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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

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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.

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.

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** 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:

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

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:

- Find the mean of the set of data.
- Subtract each number from the mean.
- Square the result.
- Add the numbers together.
- 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.

When you are analyzing the variance of a data set, the larger the variance, the larger the spread. The number 329.72 tells us that the data has a large spread, and that the numbers are very different from the mean. For more information on variance, check out our lesson on 'Population and Sample Variance.'

You can also use standard deviation to find the spread in a data set. For simplicity, standard deviation is the square root of the variance. Therefore, the standard deviation of this data set is approximately 18.15. To see this concept in depth, check out our lesson 'Standard Deviation and Shifts in the Mean.'

The **spread in data** is the measure of how far the numbers in a data set are away from the mean or the median. The spread in data can show us how much variation there is in the values of the data set. It is useful for identifying if the values in the data set are relatively close together or spread apart. There are three methods you can use to find the spread in a data set: range, interquartile range, and variance.

**Range** is the difference between the highest and lowest values in a data set. You can find the range by taking the smallest number in the data set and the largest number in the data set and subtracting them. This is how Tabatha found the age range of the actors she needed for her play.

You can also find the spread in the data set by using the **interquartile range**, which is a value that is the difference between the upper quartile value and the lower quartile value. For this method you will have to find each quartile in the data set. To find the quartiles, follow these steps:

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

You can also use **variance**, which is how far a set of numbers are spread out. To find variance, follow these steps:

- Find the mean of the set of data.
- Subtract each number from the mean.
- Square the result.
- Add the numbers together.
- Divide the result by the total number of numbers in the data set.

Each of these methods tells us something about the spread in data. Range is best for data sets where you are looking for data that is really far apart and all-encompassing. Interquartile range is best for when you are looking at a group of numbers and comparing them to the average, such as test scores or performance-based data, such as game scores. Variance is best for showing how far the numbers are spread out from one another using a single value in comparison to the mean. The larger the variance value, the farther the numbers are spread out from the mean.

After this lesson, you should be able to:

- Explain what the spread in data is and define the three methods to identify it
- List the steps involved in finding the interquartile range and variance
- Describe when it is best to use either range, interquartile range, or variance

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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

- What is the Center in a Data Set? - Definition & Options 5:08
- Mean, Median & Mode: Measures of Central Tendency 6:00
- How to Calculate Mean, Median, Mode & Range 8:30
- Calculating the Mean, Median, Mode & Range: Practice Problems 7:13
- Visual Representations of a Data Set: Shape, Symmetry & Skewness 5:22
- Unimodal & Bimodal Distributions: Definition & Examples 5:29
- The Mean vs the Median: Differences & Uses 6:30
- Spread in Data Sets: Definition & Example 7:51
- Quartiles & the Interquartile Range: Definition, Formulate & Examples 8:00
- Finding Percentiles in a Data Set: Formula & Examples 8:25
- Calculating the Standard Deviation 13:05
- The Effect of Linear Transformations on Measures of Center & Spread 6:16
- Population & Sample Variance: Definition, Formula & Examples 9:34
- Ordering & Ranking Data: Process & Example 6:54
- Go to Summarizing Data

- Go to Probability

- Go to Sampling

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