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Math Example--Measures of Central Tendency--Range: Example 24

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# Math Example--Measures of Central Tendency--Range: Example 24

## Topic

Measures of Central Tendency

## Description

This example illustrates how to find the range of the following set of numbers: 15, 21, -34, 32, -46, 15, 27, -35, -35, 15, -47, 27, 7, 38, 33, 27, 49, 39, -12. The solution involves arranging the numbers from least to greatest and finding the difference between the two extremes. The range is calculated to be 96. This example is particularly valuable as it includes a mix of positive and negative values, as well as repeated numbers, helping students understand how to handle diverse and complex datasets when calculating the range.

In the broader context of measures of central tendency, examples like this are crucial for developing a comprehensive understanding of data analysis. The range provides insight into the spread of data, complementing other measures such as mean, median, and mode. By working through multiple examples with varying types of numbers and repetitions, students can better grasp how these measures work together to provide a complete picture of a dataset's distribution.

**Teacher's Script: **Now, let's take a closer look at this example. We'll start by organizing our numbers from smallest to largest, paying special attention to the negative values and repeated numbers. Then, we'll identify the minimum and maximum values to calculate the range. Remember, when working with negative numbers, we use the absolute value of the difference. This process will help us understand how the range represents the spread of our data, especially when dealing with a mix of positive and negative numbers. Can you think of any other measures of spread that might give us more information about this dataset?

For a complete collection of math examples related to Measures of Central Tendency click on this link: __Math Examples: Measures of Central Tendency: Range Collection.__

## Range in Math

Watch this video to learn about range in math. (The transcript is also included.)

Video Transcript

Measures of central tendency tell you how a data set behaves around the middle. But the middle of what?

In this video we look at the range of a data set to look at its behavior beyond the middle. Let’s start with an example.

These are the test scores for a group of 20 students.

Notice that most of the values are unique. Also, the data is arranged randomly. In this form we can’t really make any observations about the data as a whole.

Here’s the same data organized in order from left to right.

In this form, we can quickly see one of the measures of central tendency, the median, which is 85.5.

The range, on the other hand, tells us about the spread of the data. Take the greatest value, in this case 100, and subtract from it the lowest value, or 50. The range of this data set is 50.

Let’s look at another example.

At a tee shirt store, here are 20 transactions, where customers bought different numbers of shirts.

Find the median transaction amount and the range.

First arrange the data from left to right in order from least to greatest.

The median is the middle term, which in this case is $12.45.

The range is found by subtracting these numbers, for a value of $13.94.

Let’s look at a final example.

The Bureau of Labor Statistics tracks hourly wages for different jobs. This table is from 2022.

Find the range of hourly wages and the median hourly wage.

First arrange the data from least to greatest.

The median and range are summarized below.

## Measures of Central Tendency

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

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:

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 (x_{1}…x_{n}), here is the formula for calculating the mean of a data set.

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.

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

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.

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:

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.

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.

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.

## Range

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

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.

Here are some examples of calculating the range of a data set.

In this example, there are eight numbers arranged in a random order. The first step is to arrange them in order from least to greatest. Then find the difference between the least and greatest values.

In this example there are negative numbers in the data set. The same process still applies. First, arrange the numbers in order from least to greatest. Then find the difference between the least and greatest numbers.

Common Core Standards | CCSS.MATH.CONTENT.6.SP.B.4, CCSS.MATH.CONTENT.7.SP.B.4, CCSS.MATH.CONTENT.6.SP.B.5.C |
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Grade Range | 6 - 8 |

Curriculum Nodes |
Algebra• Probability and Data Analysis• Data Analysis |

Copyright Year | 2014 |

Keywords | data analysis, tutorials, measures of central tendency, range |