BLoCK 3 ~ dAtA AnALYsIs Data Displays Lesson 14 HisTograMs ------85 Lesson 15 anaLYzing HisTograMs ------92 Explore! Predicting from Histograms Lesson 16 sTeM-and-LeaF PLoTs ------98 Explore! Making a Stem-and-Leaf Plot Lesson 17 anaLYzing sTeM-and-LeaF PLoTs ------104 Lesson 18 Box-and-wHisker PLoTs ------109 Lesson 19 anaLYzing Box-and-wHisker PLoTs ------115 Explore! Visualizing Skew review BLock 3 ~ daTa disPLaYs ------122

word wAll left skewed

eVen distriBution Box-And -whisker histogrAm Plot

stem -And -leAf distriBution Plot normAl

freQuency tABle right skewed douBle stem -And -leAf Plot Plot -whisker Box-And PArAllel

Block 3 ~ Data Analysis ~ Data Displays 83 BLoCK 3 ~ dAtA dIsPLAYs tic - tac - tOe

mAtching chAllenge grAPh switcheroo more And more stems!

Match graphs to data sets. Change from one type of Learn and apply some Challenge a friend to solve graph to another type of creative ways to make the matching problems. graph in order to change stem-and-leaf plots. the look of a data set.

See page  for details. See page  for details. See page  for details.

A skew letter surVey sAys! Prediction time!

Th e Mean writes a letter Conduct a survey of Create a brochure or poster to his friend, the Median. classmates. Display and to show how graphs can be He explains why he likes analyze the data that used to make predictions. skewed right graphs. you collect.

See page  for details. See page  for details. See page  for details.

nArrower And wider fliP-Book skew in the reAl world

Explore changes in Create a fl ip-book about Find a real-world example appearance of histograms how to make each type of of skewed data. Graph and when you increase or data display in this Block. analyze your data set. decrease the interval width.

See page  for details. See page  for details. See page  for details.

84 Block 3 ~ Data Displays ~ Tic - Tac - Toe histOgrams Lesson 14

It is oft en helpful to look at visual displays of data to fi nd patterns or compare sets of data. One helpful way to display a set of data is a histogram. A histogram is a bar graph in which data values are organized into equal intervals. While a bar graph typically displays data in categories, a histogram displays data in the form of numbers.

Ms. Sanchez collected data about students and their pets. Th e fi rst graph, a bar graph, shows the number of each type of pet her students have. Th e second graph is a histogram.

Students and their Types of Pets 12

10

8

6

4

Number of Students 2 Number of Students 0 Dogs Cats Rodents Reptiles Fish None

Th e histogram focuses on the number of pets that each student has rather than the type of pets. Th ere are two key parts of the histogram: the intervals along the horizontal axis of the graph and the number of students that fall into each interval along the vertical axis.

Each interval on a histogram includes the number on the left -hand side of an interval up to the number on the right-hand side. For example, the fi rst bar in the histogram includes all students that have 0 or 1 pet. Students who have 2 pets are included in the second interval.

Lesson 14 ~ Histograms 85 example 1 use the histogram to answer each question. a. how many students had 2 or 3 pets? b. how many students had at least 8 pets? c. do more students have fewer than 6 pets or more than 6 pets? how can you tell? d. What does it mean that the interval between 6-8 is empty? Number of Students e. how many students are in Ms. sanchez’ class? solutions a. Look at the height (number of students) of the bar which includes 2 or 3 pets. It shows that 10 students fall in that interval. b. Add the heights of the bars to the right of 8 on the horizontal axis. Th ere are 3 students who have at least 8 pets. c. Th e bars representing fewer than 6 pets are taller. More students fall into that category. d. Th is means there are 0 students who had exactly 6 or 7 pets.

e. Add the heights of all bars together. Th e sum represents the number of students in the class. 13 + 10 + 5 + 0 + 1 + 2 = 31 students

Before making the histogram, Ms. Sanchez had to make a frequency table. A frequency table shows a tally of how many times a value occurred in each interval. Ms. Sanchez tallied the students who had each number of pets in her intervals.

number of pets tally 0-2 |||| |||| ||| 2-4 |||| |||| 4-6 |||| 6-8

8-10 | Number of Students 10-12 ||

86 Lesson 14 ~ Histograms example 2 Th e table at right shows the heights of some of oregon’s tallest mountains. use the data to make a histogram.

height solution Th e minimum height is 4,097 feet. Mountain (ft ) Th e maximum height is 11,239 feet. Rock Creek Butte 9,105 Mt. Hood 11,239 Use an interval width of 1,000 to make a frequency Mt. Jeff erson 10,495 table. Mary’s Peak 4,097 Mt. Bolivar 4,319 height of Mtns. tally Mt. Ashland 7,532 4,000 - 5,000 | | Steens Mtn. 9,773 5,000 - 6,000 North Sister 10,085 6,000 - 7,000 Middle Sister 10,047 7,000 - 8,000 | | | South Sister 10,358 Mt. Washington 7,795 8,000 - 9,000 | | | Broken Top 9,175 9,000 - 10,000 | | | | Mt. Bachelor 9,068 10,000 - 11,000 | | | | Th ree-Fingered Jack 7,841 11,000 - 12,000 | Mt. Scott 8,929 Mt. McLoughlin 9,495 Put the information from the frequency table into a Mt. Mazama 8,159 histogram. Place the intervals along the horizontal Aspen Butte 8,208 Source: Oregon Blue Book axis. Th e height of the bars is determined by how many mountains are in each interval.

Number of Mountains

Height of Mountains

Any interval width works for histograms. However, it is ideal to have between 4 and 12 intervals to best display the distribution of data in a histogram. Too few intervals in a histogram will not show the distribution in the graph well. Too many bars will spread the data out so the bars will have very little height.

Lesson 14 ~ Histograms 87 example 3 Make a histogram for students’ pulse rates (in beats per minute) below. 64, 76, 60, 70, 68, 70, 88, 62, 54, 68, 70, 72, 60, 92, 64, 76, 70, 68, 62, 72 solution Identify the minimum and maximum values. Minimum = 54 Maximum = 92 Th e frequency table must start low enough to include 54 and must go high enough to include 92.

Th e overall range of the data is 92 – 54 or 38. Determine an appropriate interval width. 38 ÷ 8 ≈ 5 An interval width of 5 will create about 8 bars.

Complete the frequency table for the data.

Pulse rates number of students tally 50 - 55 | 55 - 60 60 - 65 | | | | | 65 - 70 | | | 70 - 75 | | | | | 75 - 80 | | 80 -85 85 -90 | 90 - 95 |

Use the information in the table to draw the histogram.

Number of Students

Pulse Rate (beats per minute)

Each number in the data set should only fall in one interval. If a number is on the border between intervals, count it in the uppermost interval. Add up all the tallies in your frequency table. Th ey should sum to the total number of values in the data set. In example 3, the tallies sum to 20 since there were 20 values in the data set.

88 Lesson 14 ~ Histograms exercises

1. What is the difference between a bar graph and a histogram?

2. The number of miles driven on seventeen cars was recorded and the data was displayed in the histogram below. Use the graph to answer the questions.

Number of Cars

Number of Miles (in thousands) a. Including the “empty” intervals, how many intervals are shown in the histogram? Is the graph drawn the best way possible? Explain. b. Copy down the frequency table below and use the information in the histogram above to fill in the blanks. What is the new interval width? number of Miles driven number of (in thousands) Cars 10 – 20 20 – 30 30 – 40

40 – 50 c. Use the frequency table in part b to create a new histogram.

3. Twelve newborn babies were weighed and the data was put in the frequency table below. Use the table to answer the questions. Weight of newborn tally (in Pounds) 4 – 5.5 | 5.5 – 7 | | | 7 – 8.5 | | | | 8.5 – 10 | | 10 – 11.5 | a. How many babies have been tallied so far? b. What is the interval width? c. Glen thought a 7-pound baby should be tallied in the 5.5 – 7 interval. Eliza claimed it should be tallied in the 7 – 8.5 interval. Who is correct? Explain. d. A 10-pound baby boy is born. Which interval should he be tallied in?

Lesson 14 ~ Histograms 89 4. Ben collected and weighed 20 frogs for his science class. He needs to do a report showing the weight distribution of frogs. Here is the data he collected. Follow the steps below to create a histogram for the data.

Weights of Frogs (in ounces) 5.2 5.9 6.7 4.0 6.3 7.0 6.7 7.2 7.9 5.8 6.0 6.7 7.2 6.8 6.2 6.5 7.1 7.4 6.2 6.6

a. Find the minimum and maximum values in the data set. What would be a reasonable interval width to use for this data set? b. Use your interval width in part a to create a frequency table for the data. Be sure that each data value is only included in one interval. c. Use your frequency table to create a histogram. Be sure to label both axes.

