The Five Number Summary When we look at large amounts of data, it can sometimes be more convenient to describe the data, rather than list it each time. A five number summary is one such description. We are already familiar with three of the five numbers in a five number summary. They are the minimum, median, and maximum. The other two numbers are known as the first and third quartiles . 10, 10, 11, 15, 17, 25, 27, 27, 30, 35, 38, 39, 41, 41, 41, 44, 50 Minimum: 1st Quartile (Q 1 ): Median (Q 2 ): 3rd Quartile (Q 3 ): Maximum: The 1 st quartile is the “median” of the numbers that are less than the median. The 3 rd quartile is the “median” of the numbers that are greater than the median.

Box Plots A five number summary can be used to create a visual representation of data. Can you find the members of the five number summary in this box plot? On your own, write a five number summary of the heights (in inches) of the 2007-2008 Men’s Basketball team at Haverford College, and draw a box plot to represent it. We will review in three minutes. Here are the heights: 70 70 71 71 72 72 72 72 73 73 76 77 77 79 79 0 10 20 30 40 50 60

Haverford College Men’s Basketball Heights Min: 70” Q1: 71” Med: 72” Q3: 77” Max: 79” You can see that the “box” part of the plot seems to be strongly emphasized. The size of this area is very important, and we call it the interquartile range or IQR . IQR = Q 3 – Q 1 The interquartile range, along with the range (max – min) is known as a measure of spread . 68 70 72 74 76 78 80 82 84

68 70 72 74 76 78 80 82 84 66 Calculate the range and the IQR for the heights of the three teams above. Discuss with the person sitting next to you what the different numbers mean, and what, if any, conclusions you can draw from them.

Percentiles Often, you will hear about percentiles with regards to certain test scores. For example, my nephew Jacob: Is in the 75 th percentile with regards to weight for his age (8 weeks). What does this mean?

Here are the heights (in inches) once again of last year’s Haverford College Men’s Basketball team: 70 70 71 71 72 72 72 72 73 73 76 77 77 79 79 In what percentile is a player who is 6’4”? How would you determine the height that is at the 90 th percentile for the team?

Outliers We have discussed outliers informally in class, but we haven’t had a formal definition. One of the reasons for this is that there are many different definitions for outliers. We will now discuss just one of those definitions. Often, a statistic is considered an outlier if it lies more than a certain distance from the quartiles (either less than Q1 or greater than Q3). A commonly used criteria for that distance is 1.5 times the IQR. For the Fords (yup, that’s the nickname), were there any outliers? Min: 70” Q1: 71” Med: 72” Q3: 77” Max: 79”

Let’s go back to our comparison of Babe Ruth and Roger Maris. Do any of the statistics seem to be outliers? Use the 1.5 x IQR criterion to find out. 8 4 3 6 3 3 1 0 1 2 3 4 5 6 5 5 1 1 6 6 6 7 9 4 9 0 Maris Ruth

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