9.3 Histograms and Box Plots

Transcription

1 Locker LESSON. Histograms and Box Plots Name Class Date. Histograms and Box Plots Essential Question: How can you interpret and compare data sets using data displays? Common Core Math Standards The student is expected to: S-ID. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Also S-ID., N-Q. Mathematical Practices MP. Using Structure Language Objective Explain what each part of a histogram represents. ENGAGE Essential Question: How can you interpret and compare data sets using data displays? You can estimate statistics and compare distributions using histograms, and you can compare the centers, spreads, and shapes of the distributions of two data sets by plotting box plots on the same number line. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the fact that the batting average of a baseball player is calculated by dividing the number of hits by the number of times at bat. Note that Ty Cobb, the player with the all-time highest career batting average, had a career batting average of.. Then preview the Lesson Performance Task. Explore Understanding Histograms Resource Locker A histogram is a bar graph that is used to display the frequency of data divided into equal intervals. The bars must be of equal width and should touch but not overlap. The heights of the bars indicate the frequency of data values within each interval. A B C D E F G Look at the histogram of Scores on a Math Test. Which axis indicates the frequency? The vertical axis shows the frequency for each interval. What does the horizontal axis indicate? The horizontal axis shows the test scores. How is the horizontal axis organized? It is organized in groups of. How many had scores in the interval? How many had scores in the interval? How many had scores in the interval? How many had scores in the interval? Reflect. What statistical information can you tell about a data set by looking at a histogram? What statistical information cannot be determined by looking at a histogram? Possible answer: It is easy to see the general shape and distribution of the data. It is difficult to know the actual data values.. How many test scores were collected? How do you know? Look at each interval and determine the number of test scores in each = ; test scores were collected. Score on a Math Test Test Scores Module Lesson Name Class Date. Histograms and Box Plots Essential Question: How can you interpret and compare data sets using data displays? S-ID. For the full text of these standards, see the table starting on page CA. Also S-ID., N-Q. Explore Understanding Histograms Resource A histogram is a bar graph that is used to display the frequency of data divided into equal intervals. The bars must be of equal width and should touch but not overlap. The heights of the bars indicate the frequency of data values within each interval. Look at the histogram of Scores on a Math Test. Which axis indicates the frequency? The axis shows the frequency for each interval. What does the horizontal axis indicate? The horizontal axis shows the. How is the horizontal axis organized? It is organized in groups of. How many had scores in the interval? How many had scores in the interval? How many had scores in the interval? How many had scores in the interval? vertical Reflect test scores. What statistical information can you tell about a data set by looking at a histogram? What statistical information cannot be determined by looking at a histogram?. How many test scores were collected? How do you know? Score on a Math Test Test Scores Possible answer: It is easy to see the general shape and distribution of the data. It is difficult to know the actual data values. Look at each interval and determine the number of test scores in each = ; test scores were collected. Module Lesson HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. Lesson.

2 Explain Creating Histograms When creating a histogram, make sure that the bars are of equal width and that they touch without overlapping. Create a frequency table to help organize the data before constructing the histogram. Consider the range of the data values when creating intervals. Example Create a frequency table from the data. Then use the frequency table to create a histogram. Listed are the ages of the U.S. senators at the start of the th Congress on January,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, EXPLORE Understanding Histograms INTEGRATE TECHNOLOGY Point out to students that there are many tools available for creating histograms. These include online tools, spreadsheets, and graphing calculators. An added benefit of using these tools is that they also calculate statistics for the data. Create a frequency table. The data values range from to, so use an interval width of and start the first interval at. Age Interval Check that the sum of the frequencies is = Use the frequency table to create a histogram. Ages of U.S. Senators at the Start of the th Congress Ages Image Credits: Andrea Izzotti/Shutterstock QUESTIONING STRATEGIES How is a histogram different from a bar graph? In a histogram, the bars touch, but in a bar graph the bars do not touch. In a histogram, each bar represents a range of values, while in a bar graph, each bar represents a single value or category. Histograms are used for quantitative data, while bar graphs are used to display categorical data. EXPLAIN Creating Histograms AVOID COMMON ERRORS Students may try to draw a histogram with bars representing intervals of different sizes. Point out to students that the bars of a histogram must have the same width, representing the same span of values. Module Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP., which calls for students to look for and make use of structure. By understanding the structure of a histogram, students can estimate the mean value of a data set. When students construct box plots, they understand that the basic structure of a box plot represents the center, spread, and shape of the distribution of a data set; students are then able to compare two different box plots representing two related sets of real-world data. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP. Make sure that students understand the connection between number lines, dot plots, and histograms. Histograms and Box Plots

