# Spread in Data Sets: Definition & Example

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• 0:03 The Spread in Data Sets
• 1:39 Range
• 2:32 Interquartile Range
• 3:48 Variance
• 5:14 Standard Deviation
• 5:36 Lesson Summary

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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Identifying the spread in data sets is a very important part of statistics. You can do this several ways, but the most common methods are through range, interquartile range, and variance.

## The Spread in Data Sets

Tabatha is the local community theater director. She is putting together audition sheets for the next season's plays. She is setting up auditions for two plays. One is called Wonky Willy: The Candy Maker, a musical about a mysterious candy maker that creates a contest for children and their parents to visit his mysterious candy factory. For this play, Tabatha will need a cast with actors in a variety of age categories. Tabatha knows the average ages for the play, but since some of the ages are so different, she needs a better way of identifying the variations in the age categories.

Tabatha can do this by looking at the spread in the data set. The spread in data is the measure of how far the numbers in a data set are away from the mean or median. We can calculate spread in a variety of ways using different methods known as measures of spread.

Tabatha pulls out old records from the last time her theater put on Wonky Willy. She has written down all of the ages for the actors for us: 12, 64, 11, 42, 9, 57, 13, 38, 12, 47, 43, 29, 36.

Tabatha can tell us that the mean of this data set is approximately 31.7, and the median is approximately 36. However, she can't advertise that she needs actors between 32 and 36 years old - that would be inaccurate. There are three methods Tabatha can use to find the spread in her data: range, interquartile range, and variance.

## Range

The simplest way to find the spread in a data set is to identify the range, which is the difference between the highest and lowest values in a data set. Let's arrange the ages for the last production from least to greatest: 9, 11, 12, 12, 13, 29, 36, 38, 42, 43, 47, 57, 64.

Now take the lowest number and the highest number and find the difference: 64 - 9 = 55. There is a 55-year spread in the ages for this production. Range is probably the best measure of spread for this data. Tabatha can advertise that she is looking for actors between the ages of 9 and 64 for this production. Let's look at other ways Tabatha can find the spread in her data.

## Interquartile Range

Interquartile range is a value that is the difference between the upper quartile value and the lower quartile value. For this method we will have to find each quartile in the data set. To find the quartiles, follow these steps:

1. Order the data from least to greatest.
2. Find the median of the data set, and divide the data set into two halves.
3. Find the median of the two halves.

For a more in-depth look at quartiles, check out our lesson on 'Quartiles and Interquartile Range.'

Our median is 36, which is quartile two. For each half of the data set we must find the median, the median for quartile one (the lower half of the data set) is 12, and the median for quartile three (the upper half of the data set) is 45.

To find the interquartile range, simply take the upper quartile and subtract the lower quartile: 45 - 12 = 33. The interquartile range is 33. That means that the majority of the ages in this data set are within 33 years of one another. While this information may not give Tabatha the specific age range that she is looking for, it may help her understand the variety of ages she is looking for in this production.

## Variance

Now let's look at the variance in this data set. Variance is how far a set of numbers are spread out. To find variance, follow these steps:

1. Find the mean of the set of data.
2. Subtract each number from the mean.
3. Square the result.
5. Divide the result by the total number of numbers in the data set.

Take a look at the chart below to find the variance in this data set:

The first column contains all of the numbers in the data set, the second column shows the mean of the data set. In the third column, we've taken the results of column number two and squared each number. In the fourth column, we've taken each number from column three and added them together, and in the fifth and final column, we've divided the number from column number four by the total number of values from the data set, which is 13. Our variance from this data set is 329.72.

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