What you should learn

To calculate and interpret the range, quartiles, and interquartile range of sets of data.

NCTM Curriculm Standards 5 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Measures of Variation

Range

Interquartile Range

Quartiles

Lower Quartiles

Upper Quartiles

Outlier

 

Introduction: Justin Williams has been promoted and has the option of relocationg either to Columbus, Ohio, or San Francisco California. the Williams family lives in Tampa, Florida, where it is usually warm all year long. His family's wardrobe includes basically all lightweight apparel. He wondered if moving would mean they would have to get lots of new clothes. He looked up the average high temperature for the two cities and found the following information in a table of average high temperatures of US Cities.

Mr Williams calculated the mean and median of the temperatures of each city. The mean high temperature for Columbus was 63.5 and for San Francisco, it was 62.4. The median high temeprature for Columbus was 63.5, and for San Francisco it was 63. He figured that meant the climate in teh two cities was just about the same. But then he decided to take another look at the temepratures and find another way to analyze them.

The months and temepratures for each city form a relation. Mr. Williams decided to graph the relation for each city. He used different colors for each city.

He noticed that teh pattern in teh temperatures was very different. This case shows that measures of central tendency may not give an accurate enough description of the data. Often measures of variation are also used to describe the distribution of the data.

One of the most common measures of variation is the range. Unlike the other definition of range in this chapter, the rangeof a set of data is a measure of the spread of the data.

Let's find the range for each set of temperatures. Make a table to help organize the data.

The range of temepratures for Culumbus is greater than that for San Francisco. This means that temperatures in Culumbus vary more than temperatures in San Francisco. If his family moves to Columbus, he will definitely need some new clothes for their colder winters.

Another commonly used measure of variatgion is called the interquartile range. In a set of data, the quartiles are values that divide the data into four equal parts. Statisticians often use Q1, Q2, and Q3 to represent the three quartiles. Remeber that the median separates the data into two equal parts. Q2 is the median. Q1 is the lower quartile. It divides the lower half of the data into two equal parts. Likewise, Q3 is the upper quartile. It divides the upper half of the data into two equal parts. The difference between the upper and lower quartiles is the interquartile range (IQR).

Exercise 1:The table below shows the heights and weights of 13 players on the 1999 Dallas Cowboys football team. Find the median, upper and lower quartiles, and the interquartile range for the weights of the players.

Order the 13 weights from least to greatest. Then find the median.

The lower quartile is the median of the lower half of the data, and the upper quartile is the meadian of the upper half of the data. If the median is an item in the set of data, it is not included in either half. (Q1 = 208 and Q3 = 266). The interquartile range is 266 - 208 = 58. Therefore, the middle half, or 50% of the weights of the football players, varies within 58 pounds.

In example 1, suppose the greatest weight was 355 pounds instead of 311 pounds. In a set of data, a value that is much greater or much less than the rest of the data can be called an outlier. An outlier is defined as any element of a set of data that is at least 1.5 interquartile ranges greater than the upper quartile or less than the lower quartile.

The interquartile range of the football data is 58 so 1.5(58) = 87.

From the diagram above, you can see that the only item of data occurring beyond the purple numbers is 355. So, this is the only outlier.

Exercise 2: The double stem-and-leaf plt below represents the tonnage of domastic and foreight imports being delivered at the top 19 busiest ports in the United States during a recent year.

a. Find the mean, median, lower quartiles, upper quartiles, and mode for each (domestic and foreign).

b. Find the interquartile ranges of the domestic and teh foreign tonnage.

c. Which type of imports hade a more consistent tonnage-the domestic or the foreign?

d. Find any outliers.

 

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 11 - 21 odd, 22, 23 - 29 odd, 30 - 34

Alternative Homework: Enriched: 12 - 20 even, 22 - 34

Extra Practice: Students book page 769 Lesson 5-7

Extra Practice Worksheet: Click Here.

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