5. Dierdre collects old movies. The fifteen movies in her collection have the following copyright dates. Choose an appropriate interval width. Make a frequency table and histogram for Dierdre’s movie data. 1958, 1963, 1945, 1966, 1974, 1958, 1962, 1968, 1950, 1963, 1959, 1969, 1948, 1965, 1962

6. Stephanie buys purses. She paid the following prices for twelve purses. Choose an appropriate interval width. Make a frequency table and histogram for Stephanie’s purse data. $17, $11, $22, $10, $13, $10, $27, $14, $49, $12, $18, $20

7. The histogram shows the number of hours students in Mr. Underhill’s class slept last night. Use the histogram to answer each question. a. How many students were included in the survey? b. What is the interval width? c. How many students got less than 4 hours of sleep last night? Number of Students of Number d. Which interval included the most students? e. How many students got between 4 and 8 hours of sleep last night?

8. The ages of the fourteen people at the local coffee shop on Saturday morning are given below. 15, 22, 37, 40, 8, 14, 21, 25, 32, 23, 46, 26, 30, 28 a. What are the minimum and maximum values in the data set? b. Create a histogram with an interval width of 5. c. Create a histogram with an interval width of 10. d. Which histogram would give the owner of the coffee shop the most information about his customers? Explain.

90 Lesson 14 ~ Histograms 9. The tables below show the population of all 36 counties in Oregon as estimated in July 2006. Use the data to create a histogram. County Pop. County Pop. Multnomah County 681,454 Clatsop County 37,315 Washington County 514,269 Malheur County 31,247 Clackamas County 374,230 Tillamook County 25,380 Lane County 337,870 Union County 24,345 Marion County 311,304 Wasco County 23,712 Jackson County 197,071 Crook County 22,941 Deschutes County 149,140 Curry County 22,358 Linn County 111,489 Hood River County 21,533 Douglas County 105,117 Jefferson County 20,352 Yamhill County 94,678 Baker County 16,243 Josephine County 81,688 Morrow County 11,753 Benton County 79,061 Lake County 7,473 Polk County 73,296 Grant County 7,250 Umatilla County 72,928 Harney County 6,888 Klamath County 66,438 Wallowa County 6,875 Coos County 64,820 Gilliam County 1,775 Columbia County 49,163 Sherman County 1,699 Lincoln County 46,199 Wheeler County 1,404 Source: U. S. Census Bureau, 2006 Population Estimates

review

Find the three measures of center for each data set.

10. 16, 25, 32, 36, 36 11. 33, 24, 39, 37, 37, 24, 33, 39

Find the five-number summary of the following data sets. 12. 39, 44, 47, 51, 53, 53, 58 13. 63, 72, 78, 83, 87, 93, 95, 100

14. Hector spent 65% of his savings on a new MP3 player. The MP3 player cost $200. How much did Hector have in his savings account?

15. A $36 pair of pants is on sale for 25% off. The sign above the rack advertises an additional 10% off the discounted price. How much will the pants end up costing after the discounts?

16. A $32 sweater is on sale for 15% off. A sales tax of 8% will be included in the total price. a. Anna claims that you can simply discount 7% off the original price to find the final cost with tax. Leticia says you must first find the discounted price, then add on the 8% tax. Who is correct? Explain. b. Find the final cost of the sweater, including the tax.

Lesson 14 ~ Histograms 91 analyZing histOgrams Lesson 15

Histograms provide a visual picture of the spread of a data set. Th ese graphs can be used to answer questions about the data and make predictions. Th e shape of a histogram is also helpful when making comparisons of the mean and median of data sets.

explOre! preDicting FrOm histOgrams

Th e histogram below shows the results of a survey on the number of traffi c tickets received by a random group of drivers.

step 1: How many people received either 4 or 5 traffi c tickets?

step 2: Which interval had the most people?

step 3: How many people were included in this survey? Number of People

step 4: What is the ratio of people with 4 or 5 tickets to the total number of people included in the survey?

step 5: Sixty people were at the town hall meeting. Based on the previous survey results shown in the histogram, predict how many of the 60 people might have received 4 or 5 tickets. Use the ratio from step 4 to set up a proportion like the one shown below.

In the Histogram With 60 People ______Number with 4 or 5 tickets = ______Number with 4 or 5 tickets Total people 60 people

step 6: Use a proportion and the information in the histogram to predict how many people in the group of 60 may have fewer than 4 traffi c tickets.

Th e histogram at left shows the grades of students (as percents) with no missing assignments in one math class.

step 7: What is the ratio of students with at least 80% to the total number of students with no missing assignments?

step 8: Th ere are a total of 100 students with no missing Number of Students assignments at the school. Predict how many have at least 80% in their math class. Set up a proportion to

Students’ Grades (%) solve.

92 Lesson 15 ~ Analyzing Histograms The shape of histograms can be described in the following ways:

Skewed right Skewed left Normal distribution The graph has a longer tail The graph has a longer tail The majority of the values on the right-hand side. on the left-hand side. are located in the middle of the data set. example 1 Predict the shape of the histograms representing the following data sets. explain. a. students’ shoe sizes b. number of siblings that each student has c. percents scored by students on an easy quiz d. distance that students travel to school solutions a. Normal distribution. Some students have big feet and some have small feet. More people would likely fall in the middle.

b. Skewed right. Most students probably have fewer than 3 siblings. As you get further away from 0 there are likely fewer and fewer students with that number of siblings.

c. Skewed left. Since the quiz was easy, it is likely there were more students receiving higher percentages and not very many lower percentages.

d. Normal distribution would best describe the distances. There would be fewer students who travel really far or travel very little. Most would likely be somewhere in the middle. example 2 Vicki asked some of her classmates how many children lived at their house. she put the data into the following histogram. use the graph to answer the questions.

a. how many students did Vicki include in her

survey? b. describe the shape of the histogram. c. In which interval would the median data value fall?

d. There are 25 students in Vicki’s class. owh Number of Students many would have at least 4 children living at their house? Number of Children at Home

Lesson 15 ~ Analyzing Histograms 93 example 2 a. Add the heights of each bar. 4 + 7 + 2 + 1 + 1 = 15 students solutions b. Th ere is a longer tail on the right-hand side of the graph. Th e data is skewed right.

c. Because there are 15 students in the data set, the median value is the eighth value. Th e fi rst interval only has 4 values, so the median would not fall in that interval. Th e fi rst two intervals combined have 11 values, so the median must fall somewhere in the second interval. Th e median is in the 2 – 4 interval.

d. Add the heights of the bars 2 + 1 + 1 = 4 showing 4 or more children at home. Set up a proportion and solve. __ 4 = __ x 15 25

100_ = 15x 6.6 = x

Approximately 7 students in Vicki’s class have at least 4 children living at their house.

In Lesson 10 you learned that outliers have a greater eff ect on the mean of data sets than on the median. Similarly, with skewed graphs, the mean is typically more in the direction of the skew. For example, a graph that is skewed right will have a mean greater than the median. A graph that is skewed left will have a mean less than the median.

example 3 Th is histogram shows the price of insurance premiums for eleven drivers who are considered “accident prone” by their insurance agencies. a. describe the shape of the histogram. b. In which interval would the median data value fall? c. Below are the actual premiums paid in dollars by the drivers in the graph: ($) 71, 83, 92, 93, 100, 106, 109, 111, 112, 117, 118.

Find the mean and median premium amount Number of Drivers for these drivers. d. how does the mean compare to the median? solutions a. Th e graph has a longer tail on the left -hand side. Th e graph is skewed left .

b. Since there are eleven values, the median would be the sixth value. Th e sixth value would fall in the 100 – 110 interval.

94 Lesson 15 ~ Analyzing Histograms example 3 c. Mean = ______Sum of Values = ____ 1112 ≈ 101.09 solutions Number of Values 11 (continued) On average, these drivers pay approximately $101.09.

Find the middle number of the ordered data set to find the median. 71, 83, 92, 93, 100, 106, 109, 111, 112, 117, 118 → → → → → ← ← ← ← ← The median of the data set is $106.

d. The mean is less than the median.

exercises

1. Draw a histogram that is skewed right.

2. Draw a histogram that has a normal distribution.

Based on the following data sets, predict the shape of the histogram. explain your choice.

3. number of computers at students’ homes 4. length of female students’ hair

5. number of pets that students have 6. birth weight of babies

7. The histogram shows the heights of fifteen sunflowers in inches. Use the histogram to answer each question. a. Describe the shape of this histogram. b. Which interval has the most sunflowers in it? c. In which interval will the median data value fall? d. What is the ratio of sunflowers less than 64 inches tall to total sunflowers? e. Based on the data in the histogram, if 80 sunflowers were Number of Sunflowers collected, how many would be expected to be less than 64 inches in height?

Height (inches) 8. The actual heights of the sunflowers in exercise 7 are listed below. Answer the questions with this data. heights (inches) 57, 60, 63, 65, 68, 68, 70, 71, 71, 72, 73, 74, 74, 74, 75 a. Find the median of the data set. b. The heights of these sunflowers add up to 1,035 inches. What is the mean height of the sunflowers? c. How does the mean compare to the median?