3 QUESTIONING STRATEGIES What is a difference between how data are represented on a dot plot and on a histogram? A dot plot gives each individual piece of data, while a histogram combines specific data values to find the frequency of values within intervals. B Listed are the scores from a golf tournament.,,,,,,,,,,,,,,, Create a frequency table. The data values range from to, so use an interval width of, and start the first interval at. Score Interval Golf Tournament Scores Check that the sum of the frequencies is = Use the frequency table to create a histogram. Scores Reflect. Describe the shape of the distribution of senators ages. Interpret the meaning. The data clusters around the intervals and. It appears to be roughly symmetric. This means that about half are older and half are younger than the mean. Your Turn Heights of Basketball Players. Listed are the heights of players, in inches, on a basketball team. Create a frequency table from the data. Then use the frequency table to create a histogram. Possible table and histogram shown.,,,,,,,,,,,,,, Height Interval Use the frequency table to create a histogram. Height Module Lesson COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Have each student create a data set with twenty values, calculate the mean, and create a histogram for the data set. Then have partners trade histograms and estimate the mean value for each other s histograms. After both students have found an estimated value for the mean, have them compare their estimates to the actual values. Lesson.

4 Explain Estimating from Histograms You can estimate statistics by studying a histogram. Example Estimate the mean of the data set displayed in each histogram. The histogram shows the ages of teachers in a high school. To estimate the mean, first find the midpoint of each interval, and multiply by the frequency. st interval: _ + ( )() = (.)() = nd interval: _ + ( )() = (.)() =. rd interval: _ + ( )() = (.)() = th interval: _ + ( )() = (.)() =. th interval: _ + ( )() = (.)() = Add the products and divide by the sum of the frequencies Mean: = _ =. A good estimate for the mean of this data set is an age of.. The histogram shows the Olympic results for women s weightlifting. To estimate the mean, first find the midpoint of each interval, and multiply by the frequency. st interval: _ + ( ) =. =. nd interval: _ + ( ) =. = rd interval: _ + ( ) =. = th interval: _ + ( ) =. =. th interval: _ + ( ) =. =. Add the results and divide by the total number of values in the data set Mean: = _ A good estimate for the mean of this data set is. kg.. Ages of Teachers Women s Weightlifting Ages Weight (kg) EXPLAIN Estimating from Histograms AVOID COMMON ERRORS Students who use mental math to find the midpoint of each interval on a histogram often choose the incorrect midpoint. Remind students that the midpoint of an interval can be found by adding the least and greatest values and dividing the sum by. QUESTIONING STRATEGIES Is the median of a data set always represented in one of the middle intervals of a histogram? Explain. No; if the data distribution is skewed to the left or right, the median value may be represented in an interval on either end. For example, if more than half the values are in the rightmost interval, the median would be located in that interval. Can you find the exact values of the mean and median of a data set from a histogram? Explain. No; because a histogram does not show individual data values, only an estimate can be determined. Module Lesson DIFFERENTIATE INSTRUCTION Multiple Representations Some students may be troubled by the fact that specific data values are not represented on a histogram. These students may benefit from sketching the histogram on a piece of paper and writing the actual data values into their corresponding intervals. This will help them understand how the estimates are made. For example, by locating the interval that contains the median of the data set and listing all the values in that interval, students can see how the median of the data set is some fraction of the way from the low end to the high end of the interval. Histograms and Box Plots

