## Learning Math: Data Analysis, Statistics, and Probability

# Min, Max and the Five-Number Summary Homework

**Problem H1**

Determine the Five-Number Summary for each of the remaining data sets of raisin counts from Session 2, and construct a box plot for each on the same scale as the ones you built in Problem D2. Then interpret the quantities in each Five-Number Summary. In other words, use your results to answer the question “How many raisins are in a half-ounce box of raisins?” for each brand.

Here are the raisin counts for boxes of Brand C and Brand D raisins:

**Problem H2
**Based on the interpretations you made in Problems D2 and H1, which brand of raisins would you buy? Explain.

**Problem H3
**Consider the following data on sex, height, and arm span for 24 people from Session 1, Problem B3:

**a. **Determine the Five-Number Summary and box plot for the 24 heights.**
b.** Determine the Five-Number Summary and box plot for the 24 arm spans.

**c.**Determine the Five-Number Summaries and box plots for the 12 males’ and 12 females’ heights. How do the box plots help you compare these two sets?

**d.**Determine the Five-Number Summaries and box plots for the males’ and females’ arm spans.

**TAKE IT FURTHER**

Problem H4

Describe how you would create a Four-Number Summary that divides the data into three groups with approximately one-third of the data in each group. Include instructions for determining the positions of T1 and T2 (the locations of the dividing points of the first and second thirds of the data).

**Suggested Readings:**

**Friel, Susan and O’Connor, William (March, 1999). Sticks to the Roof of Your Mouth? Mathematics Teaching in the Middle School, 4 (6), 404-411.**

Reproduced with permission from

*Mathematics Teaching in the Middle School.*Copyright ©1999 by the National Council of Teachers of Mathematics. All rights reserved.

**Download PDF File:**

Sticks to the Roof of Your Mouth?

Continued

**Kader, Gary and Perry, Mike (Summer, 1996). To Boxplot or Not to Boxplot? Teaching Statistics, 18 (2), 39-41.**

This article first appeared in

*Teaching Statistics*<http://science.ntu.ac.uk/rsscse/ts/> and is used with permission.

### Solutions

**Solution H1**

For Brand C, there are 28 data entries. Here is the Five-Number Summary:

Min = 25

Q1 = 26

Med = 28

Q3 = 29

Max = 32

Here is the box plot:

The box plot shows that the center 50% of the data lies between 26 and 29, with 25% of the data falling between 28 and 29. Compared to other brands, Brand C has less variation, although there are a few boxes that have 30 or more raisins.

For Brand D, there are 36 data entries. Here is the Five-Number Summary:

Min = 23

Q1 = 27

Med = 29

Q3 = 33

Max = 38

Here is the box plot:

The box plot shows that the center 50% of the data lies between 27 and 33, with 25% of the data falling between 29 and 33. Compared to other brands, Brand D has a large number of boxes with 30 or more raisins, but it also has far greater variation than Brand C, with boxes of as few as 23 raisins.

**Solution H2
**Answers will vary. The comparison of four box plots generally suggests that Brands A and D offer the greatest chance for a lot of raisins in a box, although Brand C offers the most consistency and the highest minimum number of raisins in a box. The answer to this question really depends on the goals of the individual purchasing the raisins!

**Solution H3**

The first step is to order the lists from lowest to highest. Please note: Although each list is ordered from lowest to highest, the height and arm span measurements at the same position in the ordered list do not correspond to the same individual. (For example, the lowest height and lowest arm span are not necessarily from the same person.) Females are marked in **bold**.

Here is the Five-Number Summary for the 24 heights:

Min = 155

Q1 = 164

Med = 176

Q3 = 184.5

Max = 193

Here is the box plot:

**b.** Here is the Five-Number Summary for the 24 arm spans:

Min = 156

Q1 = 161.5

Med = 175

Q3 = 188

Max = 200

Here is the box plot:

**
c.** Here are the Five-Number Summaries:

*Males’ Heights*

Min = 173

Q1 = 179

Med = 183

Q3 = 186.5

Max = 193

*Females’ Heights*

Min = 155

Q1 = 161

Med = 164

Q3 = 170

Max = 188

Here are the box plots:

Comparing the box plots clearly shows that the males’ heights are significantly greater than the females’. In particular, the third quartile value of females’ heights was shorter than the minimum of males’ heights, which shows that, in this survey, at least 75% of the females were shorter than the shortest male. However, the maximum height of a female is fairly close to the maximum height of a male, primarily because there was one very tall female!

**d.** Here are the Five-Number Summaries:

*Males’ Arm Spans*

Min = 173

Q1 = 177.5

Med = 188

Q3 = 191

Max = 200

*Females’ Arm Spans*

Min = 156

Q1 = 159.5

Med = 161.5

Q3 = 170

Max = 188

Here are the box plots:

Solution H4

The Four-Number Summary comprises the maximum, the minimum, and two values T1 and T2, which mark the endpoints of the first and second thirds of your data. The locations of T1 and T2 are determined by n, the number of values in the data set, and are also based on whether n is a multiple of three, one more than a multiple of three, or two more than a multiple of three.

If n is a multiple of three, then the position of T1 is (n / 3 + 1/2) and the position of T2 is (2n / 3 + 1/2). For example, if n = 12, T1 will be between the fourth and fifth data value (i.e., position [4.5]), and T2 will be between the eighth and ninth data value (i.e., position [8.5]). In this example, each of the three groups contains exactly four values.

If n is one more than a multiple of three, then the position of T1 is (n + 2) / 3, and the position of T2 is (2n + 1) / 3. For example, if n = 13, T1 will be the fifth position and T2 will be the ninth position. In this example, each group contains five values, if you include the endpoints.

If n is two more than a multiple of three, then the position of T1 is (n + 1) / 3, and the position of T2 is (2n + 2) / 3. For example, if n = 14, T1 will be the fifth position and T2 will be the 10th position. In this example, each group contains four values, if you don’t include the endpoints.

Other answers are also possible, depending on whether you include the endpoints of the intervals.