Lesson 15 ~ Analyzing Histograms 95 9. The histogram shows the heights of several eighth grade students. Use the graph to answer the questions.

a. Describe the shape of the histogram. b. How many students’ heights are recorded in the graph? c. Which interval has the most students?

d. In which interval will the median data value fall? e. What is the ratio of students taller than 66 inches to total students? f. Mrs. Zimmer’s eighth grade class has 30 students. How many Number of Students students could be expected to be taller than 66 inches? g. Two students in Mr. Tabor’s eighth grade class are taller than Height in inches 66 inches. Predict the total number of students in his class.

10. The histogram shows the annual salaries of the top seventeen salespeople at Matt’s Auto Barn. a. Describe the shape of the histogram. b. In which interval will the median data value fall? c. How does the mean salary likely compare to the median salary? Explain. Number of Salespeople Number of Salespeople d. What is the ratio of salespeople who earn over $55,000 to

those who earn less than $55,000? Salaries (Thousands of Dollars)

11. The histogram shows the number of inches of monthly rainfall in a town in Oregon. a. Describe the shape of the histogram. b. How does the mean rainfall likely compare to the median

rainfall? Explain. c. What is the ratio of months with less than 2 inches of rain to total months? d. In the next 4 years (48 months), how many months will Number of Months probably have less than 2 inches of rain? e. In the next 4 years, how many months will probably have at

least 6 inches of rain? Number of Inches of Rain review

12. Johanna kept track of her hours worked for the last fourteen weeks and recorded it in the table.

Hours Worked 24 25.5 32 22 19.5 24 23 18 27 8 23 21.5 25.5 19

a. Find the minimum and maximum values in the data set. What would be a reasonable interval width to use for this data set? b. Use the interval width from part a to create a frequency table for the data. Be sure that each data value is included in only one interval. c. Use the frequency table to create a histogram. Label both axes.

96 Lesson 15 ~ Analyzing Histograms use the partial data set and the statistics given to fi nd the missing number.

13. 34, 32, 32, 56, 30, ___ 14. 23, 17, 32, 31, 23, 38, ___ Mean = 36.5 Median = 25 Range = 21 use the IQr Method to determine if each set of data has any outliers. If so, state the outliers. If there are none, state “no outliers”.

15. 20, 26, 28, 29, 29, 31, 35 16. 30, 44, 50, 53, 55, 55, 57, 66

tic-tAc-toe ~ nArrower And wider

Histograms can have diff erent interval widths. It is ideal to have between 4 and 12 intervals to best show the distribution of a data set.

Use the data set below to make a histogram with an interval width of 10. 10, 14, 16, 19, 21, 22, 25, 26, 28, 29, 31, 31, 33, 37, 38, 40, 40, 43, 45, 46, 47, 47, 48

Use the same data set to make two more histograms with diff erent interval widths. Choose interval widths that allow between 4 and 12 intervals in your graph.

Aft er drawing your graphs, answer the following questions. 1. Which of the three histograms do you think shows the distribution of the data best? Explain.

2. Is the data set skewed or does it have a normal distribution? Does each of your three histograms have a similar shape? Does one of your histograms make the data look more skewed than another?

tic-tAc-toe ~ A skew letter

Pretend you are the Mean. You are writing a letter to your friend, the Median. In your letter to your friend you will explain why you like graphs that are skewed right better than graphs that are skewed left . Make sure you have mathematical justifi cation for your opinion. Your letter should show an understanding of the relationship between skewed graphs and their eff ect on the mean and median.

role: Th e M ean Audience: Th e Median Format: A Letter topic: Explaining why you like graphs that are skewed right better than graphs that are skewed left .

Lesson 15 ~ Analyzing Histograms 97 stem - anD - leaF plOts Lesson 16

Another type of graph that is useful for displaying data is a stem-and-leaf plot. A stem-and-leaf plot is made up of “stems”, the left most digits in the data, and “leaves”, the rightmost digits. In the example below, the fi rst digit (in the tens place) is used in the stem and the last digit (ones place) is used in the leaf.

Points scored by Players in a Basketball game 0 0 2 2 7 8 1 0 1 4 7 2 2 5 3 3 Key: 1 | 4 = 14 points scored

Stem-and-leaf plots have intervals just like histograms. Each stem represents an interval. For example, a stem of 7 might represent numbers in the seventies or seven hundreds. Each interval also has a bar representing how many numbers are in that interval. Th e diff erence from histogram bars is that stem-and-leaf plot bars are made up of numbers called leaves. Stem-and-leaf plots make identifying each and every value in the data set easier.

Th e histogram and stem-and-leaf plot below show percentages students received on the math quiz. Both data displays represent the same data. Notice how heights of bars in the histogram are equal to the number of leaves in the corresponding rows of the stem-and-leaf plot.

Histogram vs. Stem-and-Leaf Plot 7 3 7 8 0 5 8 9 1 1 1 4 8 9 10 2 6 Key: 9 | 4 = 94

Percentages

98 Lesson 16 ~ Stem - And - Leaf Plots explOre! making a stem-anD-leaF plOt

The table below shows the average monthly high temperature (in degrees Fahrenheit) from 1971-2000 in Portland, Oregon.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 46 50 56 61 67 73 79 79 74 63 51 46 Source: www.oregon.com/weather step 1: Record the minimum and maximum of the data set.

step 2: Write the stems vertically (up and down) on paper. Start with the tens digit of the minimum. Continue to the tens digit of the maximum. Draw a vertical line to the right of the stems to separate them from the leaves which will be added later.

step 3: Go through the data in the table and write in the leaves for your first stem. The numbers go to the right of the vertical line drawn in step 2. Put the leaves in order from lowest to highest. If a number appears twice, write its leaf twice. Evenly space out the leaves. Why is it important to evenly space the leaves?

step 4: Go through the rest of the data. Write in the leaves on the remaining stems (5, 6, 7). Again, be sure to put the leaves in order and evenly space them out like the leaves on the “4” stem.

step 5: Add a key to tell the reader how to interpret the numbers in the stem-and-leaf plot. For example: “Key: 7 | 3 = 73° Fahrenheit.” Box the key.

step 6: How many total “leaves” should there be in the stem-and-leaf plot?

step 7: Is the data in this plot skewed (have a longer tail) more towards the maximum or towards the minimum?

Real-world data often has more than two digits (one for the stem and one for the leaf). Stem-and-leaf plots can be made from data sets with 3 or more digits or even data sets with decimals. Just like in a histogram, it is ideal to have 4 to 12 bars in a stem-and-leaf plot. Keep this in mind as stems are chosen. example 1 The world’s 20 tallest buildings were constructed in the following years, in order of their height. Make a stem-and-leaf plot of the data. 2004, 1998, 1998, 1974, 1999, 2003, 1996, 1996, 1931, 1992, 1989, 1999, 1997, 1973, 1998, 1969, 2005, 2006, 1995, 2005 solution The minimum is 1931 and the maximum is 2006.

Decide which stems to use in the data display. Sears Tower Built 1974 If only the first digit is used, there will only be two bars (1 and 2). If the first two digits are used, again there will only be two “bars” (19 and 20). If the first three digits are used, there will be eight “bars” (193-200).

Eight bars would be best, so use three digits in each stem. Lesson 16 ~ Stem - And - Leaf Plots 99 example 1 Align the stems 193 – 200 on the 193 1 solution left . Include the vertical line to 194 (continued) separate the leaves. 195 196 9 Place the leaves in increasing 197 3 4 order and evenly space them. 198 9 199 2 5 6 6 7 8 8 8 9 9 Include a key which tells the reader how to interpret the 200 3 4 5 5 6 numbers in the plot. Key: 197 | 3 = 1973 example 2 determine the stems to use for each data set. a. data that goes from 1758 to 2007 b. data that goes from 12.5 to 44.9 c. data that goes from 0.256 to 0.987 Empire State Building Built 1931 d. data that goes from 11,580 to 12,410 solutions a. Stems of 1 and 2 would only give two bars. Stems of 175-200 would give too many bars. Stems of 17-20 would create a manageable four-bar stem-and-leaf plot. Use 17, 18, 19 and 20 as stems.

b. Using stems of 1, 2, 3 and 4 would be best. If you used a two-digit stem (12-44), there would be too many bars.

c. Stems of 02-09 would give eight bars, which would create a manageable plot. Using 025 to 098 would create too many bars.

d. Stems of 115-124 would give ten bars, which is manageable. Using a one-digit stem would only give one bar. A two-digit stem would only give two bars. exercises

1. How is a stem-and-leaf plot like a histogram? How is it diff erent?

2. Why is it important to include a key when making a stem-and-leaf plot?

3. List the data values from the stem-and-leaf plot. 0 4 7 1 2 2 5 2 9 9 3 2 Key: 1 | 2 = 12

4. example 1 states to include the stems of 194 and 195 even though they are empty. a. What does it mean to say that those stems are empty? b. Why is it important to include the empty stems?

