Lesson 4: What Does the Data Tell Us?

Transcription

1 Lesson 4: What Does the Data Tell Us? Selected Content Standards Benchmark Addressed: D-2-M Analyzing, interpreting, evaluating, drawing inferences, and making estimations, predictions, decisions, and convincing arguments based on organized data (e.g., analyze data using concepts of mean, median, mode, range, random samples, sample size, bias, and data extremes) GLEs Addressed: Grade Calculate and discuss mean, median, mode, and range of a set of discrete data to solve real-life problems (D-2-M) Grade Explain factors in a data set that would affect measures of central tendency (e.g., impact of extreme values) and discuss which measure is most appropriate for a given situation (D-2-M) Lesson Focus The mean, median, and mode are three commonly used measures of central tendency. The mean (arithmetic mean) is the average of a set of data. The median is the middle value when a set of data is arranged in order of size. The mode is the number that occurs most frequently in a set of data. The range is the difference between the highest and lowest numbers in a set of data. Students will be required to find each of the three measures of central tendency as well as the range. They will also be expected to determine which particular measure is the most appropriate for a given set of data. GEE 21 Connection Students will be required to: Calculate/find the mean, median, and mode, and recognize which measure of central tendency is most appropriate for a given set of data Solve a variety of problems involving measures of central tendency and/or ranges Translating Content Standards into Instruction There are three worksheets for this lesson. The first asks students to find the mean, median, and mode with sets of numbers and then through problemsolving activities. The second worksheet addresses the choice of an average. When is it best to choose the mean or the median or the mode? The third worksheet consists of a group of problems that have appeared on various state exams in the past few years. There are also a couple of open-ended problems with student work that you can use. 36

2 Students usually have no trouble finding the mean. This is something that they have worked with throughout school. When asked to find the average it is usually the mean that they find. They learned how to find the median in Lesson One when they constructed the box-and-whiskers plot. The mode may or may not be something new for them. It would be a good idea to conduct a quick review of all three. A teacher master is included for your convenience. (The answers are on the sheet, but can be covered while the students answer the questions.) Follow this with Student Worksheet # 1. The mean is the most useful measure of central tendency when all of the data should be taken into account. The median is the most useful as an average for data that contain a small number of highly divergent numbers. The mean is affected greatly by divergent data; the median is not. The average family income for the average price of a new home would be stated using a median, since income or new home prices have no upper limit. Thus, the mean would be too large.the median is best for any data that is skewed either to the right or to the left. The mode is a good measure of central tendency when the measurements are non-numerical, such as the answer to, What is the most popular make of automobile? Any data that is put into categories would use the mode. There are three problems on the blackline master that deal with the choice of central tendency. Go over these with the students, and then hand out Student Worksheet #2. Sources of Evidence of Student Learning Worksheet # 3, which contains the problems below should give the teacher a good idea about how well the student understands the three measures of central tendency. The open-ended problem combines a stem-and-leaf plot with the mean and median. Both the worksheet and the open-ended problem should be done individually, as if they were a test. GEE 21 Connection On the GEE 21 test, students may be given multiple choice problems like the ones shown below. These problems were taken from the websites for the Illinois and Massachusetts State tests. They may be accessed through the Louisiana Dept. of Education Web site. The correct answers are denoted with (*). These problems also appear on Student Worksheet #3 without the correct answers being given. 1. The mean salary of the ten clerks at the Corner Shop was $8.50 per hour. One of the clerks, who had been making $9.50 per hour, was given a raise of $1.00 per hour. What is the new mean salary of the ten clerks? A. $8.50 B. $8.60* C. $8.80 D. $9.00 The Massachusetts Comprehensive Assessment System 37

3 2. The same six students rated two science fiction movies on a scale of 1 to 10. The results are shown in the table to the right. The ratings for the two movies have the same A. median B. mean* C. mode D. range Student Movie 1 Rating Movie 2 Rating A 3 6 B 5 7 C 6 6 D 7 6 E 8 5 F 8 7 The Massachusetts Comprehensive Assessment System 3. In a bowling tournament, there were 2 scores in the 200s, 6 scores in the 180s, 5 scores in the 160s, 5 in the 150s, and 3 in the 130s. The median score would be in the A. 150s B. 160s* C. 180s D. There is not enough information given. The Massachusetts Comprehensive Assessment System 4. Use the graph below to answer the next two questions. Which student had the greatest difference in mathematics and science scores? M* N O P 38

