134 Lesson 4.4 ~ Histograms

hisTograms

lesson 4.4

When the numbers in a data set are more spread out and there are not a lot of repeating values, a dot plot may not be the best way to display the data. A histogram is another helpful way to display a data set. 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. The first graph, a bar graph, shows the number of each type of pet her students have. The second graph is a histogram. What do you see that is different?

Students and their Types of Pets

0

2

4

6

8

10

12

Dogs Cats Rodents Reptiles Fish None

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The histogram focuses on the number of pets that each student has rather than the type of pets. There 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 the interval up to the number on the right-hand side. For example, the first 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 4.4 ~ Histograms 135

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. The interval between 6-8 is empty. what does that mean?d. how many students are in ms. sanchez’ class?

a. Ten students have 2 or 3 pets. Look at the height (number of students) of the bar that is between 2 and 4.

b. There are 3 students who have at least 8 pets. Adding the bars to the right of 8 on the horizontal axis (1 + 2) totals 3.

c. This means that there are 0 students who had exactly 6 or 7 pets.

d. Add the heights of all the bars together. This sum represents the number of students in the class.

13 + 10 + 5 + 0 + 1 + 2 = 31 students

Before making the histogram, Ms. Sanchez made 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 the number of pets in the intervals.

number of Pets tally0-2 |||| |||| |||2-4 |||| ||||4-6 |||| 6-8

8-10 |10-12 ||

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ExamplE 1

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136 Lesson 4.4 ~ Histograms

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.

make a histogram for students’ pulse rates (in beats per minute) listed below. 64, 76, 60, 70, 68, 70, 88, 62, 54, 68, 70, 72, 60, 92, 64, 76, 70, 68, 62, 72

Identify the minimum and maximum values. Minimum = 54 Maximum = 92 The frequency table must start low enough to include 54 and must go high enough to include 92.

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

Complete the frequency table for the data.

Use the information collected in the frequency table to draw the histogram.

ExamplE 2

solution

Pulse rates number of students tally50 - 55 | 55 - 6060 - 65 | | | | |65 - 70 | | |70 - 75 | | | | | 75 - 80 | | 80 - 8585 - 90 | 90 - 95 |

Pulse Rate (beats per minute)

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Lesson 4.4 ~ Histograms 137

ExErcisEs

1. Explain the difference between a bar graph and a histogram.

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

Number of Miles (in thousands)

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f Car

s

a. How many intervals are shown in the histogram (be sure to include the empty intervals in the total)? b. Is the interval width appropriate? Explain your reasoning. c. Copy the frequency table below. Use the information in the histogram above to complete the table. number of miles driven

(in thousands)number of

Cars10 – 2020 – 30 30 – 40 40 – 50

d. Draw a new histogram with the interval width used in part c.

3. Newborn babies were weighed and the data was recorded in the frequency table below. Use the table to answer the following questions.

weight of newborn (in pounds)

tally

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 your reasoning. d. A 10-pound baby boy is born. In which interval should he be tallied?

Cars and Miles Driven

Number of Miles Driven (in thousands)

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138 Lesson 4.4 ~ Histograms

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 5.86.0 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 is 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 included in only one interval. c. Use your frequency table to create a histogram. Be sure to label both axes and title the frequency table.

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 a histogram using 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 different 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 below 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 slept less than 4 hours last night? d. Which interval included the most students? e. How many students slept between 4 and 8 hours last night?

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Lesson 4.4 ~ Histograms 139

8. The ages of the fourteen people at the local coffee shop on Saturday morning are listed 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 your reasoning.

9. The histogram below shows the number of visitors to the Children’s Museum in one hour.

Ages

a. Which age range represented the largest group? b. Approximately how many total visitors came to the Children’s Museum during that hour? c. What percent of the visitors were less than 20 years old? Show all work necessary to justify your answer.

10. A company tracked the number of songs downloaded daily from their music website. Data from the last ten days is shown in the table at right. a. Draw a histogram for the data. Let the intervals start at 0 and increase by 1,000. b. In what interval did you include the Day 7 data of 3,000 downloads? Why? c. Based on your histogram, what range of downloads was the most common? d. What percent of days had more than 2,000 songs downloaded?

11. Peter surveyed a random group of 50 students at his middle school and asked them their age in years. a. Why would a histogram NOT be the best data display for him to use? b. What is a survey question he could have used that would most likely provide data that could be displayed nicely in a histogram?

140 Lesson 4.4 ~ Histograms

12. Tia says any data that can be displayed in a dot plot can be shown in a histogram. Is she correct? Explain your reasoning.

13. Lance says he can calculate the mean, median and mode from data displayed in a histogram. Is he correct? Explain your reasoning.

rEviEw

14. The table shows the number of students in history class who have been absent each amount of days. a. Construct a dot plot to display the data. Your number line should show the number of absences. b. What is the median value for the class? c. How many students are in the class? d. How many total absences have there been? e. What is the average (mean) number of absences for students in this history class?

Find the range of each data set. 15. 19, 22, 26, 29, 33, 38, 42 16. 60, 72, 58, 65, 76, 66, 70, 65

17. Polly has run the 400-meter race in the following times this year: Time (seconds): 52, 54, 56, 53 She wants to have an average time of 53 seconds. What time does she need to get on her fifth race to have a 53-second average? Show all work necessary to justify your answer.

tic-tAc-toe ~ ex“sk e w”se my l e A n i ng Research the concept of “skew” as it relates to data displays. What does it mean for a data set to be “skewed right” or “skewed left” or to have a “normal distribution”?

Create a poster that shows examples of histograms and box-and-whisker plots with each type of skew (six examples altogether). Give an example of a real-world data set that

would have these types of data displays.

Number of Absences

Number of Students

0 31 92 83 35 28 1