100 Lesson 16 ~ Stem - And - Leaf Plots What is wrong with each stem-and-leaf plot?

5. 5 6 6. 5 6 7. 5 6 6 6 7 2 5 9 9 7 5 2 9 9 7 2 5 9 9 8 0 0 6 8 9 8 0 6 0 8 9 8 0 0 6 8 9 9 2 3 4 8 9 3 4 2 8 9 2 3 4 8 10 0 0 10 0 0 10 0 0 Key: 7 | 5 = 75 Key: 7 | 5 = 75 Key: 7 | 5 = 75

What stems would you use for each data set?

8. Data that goes from 9. Data that goes from 10. Data that goes from 1752 to 1839 2.82 to 4.00 2655 to 7802

Make a stem-and-leaf plot of each set of data.

11. Ages of people at a movie: 12. Important years in a family’s history: 9, 12, 15, 16, 19, 19, 22, 1659, 1711, 1777, 1798, 1804, 1812, 1826, 24, 25, 27, 28, 28, 30 1860, 1912, 1945, 1978, 1985, 2005, 2005

13. Cost for meals at a local fast food 14. Birth years of people at a birthday party: restaurant (in dollars): 1972, 1968, 1969, 1979, 1984, 2006, 2004, 4.50, 3.75, 5.24, 3.98, 4.29, 6.15, 7.08, 1997, 1999, 2003, 2001, 2000, 2001, 1998 3.86, 4.82, 5.17, 4.38, 3.87, 5.44, 4.93

15. Use the table below to answer the questions.

ticket Prices for the top 20 Concert tours of 2005 group ticket Price group ticket Price 1. The Rolling Stones $134 11. Motley Crue $46 2. U2 $97 12. Green Day $38 3. Celine Dion $136 13. Toby Keith $46 4. Paul McCartney $135 14. Rascal Flatts $35 5. The Eagles $104 15. Bruce Springsteen $81 6. Elton John $102 16. Gwen Stefani $54 7. Kenny Chesney $55 17. Coldplay $41 8. Dave Matthews Band $47 18. Tom Petty & The Heartbreakers $38 9. Neil Diamond $63 19. Barry Manilow $154 10. Jimmy Buffett $76 20. Anger Management Tour $64 Source: Pollstar a. What are the minimum and maximum ticket prices in the data? b. What is the range of the prices? c. What would be appropriate numbers to use as the stems? d. Use the stems from part c to make a stem-and-leaf plot of this data. e. Describe the shape of your stem-and-leaf plot. Is it skewed or does it have a normal distribution?

Lesson 16 ~ Stem - And - Leaf Plots 101 16. Create a stem-and-leaf plot for the following precipitation data for Portland, Oregon. Describe the shape of the stem-and-leaf plot. Average Monthly Precipitation Jan Feb Mar Apr May Jun Jul Aug sep oct nov dec 6.24 5.07 4.51 3.10 2.45 1.60 0.76 0.99 1.87 3.39 6.39 6.75

17. The table shows various fruits and the number of calories per serving. Choose appropriate stems and make a stem-and-leaf plot of the data. Fruit Calories per serving Apple 44 Avocado 150 Banana 107 Date 13 Grapefruit Whole 100 Guava 24 Kiwi 34 Lemon 20 Mango 40 Nectarines 42 Orange 35 Peach 35 Pear 45 Pineapple 50 Plum 25 Tangerine 26 Source: www.weightlossforall.com review

Predict the shape of the histogram representing the following data sets. explain your reasoning.

18. Shoe sizes of students 19. number of unbroken eggs 20. number of states students in a class in a carton of 12 eggs have lived in

21. Twelve cats had check ups at the veterinary clinic. Their weights were recorded. The data is shown in the histogram. a. Describe the shape of the histogram. b. In which interval does the median data value fall? c. How does the mean cat weight likely compare to the median? Explain. d. What is the ratio of cats weighing less than 6 pounds to total cats at the veterinary clinic? e. One day there were 20 cats at the clinic. How many would likely weigh less than 6 pounds? f. One week the clinic had 30 cats weighing more than 12 pounds. How many total cats likely came to the clinic that week? Weight of Cats (in pounds)

102 Lesson 16 ~ Stem - And - Leaf Plots tic-tAc-toe ~ more And more stems!

In Lesson 16 you learned how a stem-and-leaf plot is similar to a histogram. It is ideal for these types of graphs to have about 4-12 intervals to best show the spread of data. With stem-and-leaf plots, there are some creative ways you can create more intervals in your plot.

Felix made a stem-and-leaf plot for some data that went from 82 to 98. He wasn’t happy with the appearance of his plot. His friend, Erin, shared her idea with him. Below is the original plot he created along with the one Erin suggested.

Felix’s Plot erin’s suggestion

8 2 8 9 8 2 9 0 0 1 2 4 5 5 7 7 8 8 8* 8 9 9 0 0 1 2 4 Key: 8 4 = 84 9* 5 5 6 7 7 8 8

Key: 8 2 = 82 8* 9 = 89

Felix’s plot only has two intervals which do not display the spread of the data very well. Erin’s plot shows the exact same data, but she used an asterisk (*) to create more intervals. Her fi rst stem (8) contains numbers from 80-84, her second stem (8*) contains numbers from 85-89.

1. If a new number, 84, was added to the data set, in which row (1st, 2nd, 3rd, 4th) would it appear in Erin’s plot?

2. In which row would the number 99 appear?

3. In which row would the number 87 appear? use erin’s * technique to create a stem-and-leaf plot of the following data sets.

4. 4, 9, 9, 10, 12, 14, 14, 15, 17, 17, 18, 19, 19

5. 87, 94, 100, 96, 81, 84, 87, 98, 82, 93, 81, 96, 80, 90

6. 44, 45, 38, 50, 49, 46, 40, 50, 38, 47, 42, 48 1 1 1# 2 Ian made the following stem-and-leaf plot for a data set. 1^ 4 5 1~ 6 7. List the numbers in Ian’s data set. 1* 8 8 9 2 0 0 8. In which row would the number 17 appear? Key: 1 # | 2 = 12 1^ | 4 = 14 9. Using what you learned from Erin’s and Ian’s ideas, make your own creative 1~ | 6 = 16 stem-and-leaf plot. Be sure to include a key that tells the reader how to 1* | 8 = 18 interpret your plot.

Lesson 16 ~ Stem - And - Leaf Plots 103 analyZing stem - anD - leaF plOts Lesson 17

Stem-and-leaf plots are very similar to histograms. A stem-and-leaf plot shows the actual data using stems and leaves. Th is makes it easier to do more statistical analysis with the data in a stem-and-leaf plot compared to a histogram.

example 1 Maria enjoys running aft er school. she recorded the 0 8 length of her last 14 runs in the stem-and-leaf plot to 1 0 5 5 8 8 the right. 2 0 0 0 4 4 a. describe the distribution of the stem-and-leaf plot. 3 2 5 b. What is the median data value? 4 c. What is the mean of the data set? how does it compare to the median? 5 5 d. Maria wants to go on 8 more runs this month. how Key: 1 | 5 = 1.5 miles many would you expect to be over 3 miles in length?

solutions a. Th e plot has more small values and a longer tail towards the higher values. Th e plot is skewed towards the maximum value.

b. List the values out in order and fi nd the middle number. 0.8, 1.0, 1.5, 1.5, 1.8, 1.8, 2.0, | 2.0, 2.0, 2.4, 2.4, 3.2, 3.5, 5.5 Median = 2.0 miles

c. Mean = ______Sum of Values = ___ 31.4 ≈ 2.24 miles Number of Values 14 Th e mean is higher than the median.

d. Based on the data, 3 of Maria’s 14 runs have been over 3 miles in length. Use a proportion to predict how many of Maria’s next 8 runs will be over 3 miles long. Set up the proportion. __3 = _ x 14 8

Use cross products to solve. ___ 14x = __ 24 14 14 x ≈ 1.71 Of Maria’s next 8 runs, approximately 2 will be over 3 miles in length.

