1. Analyzing One-Variable Data
Lesson 1.7
Measuring Variability
Statistics and Probability with Applications, 3rd Edition
Starnes, Tabor
Bedford Freeman Worth Publishers

2. Measuring Variability
Learning Targets
After this lesson, you should be able to:
Find the range of a distribution of quantitative data.
Find and interpret the interquartile range.
Calculate and interpret the standard deviation.

3. Measuring Variability
Being able to describe the shape and center of a distribution is a great start. However, two distributions can have the same shape and center, but still look quite different.
Both distributions are symmetric and single-peaked, with centers around 150. However, there are also some noticeable differences!
The variability of these two distributions is quite different.

4. How Spread Out is the Distribution?
Variation matters, and Statistics is about variation.
Are the values of the distribution tightly clustered around the center or more spread out?
Always report a measure of spread along with a measure of center when describing a distribution numerically.

5. Measuring Variability
There are several ways to measure the variability of a distribution. The three most common are the range, interquartile range, and standard deviation.
Range
The range of a distribution is the distance between the minimum value and the maximum value. That is,
Range = Maximum - Minimum
The range is not a resistant measure of variability. It depends only on the maximum and minimum values, which may be outliers.

6. Measuring Variability
We can avoid the impact of extreme values on our measure of variability by focusing on the middle of the distribution.
Order the data values from smallest to largest.
Find the quartiles, the values that divide the distribution into four groups of roughly equal size.
The first quartile Q1 lies one-quarter of the way up the list.
The second quartile is the median, which is halfway up the list.
The third quartile Q3 lies three-quarters of the way up the list.

7. Measuring Variability
Quartiles
The quartiles of a distribution divide the ordered data set into four groups having roughly the same number of values. To find the quartiles, arrange the data values from smallest to largest and find the median.
The first quartile Q1 is the median of the data values that are to the left of the median in the ordered list.
The third quartile Q3 is the median of the data values that are to the right of the median in the ordered list.

8. Measuring Variability
The interquartile range (IQR) measures the variability in the middle half of the distribution.
Interquartile Range (IQR)
The interquartile range (IQR) is the distance between the first and third quartiles of a distribution. In symbols,
IQR = Q3 − Q1
The quartiles and the interquartile range are resistant because they are not affected by a few extreme values.

9. FINDING QUARTILES
Listed below are the lengths of the touchdown passes for the Green Bay Packers over the course of several games
28, 18, 20, 30, 32, 27, 32, 20, 22, 31, 35, 39, 33, 19, 18
Find Q1, the median, and Q3 and explain what these values tell about the distribution

10. IQR = upper quartile – lower quartile
THE IQR
The difference between the quartiles is the interquartile range (IQR), so
IQR = upper quartile – lower quartile OR
Q Q1
Find the IQR of the Green Bay data and write a sentence explaining the meaning of this value.

11. Measuring Variability
If we summarize the center of a distribution with the mean, we should use the standard deviation to describe the variation of data values around the mean.
Standard Deviation
The standard deviation measures the typical distance of the values in a distribution from the mean. To find the standard deviation sx of a quantitative data set with n values:
Find the mean of the distribution.
Calculate the deviation of each value from the mean: deviation = value − mean
Square each of the deviations.
Add all the squared deviations, divide by n − 1, and take the square root.

12. Using Technology to Find the Standard Deviation
Larson/Farber 4th ed. 12

13. Example: Using Technology to Find the Standard Deviation
Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator or a computer to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.)
Office Rental Rates
35.00
33.50
37.00
23.75
26.50
31.25
36.50
40.00
32.00
39.25
37.50
34.75
37.75
37.25
36.75
27.00
35.75
26.00
29.00
40.50
24.50
33.00
38.00
Larson/Farber 4th ed. 13

14. Measuring Variability
Properties of the standard deviation as a measure of variability:
sx is always greater than or equal to 0. sx = 0 only when there is no variability, that is, when all values in a distribution are the same.
Larger values of sx indicate greater variation from the mean of a distribution.
sx is not resistant. The use of squared deviations makes sx even more sensitive than x to extreme values in a distribution.
sx measures variation about the mean. It should be used only when the mean is chosen as the measure of center.

15. Thinking About Variation
Since Statistics is about variation, spread is an important fundamental concept of Statistics.
Measures of spread help us talk about what we don’t know.
When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small.
When the data values are scattered far from the center, the IQR and standard deviation will be large.

16. Interpreting Standard Deviation
Standard deviation is a measure of the typical amount an entry deviates from the mean.
The more the entries are spread out, the greater the standard deviation.
Larson/Farber 4th ed. 16

17. Measuring Variability
Choosing Measures of Center and Variability
The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers.
Use the mean and sx only for roughly symmetric distributions that don’t have outliers.

18. LESSON APP 1.7 Have we found the beef?
Here are data on the amount of fat (in grams) in 12 different McDonald’s beef sandwiches, along with a dotplot. The mean fat content for these sandwiches is x-bar = grams.
Find the range of the distribution.
Find the interquartile range. Interpret this value in context.
Calculate the standard deviation. Interpret this value in context.
The dotplot suggests that the Bacon Clubhouse Burger, with its 40g of fat, is a possible outlier. Recalculate the range, interquartile range, and standard deviation for the other 11 sandwiches. Compare these values with the ones you obtained in Questions 1 through 3. Explain why each result makes sense.

19. Measuring Variability
Learning Targets
After this lesson, you should be able to:
Find the range of a distribution of quantitative data.
Find and interpret the interquartile range.
Calculate and interpret the standard deviation.