Graphical Representation of Data and the Measures of Central Tendency
Graphical Representation of Data and the Measures of Central Tendency
Objectives
In this unit students will investigate data. Students will:
- analyze data.
- represent data graphically using a stem-and-leaf plot, histogram or line plot.
- describe a data set based on the shape or graphic distribution.
- calculate the mean, median, mode, and range of a data set.
- compare data sets.
- identify the measure of central tendency that best describes a data set.
Essential Questions
- How can we use the mean, median, mode, and range to describe a set of data? Why do we need three different measures of central tendency?
- How can we use mathematics to provide models that help us interpret data, make predictions, and better understand the world in which we live, and what are the limits of these models?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- NCTM Standards and Expectations for Grades 6–8, Data Analysis and Probability
- Possible PDFs to use for Performance Assessment if students do not have access to the Internet for this activity:
Formative Assessment
-
View
Multiple Choice Items
1. This stem-and-leaf plot shows the minutes recorded per individual calls made on a cell phone. How many phone calls were recorded?
A
25
B
30
C
92
D
862
2. This stem-and-leaf plot shows the minutes recorded per individual calls made on a cell phone. How many calls were more than 20 minutes?
A
3
B
16
C
20
D
22
3. Which data set has a median of 25?
4. The Bears baseball team scored a mean value of 4 runs in each of their last 6 games. Which data set can represent the possible number of runs scored in their last 6 games?
A
4, 5, 2, 2, 7
B
1, 2, 4, 4, 4, 5
C
7, 0, 2, 4, 3, 4
D
3, 6, 2, 1, 4, 8
5. Jason mowed four lawns and earned an average of $10 per lawn. Which data set can represent the amounts he charged for the four lawns?
A
$8, $10, $15, $17
B
$15, $7, $10.50, $7.50
C
$8.50, $12.50, $10, $10
D
$7, $9, $12, $16
6. During the six hours the library was open on Sunday, the circulation desk checked out an average of 145 books per hour. How many books were checked out that day?
A
640 books
B
810 books
C
840 books
D
870 books
7. Students collected data on how much television they were watching during the week. If students wanted to defend that they were not watching too much television during the week, which measure of central tendency should they use to best support their claim?
Student
A
B
C
D
E
F
G
H
I
J
Hours
15
7
5
6
5
15
17
7
15
8
A
range
B
mode
C
median
D
mean
8. The scores for Jerome’s class on a math quiz are shown in the line plot below
No partial credit was given. One person still needs to take the test. After she takes the test, which of the following class averages (mean) is possible?
A
6
B
6.5
C
6.6
D
7
9. Which data set has the same mean and median?
A
12, 6, 6, 8, 10, 10, 8, 12
B
18, 16, 3, 5, 12, 12, 15, 7
C
3, 7, 8, 2, 4, 4, 8, 4
D
5, 6, 3, 2, 4, 4, 2, 4
Multiple-Choice Answer Key
1. A
2. C
3. B
4. D
5. B
6. D
7. C
8. B
9. A
Short-Answer Items:
Use the following stem-and-leaf plot to answer item 10.
10. Record the temperatures represented in this stem-and-leaf plot.
11. Brian is a member of the bowling league. In three games, his mean average score was 125. What are the possible scores he earned in the three games if none of the three scores were the same? Show your work and explain how you solved the problem.
12. A salesperson claimed she should win the honor of Salesperson of the Week because of her average (mean) sales. Her last six sales totaled $85, $240, $60, $85, $145, $105. Which measure of central tendency would best support her claim that she deserves the honor?Short-Answer Key and Scoring Rubrics:
10. Record the temperatures represented in this stem-and-leaf plot.
The temperatures recorded in the stem-and-leaf plot are: 35, 40, 43, 48, 48, 52, 55, 60, 65, 71.
Points
Description
2
- Mathematical thinking is correct.
- All temperatures from the stem-and-leaf plot are identified and accurate.
- Student demonstrates thorough understanding of the mathematical concept.
1
- Mathematical thinking is correct.
- Most of the temperatures from the stem-and-leaf plot are identified. Minor errors may be present.
- Student demonstrates some understanding of the mathematical concept.
0
- Mathematical thinking is incorrect.
- Many of the temperatures from the stem-and-leaf plot are missing or incorrect.
- Student does not demonstrate understanding of the mathematical concept.
11. Brian is a member of the bowling league. In three games, his mean average score was 125. What are the possible scores he earned in the three games if none of the three scores were the same? Show your work and explain how you solved the problem.
Answers may vary. Be sure the sum of the three numbers equals 375. Be sure there are three values and that none of the values chosen are the same. One possible solution is: 135, 115, 125. I knew from the problem that the mean is 125. Since there are three games, I can determine the value of my three scores should equal 125 × 3 = 375. So now I can begin to guess and check three numbers that when added together equal 375.
Points
Description
2
- Mathematical thinking is correct.
- All three scores have different values.
- Explanation is thorough and complete using specific mathematical language.