104 Lesson 17 ~ Analyzing Stem - And - Leaf Plots A double stem-and-leaf plot test scores in teacher's Classes can be used to compare two Period 3 Period 6 different sets of data. The 5 2 leaves for one set of data are 9 5 5 6 4 7 8 88653702449 put on the right side of the 998644318467789 stem. The leaves for the other 87763110900478 set of data are placed on the 10 0 0 0 0 left side. The leaves are written Key: 7 | 4 = 74 from least to greatest going out from the stem. example 2 Chicago, Illinois is known as the “Windy City”. use Average Monthly Wind speed Portland, OR Chicago, IL the double stem-and-leaf plot comparing Chicago 5 5 6 and Portland to answer the questions. 6 4 2 1 1 7 a. What percent of months have an average wind 6 3 8 2 3 9 speed above 9 miles per hour in Chicago? In 5 2 9 3 Portland? 0 10 0 5 b. Find the five-number summaries for each city. 11 0 1 5 7 c. Would a month with an average wind speed of 12 0 0 Key: 8 | 9 = 8.9 miles per hour 12 mph be considered an outlier for Portland? use www.met.utah.edu the IQr Method. solutions a. Nine of the 12 months have an average __ 9 = ___ x 12 100 wind speed above 9 miles per hour in Chicago. Use a proportion to convert 900 = 12x this to a percent. 75 = x

Three of the 12 months have an average __ 3 = ___ x 12 100 wind speed above 9 mph in Portland. Use a proportion to convert this to a percent. 300 = 12x 25 = x In Chicago, about 75% of the months have an average wind speed over 9 miles per hour. In Portland, only 25% of the months fit this description.

b. List the numbers for each city. Find the five-number summaries. Portland, OR 6.5, 6.5, 7.1, 7.1, 7.2, 7.4, 7.6, 8.3, 8.6, 9.2, 9.5, 10.0 6.5 ~ 7.1 ~ 7.5 ~ 8.9 ~ 10.0 Chicago, IL 8.2, 8.3, 8.9, 9.3, 10.0, 10.5, 11.0, 11.1, 11.5, 11.7, 12.0, 12.0 8.2 ~ 9.1 ~ 10.75 ~ 11.6 ~ 12.0

c. Find the IQR for Portland. 8.9 – 7.1 = 1.8 Find 1.5 ∙ IQR. 1.5 ∙ 1.8 = 2.7 Add 2.7 to Q3 to find the upper boundary value. 8.9 + 2.7 = 11.6

A wind speed of 12 mph is above the upper boundary. It would be considered an outlier. Lesson 17 ~ Analyzing Stem - And - Leaf Plots 105 exercises

1. What is the advantage of displaying data in a stem-and-leaf plot instead of a histogram?

2. Use the stem-and-leaf plot at right to answer the questions. 0 25 a. What is the median and mode for the data? 1 15 80 b. The numbers in the stem-and-leaf plot add to 68.3. 2 30 35 50 55 55 90 95 What is the mean of the data? 3 00 10 25 25 45 60 60 85 90 c. What is the range? 4 00 00 00 00 d. What is the five-number summary? Key: 1 | 15 = 1.15 pounds

Bench Press Max 3. The table shows the maximum bench presses for twelve random players on a 18 5 5 high school football team. 19 5 a. What is the median bench press for the players? 20 0 5 5 5 b. Find the mean bench press. 21 5 5 c. There are 48 players on the football team. How many could be 22 0 5 expected to bench press at least 200 pounds? d. A new player joins the team. He can bench press 250 pounds. Is this 23 weight an outlier for the data? Use the IQR Method. 24 5 Key: 19 | 5 = 195 pounds 4. Chandra works at the zoo in the summer. She helped collect data about the age of their customers. Chandra recorded the age of sixteen random customers. She put the Zoo Customers' Ages information into a stem-and-leaf plot. 0 2 5 6 6 8 8 a. Describe the distribution of the data. Is the data skewed more towards 1 1 2 5 9 the maximum or the minimum? 2 0 b. What percent of customers were less than 30 years old? c. The zoo had 120 customers in one hour. How many of those customers 3 0 2 9 were likely to be less than 10 years old? 4 2 d. Later the zoo had 90 customers in one hour. How many of those 5 customers were likely to be between 30-40 years old? 6 8 Key: 1 | 5 = 15 years old 5. The stem-and-leaf plot shows the highest-grossing movies of all time. top grossing Movies of All time a. Describe the distribution of the data. 3 37 40 42 57 71 73 77 88 b. Find the five-number summary of the data. 4 08 23 31 35 37 61 c. What is the range of the values shown in the stem-and-leaf plot? d. A movie production executive looks at this data and exclaims, “I 5 expect that half of the movies we produce should make more than 6 01 380 million dollars!” Is this a fair expectation based on the data? Key: 4 | 08 = $408 million Explain. Source: www.movieweb.com e. “Titanic” was the highest-grossing movie of all time at approximately $601 million. Is this number an outlier for this data set? Use the IQR Method to justify the answer.

106 Lesson 17 ~ Analyzing Stem - And - Leaf Plots 6. Jenny compared the costs of her ten favorite cereals at two Mike's Market Grocery World diff erent stores. She recorded the costs in a double 99 99 1 stem-and-leaf plot. Use the data to compare the stores. 50 50 39 05 2 10 15 45 55 79 a. Which store has the larger range of prices for Jenny’s 99 99 69 3 29 29 79 89 99 favorite cereals? 29 4 b. Find the fi ve-number summary for each grocery store’s Key: 2 | 10 = $2.10 prices. Which store has the higher median price for Jenny’s favorite cereals? c. Based on the data, make an argument for why Jenny should do her cereal shopping at Mike’s Market. d. Based on the data, make an argument for why Jenny should do her cereal shopping at Grocery World.

7. Th e precipitation data for average monthly rainfall in Eugene and Corvallis Oregon is shown below. Use the information to answer the questions comparing the rainfall (in inches) for each town. Eugene: 7.65, 6.35, 5.08, 3.66, 2.66, 1.53, 0.64, 0.99, 1.54, 3.35, 8.44, 8.29

Corvallis: 6.46, 5.71, 4.59, 2.98, 2.03, 1.46, 0.57, 0.73, 1.47, 3.02, 6.94, 7.43 Source: www.weather.msn.com a. Use the data to construct a double stem-and-leaf plot. b. Which of the two towns seems to get more rainfall? Explain.

review determine the stems to use for each data set.

8. data that goes from 1.57 to 5.99

9. data that goes from 1962 to 2008

10. data that goes from 2.69 to 3.05

11. Jeremy collected data on the price of a gallon of gas from several gas stations around the state. Choose appropriate stems. Make a stem-and-leaf plot of the data. Price ($): 3.69, 3.71, 3.54, 3.69, 3.81, 3.75, 3.49, 3.67, 3.79, 3.71, 3.69, 3.76

12. A data set has the following statistics: Mean = 8 Median = 9 Mode = 11 Each number in the data set is increased by 5. What is the new mean, median and mode?

Lesson 17 ~ Analyzing Stem - And - Leaf Plots 107 13. Fift een dogs were at the veterinary clinic. Th eir weights were recorded. Weights (in pounds): 11, 28, 32, 36, 36, 40, 44, 50, 56, 59, 60, 63, 63, 78, 108 a. What was the mean weight of the dogs at the veterinary clinic? b. What was the fi ve-number summary of the dogs’ weights? c. Twenty-fi ve percent of the dogs weighed above ____ pounds. d. Does it appear there are any outliers? Explain. e. Make a histogram of the dogs’ weights. Describe the distribution of the weights.

14. Lakisha has a test average (mean) of 80% aft er 3 tests. Antonio has an average of 80% aft er 8 tests. Th ey each get 100% on the next test. Which student's average will increase the most? Explain.

15. Angie participated in cheerleader tryouts. Seven judges rated her performances on a scale of 0 to 10. Here are Angie's ratings: 4, 7, 7, 8, 8, 9, 9 a. Find the mean and median for Angie’s ratings. b. Th e highest and lowest ratings are removed. Which measure of center in part a will be aff ected the most?

tic-tAc-toe ~ surVey sAys!

Conduct a survey of at least 30 classmates. Ask them one question which requires a numerical answer. For example, “How much time do you spend per week listening to music?” Record your classmates’ answers.

Use the information from your survey to make three data displays (histogram, stem-and-leaf plot and box-and-whisker plot). Write a short refl ection about the results of your survey.

1. Find the measures of center of your data. Which measure of center best represents your data?

2. Are your graphs skewed? If so, which way?

3. Which graph do you think best showed the results of your survey? Explain.

4. Did the results of your survey surprise you? Why or why not?

5. Would you conduct your survey diff erently next time?

108 Lesson 17 ~ Analyzing Stem - And - Leaf Plots BOx - anD - whisker plOts Lesson 18

In Block 2 you learned how to fi nd the fi ve-number summary of a data set. A box-and-whisker plot is used to display the fi ve-number summary. A box-and-whisker plot provides a visual display of the spread of the data. It shows when groups of numbers are clustered Minimum Q1 Median Q3 Maximum together as well as when the numbers are spaced apart.

example 1 A grocery store manager was curious about how much each customer spent at her store. she collected data on the amount each of the next fi ft een customers spent. Amount spent: $1, $3, $5, $5, $6, $8, $10, $10, $11, $14, $19, $25, $32, $55, $68 Construct a box-and-whisker plot to display the amounts spent by the customers. solution Find the fi ve-number summary 1 , 3, 5, 5 , 6, 8, 10, 10 , 11, 14, 19, 25 , 32, 55, 68 of the data. 1 ~ 5 ~ 10 ~ 25 ~ 68

Draw a number line. Create equal intervals on your number line that include the minimum (1) and maximum (68) data values. For this data set, a number line spanning from 0 to 70 with intervals of 5 will work well.

