Math 167 Pre-Statistics. Chapter 4 Summarizing Data Numerically Section 3 Boxplots
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1 Math 167 Pre-Statistics Chapter 4 Summarizing Data Numerically Section 3 Boxplots Objectives 1. Find quartiles of some data. 2. Find the interquartile range of some data. 3. Construct a boxplot to describe a distribution without outliers. 4. Identify outliers. 5. Construct a boxplot to describe a distribution with one or more outliers. 6. Compare the boxplots of two groups of data. Vocabulary 1. quartile 2. interquartile range 3. boxplot 4. whisker 5. left/right fence 6. outlier 1
2 OBJECTIVE 1 Find quartiles of some data. Definition: First quartile, second quartile, and third quartile The first quartile, second quartile, and third quartile are the 25th, 50th, and 75th percentiles, respectively. We use the symbols Q 1, Q 2, and Q 3 to stand for the first, second, and third quartiles, respectively. 9 observations: 1, 2, 3, 4, 5, 6, 7, 8, 9 Q + = 2.5 Q 0 = 5 = M Q 2 = observations: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Q + = 3 Q 0 = 5.5 = M Q 2 = 8 11 observations: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Q + = 3 Q 0 = 6 = M Q 2 = 9 12 observations: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Q + = 3.5 Q 0 = 6.5 = M Q 2 = 9.5 If some data have an odd number of data values, do not include the median in the lower half of the data when finding Q 1. Likewise, do not include the median in the upper half of the data when finding Q 3. 2
3 1. Students in one of the author s statistics classes were surveyed about the number of hours they spend watching videos, movies, and television shows on a weekday. Here are the anonymous responses (in hours) of ten of the students: 4, 10, 3, 2, 0, 2, 1, 2, 4, 7. a. Construct a histogram to determine whether the mean or the median is the better measure of the center. Explain. b. Find Q 1, Q 2 (M), and Q 3. c. Use technology to verify the results you found in Part b. d. Describe the meaning of the results you found in Part b. 3
4 OBJECTIVE 2 Find the interquartile range of some data. Definition: Interquartile Range The interquartile range (IQR) is given by IQR = Q 2 Q +. The IQR measures the spread of the middle 50% of the observations (from Q + to Q 2 ). 2. In Problem 1, we found Q + = 2, Q 0 = 2.5, and Q 2 = 4 for the data about the number of hours students spend watching videos, movies, and television shows per weekday: 4, 10, 3, 2, 0, 2, 1, 2, 4, 7. a. Calculate the IQR. b. Calculate the range. c. What do the IQR and the range mean in this situation? d. Discuss whether the IQR and the range are sensitive to the outlier 10 hours. The IQR is resistant to outliers. The range is sensitive to outliers. 4
5 OBJECTIVE 3 Construct a boxplot to describe a distribution without outliers. 3. The number of annual shark attacks in Hawaii are shown in the table for the years a. Construct a boxplot. b. Which is longer, the left part of the box or the right part? Which is longer, the left whisker or the right whisker? Assuming the distribution is unimodal, is the distribution skewed left, skewed right, or symmetric? c. Refer to the boxplot to estimate the range. d. Refer to the boxplot to estimate the IQR. What does it mean in this situation? What does it mean in this situation? 5
6 If one whisker is longer than the other, that does not mean more observations are represented by the longer whisker. It means that the observations represented by the longer whisker are more spread out than the approximately equal number of observations represented by the shorter whisker. Meaning of Boxplots That Describe Distributions without Outliers A boxplot that describes a distribution without outliers consists of four parts: the left whisker, the left part of the box, the right part of the box, and the right whisker. Each part represents 25% of the observations. The full length of the box is equal to the IQR. Boxplots may be drawn horizontally or vertically. The following two statements apply to unimodal distributions: If the left part of the box is at least as long as the right part, and the left whisker is quite a bit longer than the right whisker, then the distribution is skewed left. If the right part of the box is at least as long as the left part, and the right whisker is quite a bit longer than the left whisker, then the distribution is skewed right. 6
7 4. The percentages of residents who are nonreligious are described by the boxplot for the fifty states. a. It turns out the distribution is unimodal. On the basis of the boxplot, is the distribution symmetric, skewed left, or skewed right? Explain. b. Estimate the range. c. Estimate the IQR. What does it mean in this situation? What does it mean in this situation? 7
8 OBJECTIVE 4 Identify outliers. Definition: Fences The left fence of some data values are given by: left fence = Q IQR The right fence of some data values are given by: right fence = Q IQR Definition: Outlier An outlier is an observation that is less than the left fence or greater than the right fence. We say that the outlier lies outside the fences. 5. In Problems 1 and 2, we found Q + = 2, Q 2 = 4, and IQR = 2 for the data about the number of hours students spend watching videos, movies, and television shows per weekday: 4, 10, 3, 2, 0, 2, 1, 2, 4, 7. Identify and interpret any outliers. 8
9 OBJECTIVE 5 Construct a boxplot to describe a distribution with one or more outliers. 6. Construct a boxplot that describes the data about the number of hours students spend watching videos, movies, and television shows per weekday: 4, 10, 3, 2, 0, 2, 1, 2, 4, Which is longer, the left part of the box or the right part? Which is longer the left whisker or the right whisker? What do your responses tell you about the shape of the distribution? 9
10 8. The number of annual shark attacks in California, Florida, and Hawaii are described by the boxplots for the years OBJECTIVE 6 Compare the boxplots of two groups of data. a. Compare the centers of the three distributions. What does this mean in this situation? California Florida Hawaii 10
11 b. Compare the spreads of the three distributions. What does this mean in this situation? California Florida Hawaii c. Compare the IQR of Florida s distribution to the range of California s distribution. What does this mean in this situation? Florida California 11
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