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Math Example--Measures of Central Tendency--Range: Example 12
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Range in Math
This is part of a collection of math examples on the topic of range in math. These downloadable images are easily incorporated into a lesson plan.
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Below is a re view of measures of central tendency, with a focus on range. This includes an instructional video (with transcript). Use this material to supplement the math examples.
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 (x1…xn), 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 in Math
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.
The downloadable image is part of a collection of math examples that focus on calculating measures of central tendency: mean, median, and mode. This example is about range in math.
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| 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 |
|---|---|
| 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 |