5 EXPLAIN Constructing Box Plots INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP. Help students better understand the connection between dot plots and box plots by drawing a dot plot directly above a box plot for the same set of data. Compare how both data displays show measures of center and spread. Your Turn. The histogram shows the length, in days, of Maria s last vacations. Estimate the mean of the data set displayed in the histogram. st interval: ( + )() = () () = Explain nd interval: ( + )() = ()() = + rd interval: ( )() = ()() = + th interval: ( )() = ()() = Mean: = =. days Constructing Box Plots Maria s Vacations A box plot can be used to show how the values in a data set are distributed. You need values to make a box plot: the minimum (or least value), first quartile, median, third quartile, and maximum (or greatest value). Length (in days) QUESTIONING STRATEGIES A box plot combines some of the advantages of a dot plot and of a histogram. What useful characteristics of each does a box plot share? All three diagrams show the shape of a data distribution. Like a dot plot, a box plot displays some specific numbers (the minimum, maximum, median, and first and third quartiles). Like a histogram, a box plot summarizes the data so you can see the shape of the distribution without looking at many individual data points. How can you tell that the box in a box plot represents about % of the data in a data set? The interval from the first quartile to the median contains about one-fourth of the values in the data set, and the interval from the median to the third quartile also contains about one-fourth of the values. This means the section from the first quartile to the third quartile contains about half the values in the data set. Example Use the data to make a box plot. The numbers of runs scored by a softball team in games are given.,,,,,,,,,,,,,,,,,,, Order the data from least to greatest.,,,,,,,,,,,,,,,,,,, Identify the needed values. Those values are the minimum, first quartile, median, third quartile, and maximum.,,,,,,,,,,,,,,,,,,, Minimum Q Median Q Maximum. Draw a number line and plot a point above each of the needed values. Draw a box whose ends go through the first and third quartiles, and draw a vertical line segment through the median. Draw horizontal line segments from the box to the minimum and maximum. Minimum First quartile Third quartile Median Maximum. Module Lesson LANGUAGE SUPPORT Connect Vocabulary Students might be familiar with dot plots, measures of center (mean, median), and measures of spread (range) from previous courses, but it is important to make sure that students also know the terms used to describe these measures. When creating box plots and comparing two sets of data, students must find the first quartile, third quartile, and interquartile range. Point out that the English word quartile has a Spanish cognate, cuartil. Be sure students are aware that first quartile and third quartile refer to specific values, not to the lower and upper quarters of the data set. Lesson.

6 B,,,,,,,,,,,,,,, Order the data from least to greatest.,,,,,,,,,,,,,, Identify the needed values. Those values are the minimum, first quartile, median, third quartile, and maximum.,,,,,,,,,,,,,,, Minimum Q Median Q Maximum Draw a number line and plot a point above each of the needed values. Draw a box whose ends go through the first and third quartiles, and draw a vertical line through the median. Draw horizontal lines from the box to the minimum and maximum. Reflect. The lines that extend from the box in a box plot are sometimes called whiskers. What part (lower, middle, or upper) and about what percent of the data does the box represent? What part and about what percent does each whisker represent? Box: middle %; left whisker: lower %; right whisker: upper %. Which measures of spread can be determined from the box plot, and how are they found? Calculate each measure. The range is the length of the entire box plot: =. The IQR is the length of the box: - =. Your Turn Use the data to make a box plot..,,,,,,,,,. The numbers of goals scored by Lisa s soccer team in games are listed below.,,,,,,,,, Min Q Median Q Max,,,,,,,,,,,,,,,,,,,,,,,,. Module Lesson Histograms and Box Plots

