involved are beyond the abilities of many computers.

So, with extremely large data sets, there’s a different “average” to use, called the median. If you arrange your data set from least to greatest, the median is the term in the middle of the data set.

An illustration showing the median of a data set that consists of an odd number of terms.

If there are an odd number of terms, the median is the middle term. For example, the median of five terms is the third term:

An illustration showing the median of a data set that consists of an odd number of terms, with the middle term highlighted.

If there are an even number of terms, the median is the mean of two of the terms. For example, the median of six terms is the mean of the third and fourth terms.

An illustration showing the median of a data set that consists of an even number of terms.

To see some worked-out examples of finding the median, click on this link to see a slide show. It includes a video, definitions, formulas, and examples.

Mode

A third type of “average” is the mode. It’s especially useful when working with categorical data. For example, suppose you have survey data in which people select their favorite flavor of ice cream. A graph of the data set might look like this.

 

A bar graph showing preferences for ice cream flavors.

The mode is the item that occurs most often. The mode can also work with numerical data, but when all you have is categorical data, use the mode.

To see some worked-out examples of finding the mode, click on this link to see a slide show. It includes a video, definitions, formulas, and examples.

Range

Going back to the fictitious data set we’ve been looking at:

 

A data set consisting of x1, x2, etc., up to x-n.

 

If this data set is arranged in order from least to greatest, then the range of values is the difference between the greatest value and the least value.

Range. For an ordered data set, the difference between the largest value and the least value.

Special Case: Weighted Average

A particular case of the mean of a data set is what’s referred to as the “weighted average.” This is what such a question would look like:

Look at the following set of coins. What is the average amount of money?

A collection of coins consisting of three pennies, two dimes, three nickels, and one quarter.

This is an example of a weighted average. If there were only one of each coin, the average, or mean, of the coins would be this:

The mean formula without any weightings.

But the average should take into account that there are three pennies, two dimes, three nickels, and one quarter. This is how to calculate the weighted average:

The mean formula with weightings.

The numbers in red represent the number or “weight” of each coin. See how the weighted average is lower than the other average calculated? The reason is because of the different number of coins.

SAT Skill: Measures of Central Tendency

Example 1

A student’s mean score for 10 tests is 85. If she is allowed to remove her lowest score, then her average increases to 90. What is the value of that lowest score?

Recall the formula for the mean:

The mean formula 

For this scenario, the formula becomes this:

The mean formula for all 10 tests.

We also know the following:

The mean formula for nine scores.

The sum of the nine test scores plus the lowest score, x, equals the sum of the ten test scores. We can then write this equation:

Solving an equation to find the lowest score.

Example 2

At a pizza party, students ate different numbers of pizza slices. This graph summarizes the number of slices eaten and the number of students that ate that number of slices. Based on the data, what is the average number of slices of pizza eaten?

A bar graph showing the number of slices of pizza eaten by students and the number of students who ate that number of slices. The graph can be used to find the average number of slices eaten.

This is an example of a weighted average. We want to find the average, or mean, number of slices eaten, but the number of students who ate that number of slices needs to be taken into account. Here’s the weighted average:

Weighted average for the mean number of pizza slices eaten.

 

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