Create a box just above the number line that goes from the Q1 value (5) to the Q3 value (25). Draw a vertical line through the box where the median value (10) lies.

Add “whiskers” to the ends of the box that extend out to the minimum and maximum values.

Lesson 18 ~ Box - And - Whisker Plots 109 Th e fi ve-number summary divides a data set into four quartiles. Each section represents 25% of the data. A box-and-whisker plot for a set of data makes it easier to answer questions about the distribution of the data.

Minimum Q1 Median Q3 Maximum

25% 25% 25% 25%

example 2 Th e following box-and-whisker plot shows the number of siblings each student in Mrs. grady’s class has. a. twenty-fi ve percent of the students have more than ___ siblings. b. Fift y percent of the students have between ___ and 4 siblings.

solutions a. Th e question is asking about the top 25% of the data set. Th e top 25% starts at 4 siblings.

b. Th is question is asking about the middle 50% of the data set, which runs between Q1 and Q3. Th e 3rd quartile (Q3) is 4 and the missing value (Q1) is 1.

Creating parallel box-and-whisker plots can be helpful for comparing two or more data sets. Th ey can be created by placing one box-and-whisker plot above another on the same number line.

example 3 two shoe salesmen, tim and Caleb, compared the number of shoes they each sold over the last 14 days. tim: 0, 1, 3, 5, 5, 6, 8, 10, 11, 12, 13, 13, 14, 16 Caleb: 2, 4, 5, 6, 9, 10, 11, 11, 11, 11, 13, 14, 15, 15 Create a parallel box-and-whisker plot for the data.

solution Find the fi ve-number summary for each person. Tim: 0 ~ 5 ~ 9 ~ 13 ~ 16 Caleb: 2 ~ 6 ~ 11 ~ 13 ~ 15

Draw and label a number line that will include both the minimum and maximum values of both data sets. Use the fi ve-number summaries to create two box-and-whisker plots. Place the plots one above the other and label each with their salesman’s name.

Th e plots show that Tim’s maximum sales day was larger, but his minimum was also smaller than Caleb’s minimum. Although Caleb’s maximum is lower, 50% of the days he sold more than 11 pairs of shoes. For comparison, Tim sold more than 9 pairs of shoes on 50% of the days.

110 Lesson 18 ~ Box - And - Whisker Plots exercises

1. What are the fi ve numbers that make up the key parts of a box-and-whisker plot?

2. What does the vertical line in the middle of the box represent?

3. Callyn found the fi ve-number summary of a data set. It was 7 ~ 9 ~ 15 ~ 16 ~ 21. She made the box-and-whisker plot shown at the right. a. What did Callyn do wrong in her box-and-whisker plot? b. Draw the correct box-and-whisker plot for the fi ve-number summary that Callyn found.

Create a box-and-whisker plot for each fi ve-number summary below.

4. 5 ~ 8 ~ 10 ~ 18 ~ 29 5. 62 ~ 78 ~ 84 ~ 92 ~ 100 6. 49 ~ 59 ~ 62 ~ 64 ~ 72

Find the fi ve-number summary. Create a box-and-whisker plot for each set of data.

7. 0, 0, 2, 6, 7, 7, 8, 8, 8, 9 8. 17, 20, 24, 27, 27, 27, 30

9. 220, 112, 120, 305, 360, 198, 380, 395, 205, 230 10. 2.1, 0.9, 3.9, 2.75, 3.3, 3.5, 3.5, 3.0

11. Th e box-and-whisker plot shows the cost of tickets for the opera rounded to the nearest dollar. Use the information to answer the questions. a. Th e cheapest ticket to the opera costs ___ dollars. b. Half of the tickets cost less than ___ dollars. c. Twenty-fi ve percent of the tickets cost more than ___ dollars. d. What is the range of the ticket prices? e. What is the interquartile range (IQR) of the ticket prices?

12. Rodolfo takes his dog for walks. He kept track of the hours per week he walked his dog for the last nine weeks.

a. Create a box-and-whisker plot of the data. b. What is the range and IQR of the data? c. Rodolfo walks his dog at least ___ hours on approximately 25% of the weeks. d. Th e middle fi ft y percent of the weeks Rodolfo walks his dog between ___ and ___ hours.

Lesson 18 ~ Box - And - Whisker Plots 111 13. Kristen makes stuff ed animals and sells them at the student store. She sells her stuff ed animals at diff erent prices based on the amount of material needed and the time they take to make. Make a box-and-whisker plot for the prices of Kristen’s stuff ed animals.

14. Th e parallel box-and-whisker plot shows the number of rebounds per game by two players on a basketball team. a. Which player had the higher maximum rebounds per game? b. Which player had the higher minimum rebounds per game? c. Which player’s median number of rebounds was larger? d. Based on their rebounding ability, which player would be a better teammate? Explain.

15. Nicole and Fatima compared the costs of jeans at their two favorite stores: Denim Factory and Dream Jeanie’s. Th ey chose random jeans of various styles at each store. Th ey recorded the prices.

denim Factory Prices ($): 50, 30, 35, 60, 45, 50, 30, 38, 40, 40 dream Jeanie’s Prices ($): 55, 50, 50, 28, 50, 55, 18, 58, 25, 45

a. Make a parallel box-and-whisker plot to compare the prices at the two stores. b. Nicole says that Denim Factory has better prices. Fatima prefers Dream Jeanie’s prices. Who do you agree with?

review

16. Th e stem-and-leaf plot shows the heights of the twenty tallest buildings in the world in feet. a. Describe the distribution of the data. b. What is the mode of the data? heights of the tallest c. What is the fi ve-number summary? Buildings in the World d. Is a 1,667 feet building an outlier for this data set? Use the IQR method. 10 58 83 87 93 11 27 35 36 40 65 12 05 27 50 60 83 13 62 81 14 51 83 83 15 16 67 Key: 13 | 62 = 1,362 feet

112 Lesson 18 ~ Box - And - Whisker Plots Jon’s Movies Scott’s Movies 17. Scott and Jon both collect movies. Th e double stem-and-leaf 7 195 plot shows the copyright dates of twelve of the movies they 7 7 5 4 1 0 196 7 8 own. 2 197 a. Compare Scott and Jon’s movie collections. 4 4 198 4 5 8 8 b. Which person’s copyright dates has the larger range? 4 199 1 3 4 7 9 c. Jon has a total of 40 movies. Predict how many might 1 200 2 have a copyright date before 1970. Key: 195 | 7 = 1957 copyright date d. Scott has a total of 30 movies. Predict how many might have a copyright date aft er 1980.

18. Th e fi rst fi ft een customers at a convenience store spent the following amounts of money:

Amount spent $0.89, $2.99, $5.15, $1.19, $0.99, $1.99, $12.25, $4.50, $0.99, $1.59, $3.99, $6.29, $1.79, $4.09, $8.15

a. Choose an appropriate interval width and make a histogram of the amounts spent by the fi ft een customers. b. Describe the distribution of the data. Which way is the graph skewed? c. Th e next hour the convenience store expects 60 customers. How many customers would likely spend more than $5?

tic-tAc-toe ~ fliP-Book

Create a fl ip-book that instructs students on how to make each of the types of data displays in this Block. Each page of your fl ip-book should have a sample data set and step-by-step instructions on how to make the display. Include helpful hints in your fl ip- Making data displays book (for example, “be sure to put the ‘leaves’ in order in histograms your stem-and-leaf plots”). stem-and-Leaf Plots Box-and-Whisker Plots

Lesson 18 ~ Box - And - Whisker Plots 113 tic-tAc-toe ~ mAtching chAllenge

Match the four graphs below with one of the data sets below. Th ere is only one correct match for each and some data sets will not be chosen.

1. 2.

3. 4.

Create four graphs of your own (two box-and-whisker plots and two histograms) and draw them on index cards. Write the four data sets that match the graphs on separate index cards. Also, make up two additional distracter data sets and put them on index cards. Give all ten cards to a classmate. Have the classmate match the graphs to the data sets. Turn in your answers, as well as your cards, to your teacher.

114 Lesson 18 ~ Box - And - Whisker Plots analyZing BOx - anD - whisker plOts Lesson 19

explOre! visualiZing skew

In the 2006 Winter Olympics, twenty countries won 2 or more medals. Th e following list shows the number of Gold medals won by those countries (listed from least to greatest). 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 5, 5, 6, 7, 7, 8, 9, 9, 11 Source: Time Almanac step 1: Find the fi ve-number summary of the data.

step 2: Create a box-and-whisker plot of the data.

step 3: Using an interval width of 2, create a histogram of the same data.

step 4: Describe the distribution of the data in your histogram. Is the graph skewed? If so, which way is the data skewed?

step 5: Compare the box-and-whisker plot in step 2 to the histogram in step 3. a. How can someone tell if data is skewed by looking at the box-and-whisker plot? b. How can someone tell which direction it is skewed?