7 EXPLAIN Comparing Box Plots AVOID COMMON ERRORS When comparing box plots, some students may draw each box plot using a number line with a different scale. Remind students that to compare data sets by placing box plots one above the other, both box plots must be drawn using the same number line. QUESTIONING STRATEGIES What effect would outliers have on the center and shape of a box plot? Outliers increase the maximum value or decrease the minimum value of a data set, but they have relatively little effect on the median and the first and third quartiles. Therefore, the length of one or both lines extending from the box would be longer for a data set with outliers, but the box itself would have nearly the same size and position as it would in a box plot for the same data set without the outliers. Explain Comparing Box Plots You can plot two box plots above a single number line to compare two data sets. Example Construct two box plots, one for each data set. Compare the medians and measures of variation for each distribution. The tables show the total gross earnings, in dollars, for the top movies of and. Total Gross Earnings by the Top Movies of & Rank Total Gross in Rank Total Gross in $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, $,, Analyze Information For each set of data, identify the five values you need to make a box plot: the minimum, first quartile, median, third quartile, and maximum. In this case, the data is from least to greatest reading from the bottom to the top. Formulate a Plan With the needed values for each data set, construct box plots on the same number line. The number line for both plots can go from $ million to $ or million. Interpret the box plots to compare the gross earnings for the top movies in and. Module Lesson Lesson.

8 Solve Using the statistics feature of a graphing calculator, find the five needed values, rounded to the nearest hundred million. Minimum Q Median Q Maximum Top Movies Gross Earnings ($,,s) Top Movies Gross Earnings ($,,s) The medians for both years are very close together, but gross earnings for did not vary as much as in. The range and interquartile range were greater for the data than for the data. The maximum value in the data appears to be an outlier, but because is not greater than, it is not an outlier. Justify and Evaluate Considering the little difference between minimum values but the great difference between maximum values the data sets, it makes sense that their measures of variation would / would not be alike. Your Turn Construct two box plots, one for each data set. Compare the medians and measures of variation for each distribution.. The net worth of the richest people in the world for and (in billions) are: :,,,,.,,,,.,. :,,,.,,,,,,... The medians are close, but the median for is slightly greater than that of. The ranges are close, but the interquartile range for is much greater than that of. Module Lesson Histograms and Box Plots

9 ELABORATE QUESTIONING STRATEGIES Describe what a box plot representing a symmetric distribution and a box plot representing a skewed distribution would look like. A box plot for a symmetric data distribution would be symmetrical: the median would be in the center of the box, and the lines extending to each side of the box would be about the same length. In a box plot for a skewed distribution, the median would be closer to one end of the box, and the line extending from that end of the box might be shorter than the line at the other end. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP. Have students explore how an outlier affects a box plot. Have them first draw a box plot for a data set with no outliers, then add an outlier to the data set and draw a new box plot. Have them compare the two plots. Discuss which changes will occur with any outlier, and which might be different for a different outlier. SUMMARIZE THE LESSON What are the steps for creating a box plot? Order the data. Determine the minimum, median, first quartile, third quartile, and maximum. Draw a number line and plot the values. Draw a vertical line at the median, a box from Q to Q, and lines from the box to the minimum and maximum points.. The ages of the richest people in the world for and (in years) are: :,,,,,,,,, :,,,,,,,,, Elaborate. The median for is much greater than the median for. Both the range and interquartile range for are less than those of, so the ages were more varied in.. How can you create a histogram from a data set? First, create a frequency table using the data. Look at the range of the data to determine the interval width. Then, use the frequency table to create the histogram, where the bar heights for each interval correspond to the frequency.. How can you create a box plot from a data set? First, order the data from least to greatest. Find the minimum and maximum values, the median, and the first and third quartiles. Draw a box whose ends go through the first and third quartiles, and then draw a vertical line through the median. Draw horizontal lines from the box to the minimum and maximum.. Essential Question Check-In How can you use histograms and box plots to interpret and compare data sets? You can use histograms to estimate statistics and compare distributions. You can use box plots to compare the centers, spread, and shape of the distribution. Module Lesson Lesson.