- Student demonstrates thorough understanding of the mathematical concept.
1
- Mathematical thinking is correct.
- Answer may have two scores with the same value.
- Explanation is complete with some use of mathematical language.
- Student demonstrates some understanding of the mathematical concept.
0
- Mathematical thinking is incorrect.
- Values may be present but are not appropriate for the problem.
- Explanation is incomplete or missing.
- Student does not demonstrate understanding of the mathematical concept.
12. A salesperson claimed she should win the honor of Salesperson of the Week because of her average (mean) sales. Her last six sales totaled $85, $240, $60, $85, $145, $105. Which measure of central tendency would best support her claim that she deserves the honor?
The mean would be the best measure of central tendency to support her claim. If she is trying to show a higher average for total sales then the mean is the greatest. It is $25 greater than the median and $35 greater than the mode.
Mean: 85 + 240 + 60 + 85 + 145 + 105 = 720 ¸ 6 = $120
Median: 60 85 85 105 145 240; median 85 + 105 = $95
Mode: $85
Points
Description
2
- Mathematical thinking is correct.
- Values to represent measures of central tendency are calculated correctly.
- Explanation is thorough and complete using specific mathematical language.
- Student demonstrates thorough understanding of the mathematical concept.
1
- Mathematical thinking is correct.
- Values to represent measures of central tendency are calculated. Minor errors may be present.
- Explanation is complete with some use of mathematical language.
- Student demonstrates some understanding of the mathematical concept.
0
- Mathematical thinking is incorrect.
- Values to represent measures of central tendency may be present but are not appropriate for the problem.
- Explanation is incomplete or missing.
- Student does not demonstrate understanding of the mathematical concept.
Performance Assessment:
Suppose you are looking to make a presentation about the nutritional value of food on the dollar menu at a particular restaurant. Use the information below to do so.
Sausage biscuit: 430 calories, 27g of fat, 1,080 mg of sodium; Hash brown: 150 calories, 9 g of fat, 310 mg of sodium; Small Fries: 380 calories, 19 g of fat, 270 mg of sodium; Small double burger (no cheese): 390 calories, 19 g of fat, 920 mg of sodium; Small chicken sandwich:360 calories, 16 g of fat,
830 mg of sodium; Hot fudge sundae:330 calories, 10 g of fat,
180 mg of sodium; Fruit and yogurt parfait (7oz): 160 calories, 2 g of fat, 85 mg of sodium; 1 Baked apple pie (77g): 250 calories,13 g of fat, 170 mg of sodium; Side salad (no dressing):
20 calories, 0 g of fat, 10 mg of sodium; Medium Coke (21oz):
210 calories, 0 g of fat, 15 mg of sodium
http://www.mcdonalds.com/us/en/food/meal_bundles/dollar_menu.html
- Create a chart of the items in the list above. Then record either the amount of calories, fat, or sodium found in each of the items. Choose only one of the three to compare.
- Construct a stem-and-leaf plot to represent the information.
- Construct a line plot to represent the information.
- Create a list of similarities and differences between the two displays of data.
- Calculate the measures of central tendency: mean, median, and mode.
- Write a description about the data you analyzed. Be sure to:
- describe the shape of the data
- list similarities and differences about the two displays
- draw conclusions from the data you analyzed
- find the measures of central tendency (mean, median, mode, range) and state which would best represent the data and why
- state what could happen to the measures of central tendency if a new piece of data were added
- use specific math language in your descriptions
Performance Assessment Scoring Rubric:
Points
Description
4
- Mathematical calculations are correct with complete work shown.
- Written explanation is thorough, clear, and uses specific math language.
- Work displays advanced understanding of the questions, mathematical concepts, and processes related to the measures of central tendency.
- Graphic representations and chart are appropriate and constructed with clarity and accuracy.
3
- Mathematical calculations are correct with some evidence of work shown.
- Written explanation is clear and uses some specific math language.
- Work displays good understanding of the questions, mathematical concepts, and processes related to the measures of central tendency.
- Graphic representations and chart are appropriate and constructed with accuracy.
2
- Mathematical calculations are correct with minimal work shown.
- Written explanation is present but lacks some detail. Specific math language may be present.
- Work displays partial understanding of the questions, mathematical concepts, and processes related to the measures of central tendency.
- Answer meets a significant amount of the problem requirements.
- Graphic representations and chart is complete but may have errors.
1
- Mathematical calculations are incorrect with minimal work shown.
- Written description is incomplete and lacks detail.
- Work displays little understanding of the questions, mathematical concepts, and processes related to the measures of central tendency.
- Answer does not meet the majority of the requirements of the problem.
- Graphic representations and chart, if used, are inappropriate for the problem.
0
- Mathematical calculations are incorrect or incomplete. No work shown.
- Written description is illogical or not present.
- Work displays no real understanding of the questions, mathematical concepts, or processes related to the measures of central tendency.
- Answer does not meet the requirements of the problem.
- Graphic representations and chart are incomplete or missing.
DRAFT 10/07/2011