4 Which is the best estimate of the mean of the science test scores for all five students? A. 75* B. 80 C. 85 D. 95 The Massachusetts Comprehensive Assessment System 5. Note: This question combines a frequency table with the question on the mean. Students should understand that they must multiply the number of employees by the hourly wage and divide the result by 50, the number of employees. Which represents the mean (arithmetic average) of the hourly wages for the 50 employees? Trex Trucking Employees Hourly Wage 15 $ $ $15.70 A. $14.60 B. $14.90 C. $15.20 * D. $15.25 E. $15.70 llinois Standards Achievement Test And an open-ended problem: 6. A class of 25 students is asked to determine approximately how much time the average student spends on homework during a one-week period. Each student is to ask one of his/her friends for information, making sure that no one student is asked more than once. The numbers of hours spent on homework per week are as follows: 8, 0, 25, 9, 4, 19, 25, 9, 9, 8, 0, 8, 25, 9, 8, 7, 8, 3, 7, 8, 5, 3, 25, 8, 10 a. Find the mean, median, and mode for these data. Explain or show how you found each answer. answer: mean - 10; median 8; mode - 8 b. Based on this sample, which measure (or measures) found in part (a) best describes the typical student? Explain your reasoning. ans: the median and/or mode The 4 answers of 25 skewed the mean so that it is not representative of those surveyed. Only 5 of the 25 studied more than 10 hours. 39

5 Attributes of Student Work at the Got-It level Students need to know how to find the three measures of central tendency and how to choose the one best suited for the data set. The open-ended problem on worksheet #3 should give you a good idea of how well they understand these concepts. In part (a) they should either show the work done in finding the mean or explain that they added up all of the hours using their calculators and divided by 25. To find the median, they should arrange the hours in order, either increasing or decreasing, and explain that 8 is the middle number. With the mode, they should explain that 8 is given most frequently. If students can work this problem without help, then they should be ready to answer any questions on mean, median, and mode found on the GEE

6 Lesson 4: What Does Data Tell Us? Teacher Blackline #1 Find the mean, median, and mode for problems 1 and The test scores for a math student were: 85, 74, 92, 87, 84 78, The number of items produced in one hour by twelve different machines were: 0, 1, 3, 5, 5, 5, 7, 9, 9, 11, 15, John has exam grades of 84, 91, 72, 68, 78. What is the minimum grade John must obtain on the next exam to obtain an average (mean) score of 80? 4. Find the median salary of the following annual salaries: $10,000, $12,000, $15,000, $60,

7 Lesson 4: What Does Data Tell Us? Teacher Blackline #2 1. Bill s class was studying mean, median, and mode. Each of the 35 students was given a different sampling problem. Bill s problem was to sample the class and determine the class favorite color. The five categories with the number of responses were: Red, 9 Blue, 12 Yellow, 7 Green, 4 Other, 3 When it came Bill s turn to report on his project, he showed them a bar graph picturing the information. The teacher was quite pleased. Bill then turned the chart over to show that he had also calculated the average of the categories: = 7 The class laughed. Why? What should he 5 have reported? 2. Ray got scores of 80, 96, 84, 95, and 90 on five math tests. What is his mean score? His median score? On his next test he earned a 25. What is his mean score now? The teacher uses the following grading scale: A ; B 80-89; C 70-79; D 60-69; and F Knowing this, Ray went to the teacher and suggested that she use the median to determine his score. Why? 3. Why is the median used in these statements? According to the U.S. Bureau of the Census, the median age at first marriage for men is 26.3 years and for women is 24.1 years. The median-priced single family home in the United States is $79,

8 Lesson 4: What Does Data Tell Us? Student Worksheet #1 1. Alice bowled games of 250, 300, and 200. Find her average (mean) score. 2. Jeron has grades of 50, 60, 90, 100, and 70. If his next grade is 40, what is his average (mean) score? 3. To earn a grade of B in math, Ross must achieve an average score of 85 on five math tests. His scores on the first three tests are 87, 92, and 82. What is the lowest total score that Ross must have on the last two tests to earn a B test average? 4. Mr. Carson earned $10,000 per month for 11 months. What is his salary for the 12 th month, if his average salary for the 12 months is $12,000? 5. Arrange the following set of data in a frequency distribution table; then find the mean, median, mode, and range. 48, 47, 43, 42, 47, 49, 43, 45, 47, 48, 46, 47, 44, 47, 44, 40, 49, Use the table below to find the median high temperature, the median low temperature. TEMPERATURE RANGES CITY HIGH LOW Biloxi 87 degrees 73 degrees Denver 89 degrees 58 degrees Los Angles 80 degrees 70 degrees Orlando 93 degrees 76 degrees Philadelphia 90 degrees 68 degrees Toledo 85 degrees 63 degrees 43