10 Evaluate: Homework and Practice Use the histogram to answer the following questions.. What does each axis indicate? The vertical axis shows the frequency for each interval, and the horizontal axis shows the bowlers scores.. How is the horizontal axis organized? The scores are organized in groups of.. How many bowlers competed? = bowlers competed.. Describe the general shape of the distribution. The distribution is skewed left. Create a histogram for the given data.. Listed are the ages of the first U.S. presidents on the date of their first inauguration.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Ages Possible tables and histograms shown. Online Homework Hints and Help Extra Practice Bowlers' Scores. Listed are the breathing intervals, in minutes, of gray whales.,,,,,,,,,,,,,,,,,,,,,,,, Time (min) Module Lesson EVALUATE ASSIGNMENT GUIDE Concepts and Skills Explore Understanding Histograms Example Creating Histograms Example Estimating from Histograms Example Constructing Box Plots Example Comparing Box Plots Practice Exercises Exercises Exercises, Exercises, Exercises, INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP. Have students conduct a survey with quantitative results that can be displayed in intervals. Then have them choose at least three different-sized intervals for grouping the data, and make both a frequency table and a histogram for each interval choice. Have students estimate the mean and median from each histogram and compare the estimates to the mean and median calculated from the raw data. Then have students explain which interval they think best displays the survey results. Exercise Depth of Knowledge (D.O.K.) Mathematical Practices Recall of Information MP. Modeling Recall of Information MP. Using Tools Skills/Concepts MP. Modeling Strategic Thinking MP. Patterns Strategic Thinking MP. Logic Histograms and Box Plots

11 . Listed are the heights, in inches, of the students in Marci s karate class.,,,,,,,,,,,,,,. Listed are the starting salaries, in thousands of dollars, for college graduates.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Height (in.) Salary Range (thousands $) Estimate the mean of the data set displayed in each histogram.. The histogram shows the GPAs of the students in George s class. st interval: (_. +. )() = (.)() =. nd interval: (_. +. )() = (.)() =. rd interval: (_. +. )() = (.)() =. th interval: (_. +. )() = (.)() =. Mean: _ = =. The mean GPA is GPA Module Lesson Lesson.

12 . The histogram shows long jump distances, in feet, for a track and field team. Distance (ft) _ + st interval: ( )() = ()() = + nd interval: (_ )() = ()() = + rd interval: (_ )() = ()() = + th interval: (_ )() = ()() = Mean: = =. The mean long jump distance is. ft.. The histogram shows the speeds of downhill skiers, in miles per hour, during a competition. Speed (mi/h) _ + st interval: ( )() = (.)() = + nd interval: (_ )() = (.)() = + rd interval: (_ )() = (.)() =. + th interval: (_ )() = (.)() =. + th interval: (_ )() = (.)() = Mean: =.. The mean speed is about. mi/h. Image Credits: Design Pics Inc./Alamy MANIPULATIVES To help students understand how a histogram represents data, allow each student to grab a handful of integer chips or other small objects. Have each student count the number of integer chips he or she holds and write that number on a sticky note. Write intervals of across the board and have students place their numbers above the correct interval. Numbers within the same interval should be lined up vertically. The result will resemble a histogram. AVOID COMMON ERRORS Remind students that each interval of a histogram should represent the same span of values, so all the bars must have the same width. Some students may think that each bar of a histogram can represent a range of a different size. Module Lesson Histograms and Box Plots

13 . The histogram shows the depths a diver has been to, in meters. + st interval: (_ )() = (.)() =. + nd interval: (_ )() = (.)() = + rd interval: (_ )() = (.)() =. + th interval: (_ )() = (.)() = Mean: = =. The mean depth is. m. Use the data to make a box plot.. The numbers of points Julia s basketball team scored in games are listed below.,,,,,,,,,, Depth (m). The numbers of baskets Kelly s team scored in games are listed below.,,,,,,,,. The numbers of runs Jane s baseball team scored in games are listed below.,,,,,,,, Module Lesson Lesson.

