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.