Twelve students competed in the shot put at a track meet. Th e following list shows the distances of their best throws in feet. 22, 16.5, 25, 22.5, 14, 23.5, 25.5, 22, 21, 20, 24, 24

step 6: Find the fi ve-number summary of the data.

step 7: Create a box-and-whisker plot of the data.

step 8: Using an interval width of 2, create a histogram of the same data.

step 9: Describe the distribution of the data in the histogram. Which way is the graph skewed?

step 10: Compare the box-and-whisker plot in step 7 to the histogram in step 8. How can someone tell if data is skewed by looking at the box-and-whisker plot? How can someone tell which direction it is skewed?

Lesson 19 ~ Analyzing Box - And - Whisker Plots 115 Based on the shape of a box-and-whisker plot, you can make statements as to how the data is distributed.

skewed left skewed right The 50% “tail” on the left The 50% “tail” on the right side is longer. side is longer.

normal distribution Even distribution 50% of the data is clustered There is no clustering as each closely in the middle, while quartile is evenly spread across the whiskers stretch longer the range of the data. to the left and the right.

The information in box-and-whisker plots can also be used to make predictions. Remember, each quartile represents 25% of the data. example 1 Mr. Jansen, the Pe teacher, collected data about his students’ mile times for several years. he put the data into the box-and-whisker plot below. a. describe the distribution of the data. b. Approximately what percent of Mr. Jansen’s students can run a mile in less than 7.5 minutes? c. If one of Mr. Jansen’s classes has 36 students, about how many would you expect to run the mile in more than 8.5 minutes?

solutions a. The data has a normal distribution. Fifty percent of the students are clustered in the middle with longer whiskers on the left and the right.

b. The first quartile (Q1) value is 7.5. Twenty-five percent of the students can run a mile faster than 7.5 minutes.

116 Lesson 19 ~ Analyzing Box - And - Whisker Plots example 1 c. Twenty-fi ve percent of Mr. Jansen’s ___ 25 = __ x 100 36 solutions students run a mile in more than (continued) 8.5 minutes. Use a proportion to predict the number of students out of 900 = 100x 36 that run a mile in this time frame. 9 = x

Approximately 9 of the students would run a mile in more than 8.5 minutes. example 2 two diff erent fertilizers were applied to several plants of the same breed. Th e parallel box-and-whisker plot shows the distribution of the heights (in inches) of the plants aft er 6 weeks. a. describe the distribution of the plants for each type of fertilizer. b. eighty plants were given Fertilizer A. About how many would likely grow at least 14 inches in height? heights of Plants in Inches c. eighty plants were given Fertilizer B. About how many would likely grow at least 14 inches in height? d. What are the advantages of each fertilizer? solutions a. Since there is a longer tail on the left , Fertilizer A’s graph is skewed left . Each quartile in the plot for Fertilizer B is approximately the same size. Fertilizer B has an even distribution.

b. According to the graph, 75% of the ___75 = __ x 100 80 plants given Fertilizer A grow to be at least 14 inches in height. x = 60

About 60 of the plants should grow to at least 14 inches tall if given Fertilizer A.

c. According to the graph, 50% of the ___ 50 = __ x 100 80 plants given Fertilizer B grow to be at least 14 inches in height. x = 40

About 40 of the plants should grow to at least 14 inches tall if given Fertilizer B.

d. Plants given Fertilizer B have the potential to grow the most. Plants given Fertilizer A seem to be more consistent in their growth. Fertilizer B plants have a range of 18 inches. Fertilizer A plants have a range of 12 inches.

Lesson 19 ~ Analyzing Box - And - Whisker Plots 117 exercises

1. What percent of the data is “in the box” in a box-and-whisker plot?

2. Explain how to tell if a data set is skewed right by looking at the box-and-whisker plot.

3. Use the data set 0, 6, 10, 15, 15, 15, 21. a. Find the five-number summary of the data set. b. Janelle created the following box-and-whisker plot on the right. She did everything correctly. It appears to be missing the “median” line in the middle of the box. Explain how this is possible.

4. Ishmael created the box-and-whisker plot to the left. He did everything correctly. It appears to be missing the “whisker” on the right hand side. Explain how this is possible.

describe the distribution of each box-and-whisker plot as skewed right, skewed left, normal distribution or even distribution.

5. 6.

7. 8.

9. Use the box-and-whisker plot at right to answer each question a. What is the five-number summary of the data? b. Describe the distribution of the data. c. What percent of the values are less than 30? d. Which is likely larger, the mean or the median of this data set? Explain.

118 Lesson 19 ~ Analyzing Box - And - Whisker Plots 10. The parallel box-and-whisker plot shows the results of the latest test for two of Mrs. Kessler’s classes. a. Which class had the larger range? b. Describe the distribution of each period’s test scores. c. There are 28 students in Mrs. Kessler’s th6 period class. How many students had a score of at least 85%? d. In future tests, which class is likely to have more students earning grades of at least a B (80% or above)? Explain.

11. Lumi was curious about how many hours per week her classmates used the internet. She surveyed several classmates and made a box-and-whisker plot to show the results. a. Describe the distribution of the graph. b. Lumi surveyed 48 students to collect her data. How many of those students use the internet at least 10 hours per week? c. There are 300 students in Lumi’s school. How many would likely use the internet at least 14 hours per week? d. At another school, 80 students use the internet less than 6 hours per week. About how many students would there likely be at that school?

12. The plot at the right shows the average speed of drivers on a section of a highway. a. Describe the distribution of the data. b. During a ten-minute span, 40 cars passed through this section of highway. About how many would likely be traveling 62 mph or less? c. A group of cars passes through this stretch of highway. Thirty of the cars were traveling more than 52 mph. About how many cars would likely be in the group?

13. In 2005, Oregon ranked second in the nation in number of people who regularly use their seat belts. The following lists show the percentages (rounded to the nearest percent) of people in various states who used seat belts in 1998 and in 2005. Use the information to answer the following questions. 1998: 46, 54, 56, 57, 58, 60, 61, 61, 63, 70, 73, 77, 77, 78, 79, 83, 83 2005: 61, 67, 69, 74, 76, 77, 79, 80, 82, 83, 85, 86, 87, 90, 93, 93, 95 Source: National Highway Traffic Safety Association

a. Make a parallel box-and-whisker plot for the two data sets. b. Describe the distribution in each. c. Which data set has the larger range? Why might this be so? d. In 1998, one-fourth of the states had over ____% of their population wearing seat belts. e. In 2005, one-fourth of the states had over ____% of their population wearing seat belts.

Lesson 19 ~ Analyzing Box - And - Whisker Plots 119 review

14. Use the stem-and-leaf plot to fi nd the mean, median and mode of the data set.

0 5 9 9 1 2 2 2 2 5 8 9 3 0 1 7 4 Key: 2 | 5 = 25

15. Copy and complete the following table. Create a pie chart of the data.

students’ Favorite Food Pizza Hamburger Tacos Steak Total Frequency 12 3 8 2 Percent

16. Use the given information to fi nd the missing number in this data set: 10, 22, 40, 32, 9, 2, ___ Mean = 18

17. Find the three missing numbers in the data set: 25, 9, 5, 20, 23, 10, 14, ___, ___, ___

Median = 12 Range = 25 IQR = 13 Mode = 10 Minimum = 5

18. Simon earned an average of $12 per lawn for the fi ve lawns he mowed this summer. Aft er mowing his sixth lawn, his average rose to $15. How much did he get paid for mowing the sixth lawn?

tic-tAc-toe ~ Prediction time

Create a brochure or a poster that shows how the graphs you have learned in this block can be used to make predictions. Give examples from each type of graph.

Example: If 80 students were asked about the amount of sleep they get each night, how many would you expect to get at least 8 hours of sleep?

Number of Hours of Sleep

120 Lesson 19 ~ Analyzing Box - And - Whisker Plots tic-tAc-toe ~ grAPh switcheroo

You can use the information in one type of graph to make a graph of another type. For example, this histogram can be used to approximate a matching box-and-whisker plot.

Possible Data Set: 20, 21, 24, 25, 25, 27, 28, 30, 31, 32, 32, 33, 40, 42

Use the information in each graph below to create a possible data set with at least 10 values. Transform your data set into the requested type of graph.