14 . The numbers of field goals James football team scored in games are listed below.,,,,,,,,,,,.. The numbers of points scored by Jane in basketball games are listed below.,,,,,,,... The numbers of goals Claudia s soccer team scored in games are shown below.,,,,,,,,,,,,,,,,,,,,. Mario and Carlos, two brothers, play for the same basketball team. Here are the points they scored in games: Game Mario X X Carlos X X (Xs mark games each one missed.) Which brother had the highest-scoring game? How many more points did he score in that game than the other brother did in his highest-scoring game? Mario Carlos... Carlos scored a maximum number of points in a game versus Mario s maximum of. Between these two games, Carlos scored more points than Mario. Module Lesson Histograms and Box Plots

15 JOURNAL Have students write a paragraph explaining how to compare data sets using box plots. Students should include how to construct a box plot, with instructions on how to find the minimum and maximum values, as well as the first quartile, median, and third quartile. H.O.T. Focus on Higher Order Thinking. Communicate Mathematical Ideas Describe how you could estimate the IQR of a data set from a histogram. Count the number of values, and determine which intervals will contain Q and Q. Use the method for estimating the median to estimate Q and Q, and then find the difference of the two values.. Critical Thinking Suppose the minimum in a data set is the same as the first quartile. How would this affect a box plot of the data? Explain. Because the minimum and the first quartile are the same, the box plot would have no whisker on the left side.. Draw Conclusions Dolly and Willie s scores are shown. Dolly claims that she is the better student, but Willie claims that he is the better student. What statistics make either Dolly or Willie seem like the better student? Explain. Median and IQR make Dolly seem like the better student. Dolly has a higher median score. She also % has the smaller IQR, which means that her scores % % % % % are more consistent. Third quartile and maximum make Willie seem like a better student. Willie s set has the higher third quartile and maximum. This means that his greatest % of scores are all higher than Dolly s highest score. Dolly Willie Module Lesson Lesson.

16 Lesson Performance Task The batting averages for the starting lineup of two competing baseball teams are given in the table. Batting Averages for Team A Batting Averages for Team B a. Create a box plot for each data set on the same number line. Batting Averages Team A CONNECT VOCABULARY Some students may not know what a starting lineup is. Ask students in the class with knowledge of baseball to explain what a starting lineup is a list of players who will participate in the game when it begins and the criteria that might be used for including a player on the starting lineup list..... Team A.,.,.,.,.,.,.,.,.,. Minimum =. Q =. Q =. Maximum =. Maximum =. b. Compare the distributions of the batting averages for each team. c. Which team has a better starting lineup? Explain. Team B.. Team B.,.,.,.,.,.,.,.,.,. Minimum =. Q =.. +. Median = =.. +. =. Median = =. =. Q =. The distribution for Team A is more symmetric and the distribution for Team B is more skewed to the right. Team A has a higher median batting average than Team B. The batting averages are more spread out for Team B than they are for Team A. Team A has a higher median and a smaller spread. So, the batting averages of Team A s starting lineup are more consistent. Team A also has a higher maximum, so the highest batting average of Team A s starting lineup is higher than the highest batting average in Team B s starting lineup. So, Team A has a better starting lineup. INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP. Have students use the one-variable statistics function on a graphing calculator to check the values they calculated for the median and first and third quartile of each data set. Students can also use graphing calculators to generate the double box plot of the data sets. Have them research the procedure online, apply it to the data from the Lesson Performance Task, and check whether the box plots they drew by hand match the ones shown by the calculator. Module Lesson EXTENSION ACTIVITY Have students research the batting averages of the top players on two of their favorite baseball teams. Then have students use the data they collect to create a double box plot of the data and compare the performance of the two sets of players. Students who are not baseball fans may choose to analyze statistics from a different sport or competition, such as basketball, football, or chess. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. points: Student does not demonstrate understanding of the problem. Histograms and Box Plots