9 Lesson 4: What Does Data Tell Us? Student Worksheet #2 1. Find the mean, median, and mode and determine which measure(s) of central tendency would best represent each of the following sets of data? a. 51, 56, 48, 51, 97, 52, 123, 51 b. 78, 84, 93, 87, 82, 85, 91, 79 c. 23, 25, 22, 23, 48, 45, 46, Mrs. Johnson owns a small dress shop and it is time to make her order for the next season. Space constraints are such that she will have to cut back on her stock. Looking at her sales figures, she finds she sold the following dresses: Size Number sold a) What is the mean? The median? The mode? b) Why are the mean and median sizes meaningless to Mrs. Johnson? Explain. c) What sizes would you recommend she stock? 2. The table below shows the number of moons that circle each planet in our solar system. Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Find the mean and median number of moons. How many planets have more than the mean number of moons? 3. The table below shows the salaries of the six players who have started more than 25 games for the Minnesota Timberwolves this year. Kevin Garnett Terrell Brandon Joe Smith Chauncy Billups Wally Szczerbiak Radoslav Nesterovic 22.4 million 9.3 million 4.5 million 2.5 million 2.3 million 1.7 million Suppose the players were negotiating for raises for Would they use the mean or median in their arguments for raises? Which would the owners of the team use? 5. Which measure of central tendency (mean, median, or mode) can be most greatly affected by the range of a set of data. Explain. 44

10 Lesson 4: What Does Data Tell Us? Student Worksheet #3 1. The mean salary of the ten clerks at the Corner Shop was $8.50 per hour. One of the clerks, who had been making $9.50 per hour, was given a raise of $1.00 per hour. What is the new mean salary of the ten clerks? A. $8.50 B. $8.60 C. $8.80 D. $ The same six students rated two science fiction movies on a scale of 1 to 10. The results are shown in the table to the right. The ratings for the two movies have the same A. median B. mean C. mode D. range Student Movie 1 Rating Movie 2 Rating A 3 6 B 5 7 C 6 6 D 7 6 E 8 5 F In a bowling tournament, there were 2 scores in the 200s, 6 scores in the 180s, 5 scores in the 160s, 5 in the 150s, and 3 in the 130s. The median score would be in the A. 150s B. 160s C. 180s D. There is not enough information given. 4. Use the graph below to answer the next two questions. Which student had the greatest difference in mathematics and science scores? A. M B. N C. O D. P 45

11 Which is the best estimate of the mean of the science test scores for all five students? A. 75 B. 80 C. 85 D Which represents the mean (arithmetic average) of the hourly wages for the 50 employees? Trex Trucking Employees Hourly Wage 15 $ $ $15.70 A. $14.60 B. $14.90 C. $15.20 D. $15.25 E. $15.70 And an open-ended problem: 6. A class of 25 students is asked to determine approximately how much time the average student spends on homework during a one-week period. Each student is to ask one of his/her friends for information, making sure that no one student is asked more than once. The numbers of hours spent on homework per week are as follows: 8, 0, 25, 9, 4, 19, 25, 9, 9, 8, 0, 8, 25, 9, 8, 7, 8, 3, 7, 8, 5, 3, 25, 8, 10 a. Find the mean, median, and mode for these data. Explain or show how you found each answer. b. Based on this sample, which measure (or measures) that you found in part a best describes the typical student? Explain your reasoning. 46

12 Lesson 4: What Does Data Tell Us? Answer Keys Teacher Blackline #1 1. mean is 84.3, median is 85, mode none 2. mean is 14.1, median is 6, and mode is $13, 500 Teacher Blackline # /5 = 7 regardless of the categories. Finding the mean says nothing about any average when it comes to categories. Bill should have reported the mode, the color blue. 2. mean 89, median mean the median is 87 which would give a B 5. In both cases the data is skewed. Most women marry 18 26, but there are some first time brides in their 50 s for instance. Worksheet # is the mean is the mean x = 85, x = ,000 + x = 144,000 x = $34, Frequency Table high 88 o low 69 o Mode 47 Median 47 Mean 46 Range 9 Worksheet #2 1. a) mean, 51 mode, 51.5 median The median is a better choice because of the dispersion of data b) 84.5 median, mean, no mode; the median or mean c) 34.5 median and mean, mode 23 The median or mean 47

13 2. The mode is size 10, the median is size 12, and the mean is 11.8; the mode is the best choice, when the data set is non-numerical, in this case dress sizes. Answers will vary in part (c) Most will choose sizes The mean is 5.6 and the median is 2. Only 2 planets have more moons than the mean. 4. The median is 3.5 million and the mean is 7.1 million. The team would use the median in their argument for more salary; the team owners would use the mean. 5. The mean. A range of numbers that is skewed either left or right will not give a good measure of central tendency. Worksheet #3 1. B 2. B 3. B 4. A, A 5. C 6. See answers on page