1. Histogram → stem-and-leaf plot 2. Box-and-whisker plot → histogram

3. Box-and-whisker plot → Stem-and-leaf plot and histogram

Lesson 19 ~ Analyzing Box - And - Whisker Plots 121 review BLoCK 3

vocabulary box-and-whisker plot frequency table skewed left double stem-and-leaf plot histogram skewed right even distribution normal distribution stem-and-leaf plot parallel box-and-whisker plot

Lesson 14 ~ Histograms

1. Explain the diff erence between a histogram and a bar graph.

2. An ideal histogram has between ___ and ___ intervals.

3. Use the histogram to answer the questions. a. What is the interval width? b. How many total tickets were sold? c. Which price interval has the most tickets sold? d. A ticket sold for $24. Which interval would it fall in? Number of Tickets Sold

e. How many tickets were sold for less than $28? f. Th e 28-32 interval is empty. What does this mean? Cost of Tickets ($)

4. Cory compared the cost of a large coff ee at several coff ee shops. He went to fourteen diff erent shops and recorded the prices.

Price of a 20-ounce Coff ee $1.10 $1.49 $2.15 $1.00 $1.15 $1.25 $1.50 $1.79 $1.59 $1.75 $1.15 $1.30 $1.50 $1.40 a. Find the minimum and maximum values in the data set. What would be a reasonable interval width to use for this data set? b. Use the interval width from part a to create a frequency table for the data. Be sure that each data value is included in only one interval. c. Use the frequency table to create a histogram. Label both axes.

5. Make a frequency table and histogram from this data set. 85, 102, 99, 121, 90, 88, 95, 97, 88, 104, 99, 101, 112, 100, 92, 81, 95, 109, 111, 120

122 Block 3 ~ Review Lesson 15 ~ Analyzing Histograms

6. Draw a histogram which is skewed left.

7. Draw a histogram with a normal distribution.

Based on the following data sets, predict the shape of the histogram. explain your choice.

8. the heights of all students 9. number of TVs students 10. number of days per week in the 8th grade have at home students watch TV

11. Use the histogram below to answer the questions.

a. Describe the shape of the histogram. b. How many data pieces are included in the graph? c. In which interval does the median lie? d. How does the mean likely compare to the median? Explain.

12. Jake taught himself to play the guitar. He kept track of the hours he practiced each week for 17 weeks. He put the information into the histogram at right. a. Describe the shape of the histogram. b. What is the ratio of weeks Jake practiced for more than

6 hours to total weeks? Number of Weeks c. Jake practiced for 10 more weeks. About how many weeks did he likely practice for at least 6 hours during those 27 weeks? Number of Hours Per Week d. Jake practiced for 10 more weeks. About how many weeks did he likely practice for less than 2 hours during those 37 weeks?

Lesson 16 ~ Stem-and-Leaf Plots

13. How is a stem-and-leaf plot like a histogram? How is it different?

14. List the data values in this stem-and-leaf plot.

0 8 9 1 0 4 2 1 1 2 7 3 4 Key: 1 | 4 = 14

Block 3 ~ Review 123 What stems would you use for each data set?

15. Data that goes from 16. Data that goes from 17. Data that goes from 1582 to 2007 2.9 to 6.7 1958 to 2007

Make a stem-and-leaf plot for each set of data.

18. 82, 93, 80, 77, 89, 64, 88, 92, 100, 98, 85, 72, 79, 68, 88

19. 1958, 1963, 1966, 1968, 1968, 1969, 1970, 1974, 1979, 1983, 1988, 1993, 1993, 2007

20. Riley investigated the cost of a gallon of milk at various stores. He recorded the prices. Make a stem-and-leaf plot of Riley’s data.

Cost of a gallon of Milk $3.50, $3.62, $3.39, $3.55, $3.59, $3.91, $3.47, $3.65, $3.49, $3.57, $3.79

Lesson 17 ~ Analyzing Stem-and-Leaf Plots

21. What is the advantage of displaying data in a stem-and-leaf plot instead of a histogram?

22. Use the stem-and-leaf plot to answer each question. shoe size of 8th grade Boys a. What is the range of the data? 6 0 5 5 b. What is the mode of the data? 7 0 0 5 c. What is the mean? 8 0 0 0 5 5 d. What is the five-number summary of the shoe sizes? 9 0 0 5 5 e. A group of 40 eighth grade boys are randomly chosen. 10 0 5 How many would likely wear a shoe size less than 7? Key: 9 | 5 = 9.5 shoe size f. How many of the group of 40 eighth grade boys would likely wear a shoe size of at least 9?

23. The parallel stem-and-leaf plot shows the average amount of monthly rainfall for two towns. a. Which town’s average monthly rainfall is skewed towards the town A town B minimum? 9 9 0 7 b. Find the median monthly rainfall for each town. Which town has 8 6 2 2 1 1 the higher median monthly rainfall? 9 5 2 6 c. Find the mean monthly rainfall of Town A. How does it compare to 7 5 3 1 the median? 4 4 3 6 d. In the future, Town A will have a month where it rains 6 inches. Will 0 5 2 8 9 that amount of rainfall be an outlier? Use the IQR Method. 6 0 2 6 Key: 2 | 6 = 2.6 inches of rainfall

124 Block 3 ~ Review Lesson 18 ~ Box-and-Whisker Plots

24. Use words or pictures to describe how the five-number summary is used to create box-and-whisker plots.

25. The five-number summary breaks a data set into quartiles. Each section represents ___% of the data.

Create a box-and-whisker plot for each five-number summary.

26. 20 ~ 32 ~ 41 ~ 46 ~ 50 27. 50 ~ 57 ~ 62 ~ 66 ~ 74

Create a box-and-whisker plot for each data set.

28. 0, 0, 2, 3, 7, 10, 10, 15 29. 17, 23, 28, 29, 29, 30, 31, 35, 36, 38, 39, 40, 45

30. The lists show the “talk-time” battery life of ten cell phones offered by two different cell phone producers. Use the information to answer each question. talk time (in hours) MotoPhone: 10, 7, 16, 18, 13, 4, 8, 20, 15, 15 DuraFone: 20, 10, 12, 2, 11, 14, 11, 23, 5, 8

a. Make a parallel box-and-whisker plot to compare the battery lives of the two companies’ phones. b. Which company’s phones has a higher median battery life? c. Twenty-five percent of MotoPhone’s phones have a battery life of at least ___ hours. d. Twenty-five percent of DuraFone’s phones have a battery life of at least ___ hours. e. Which company offers the best phones in terms of battery life? Explain.

Lesson 19 ~ Analyzing Box-and-Whisker Plots

31. Explain how to tell if a data set is skewed left by looking at the box-and-whisker plot.

describe the distribution of each box-and-whisker plot as skewed right, skewed left, normal distribution or even distribution.

32. 33. 34.

Block 3 ~ Review 125 35. A teacher surveyed her students to fi nd out how many other states each has lived in. Use the box-and-whisker plot to answer the questions. a. Twenty-fi ve percent of the students have lived in more than ___ other states. b. What percent of the students have lived in 1 to 6 other states? c. Which is likely larger, the mean or the median of the data set? d. Th ere are 26 students in the class. How many have lived in at

least 3 other states? Number of Other States e. One student in the class has lived in 13 other states. Is that number an outlier?

36. Th e parallel box-and-whisker plot shows the minutes spent on the April internet per day for two students. Use the plot to answer the questions. a. In a 30-day month, about how many of those days will April spend at least 35 minutes on the internet? b. Write the fi ve-number summary for April’s time spent on the Sonya internet. c. Based on Sonya’s plot, there are two possible medians for her data set. What are they?

Number of Minutes d. Describe the distribution for each student’s data set. e. Which student’s data set has outliers?

tic-tAc-toe ~ skew in the reAl world

Find a set of data in a magazine or newspaper. Th e data set should be skewed right or skewed left . Find the mean and median of the data set. Describe how the skew aff ects each of these measures of center. Make three diff erent data displays for the data set: a histogram, a box-and-whisker plot and a stem-and-leaf plot.

126 Block 3 ~ Review CAreer

JAmie FoCus dentist centrAl Point, oregon

I am a dentist. Dentists care for people’s oral health. Oral health includes teeth, gums, mouth, lips, head and neck. Dentists do exams to make sure that people’s oral health is good. If something is wrong, I may have to put in a filling. I also might have to do surgery or provide braces. Dentists do many different tasks in their job to make sure their patients have good oral health.

I use math in many aspects of my job as a dentist. The metric system is used for everything in dentistry. That means I make many conversions during my day. I use fractions and ratios for mixing filling materials and cements. I even use geometry to take x-rays at the correct angles. Many dentists are also small business owners. They use math for accounting, ordering and budgeting in their practices.

In order to become a dentist, you need a four year college degree. An additional four years of dental school is needed after that. With an additional two to six years of schooling, a dentist may become an orthodontist, an oral surgeon or another type of specialist. A person pursuing a dental career will need eight to fourteen years of schooling after high school.

Dentists earn good salaries. Most dentists make over $125,000 per year. This can be higher or lower depending on how many years a dentist has been practicing and what kinds of special training they have. Many dentists own their practice which allows them to be their own boss.

The thing I like most about my job as a dentist is helping people. It feels good to help create beautiful smiles that make people happy. I also enjoy being my own boss and setting my own schedule.

Block 3 ~ Review 127