YOUNGSTOWN CITY SCHOOLS
MATH: ALGEBRA 1
UNIT 3 - - DESCRIPTIVE STATISTICS (3 WEEKS) 2013-2014
Synopsis: In this unit, students will review and extend their knowledge of dot plots, histograms, box plots, finding
measures of central tendency and spread. They will examine frequency tables, paying particular attention to joint,
marginal, and conditional relative frequencies and look for trends in data. To complete the unit, the students will work
with line of best fit, assessing it with the correlation coefficient and residuals. Students will also discuss the relation
between causation and correlation, citing examples of each.
STANDARDS
S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
S.ID.2 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.
S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of
extreme data points (outliers).
S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of
the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or
choose a function suggested by the context. Emphasize linear and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
S.ID.9 Distinguish between correlation and causation.
MATH PRACTICES:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with Mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
LITERACY STANDARDS:
L.2
L.3
L.4
L-6
L.7
L.8
Communicate using correct mathematical terminology
Read, discuss, and apply the mathematics found in literature, including looking at the author’s purpose
Listen to and critique peer explanations of reasoning
Represent and interpret data with an without technology
Research mathematics topics or related problems
Read appropriate text, providing explanation for mathematical concepts, reason or procedures
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MOTIVATION
TEACHER NOTES
1. When working with a unit involving statistics, having data available for students to use is crucial. The
web site, http://www.amstat.org/censusatschool/, provides copious amounts of data from students
across the world. When logging on to this site, go to the teacher selection and use the user id
(melissa.rhode@youngstown.k12.on.us) and the password (password). Then proceed to class
registration where your class should be registered with an ID number. Make note of the ID number for
each of your classes and the password. To participate in this collection of data, go to the student
section, print out the questionnaire, copy enough for your students, pass out the questionnaire to the
students, have students complete the survey in class working in pairs and then have students enter their
data on the “Census at School” web site. (Note – do not hit enter, students will need class identification
number and password). After they have entered their data, give the students some time to become
familiar with the web site and look at the data. (MP.5)
2. Preview expectations for the end of the Unit
3. Have students set both personal academic goals for this Unit.
VOCABULARY
Data
Histogram
Negative Correlation
Quartile
Measures of Spread
Standard Deviation
Measures of Center
Measurement Classes
Dot Plots
Positive Correlation
Upper Quartile
Median
Range
Interquartile Range
Frequency
Sample
Box-and-Whisker Plot
Lower Quartile
Mean / Average
Outliers
Population
Frequency Table
Random Sample
Extreme Values
Data Sets
Sample Size
Skew
Joint frequency
Marginal frequency
Conditional relative frequency
Mean Absolute Deviation (MAD)
The following are additional web sites you might find helpful when teaching this section:
1. http://www.mathsisfun.com/data/standard-deviation.html (used to find variance and standard deviation)
2. http://www.meta-calculator.com/online/ (graphing, scientific, matrix, and statistics calculators)
3. http://nlvm.usu.edu/en/nav/frames_asid_145_g_4_t_5.html?open=instructions&from=category_g_4_t_5.html
(This site allows you to enter data and a box plot and histogram are formed using the data)
4. www.oh.algebra1.com (Alg 1 Textbook: user name: ALG105; password: bep8emek)
TEACHING-LEARNING
TEACHER NOTES
1. Discuss the five step process used to collect and analyze data to solve a problem or answer a question:
a) Develop a plan: state goals, describe the model, stating what is being measured and how it is to be.
b) Collect the data: use the “Data Census At School” site.
c) Analyze the data: histograms, scatter plots, box and whisker plots, box plots, etc.
d) Interpret the data
e) Make a conjecture about a future event and support it with information collected and analyzed.
After discussing the five steps have students read “Motivating Students in the High School Mathematics
Classroom” posted at http://scholar.valpo.edu/cgi/viewcontent.cgi?article=1002&context=sarp and
then have them answer the following questions:
 Is the sampling size appropriate? Why or why not?
 What was the data collection method?
 What needs to be done to apply this to YCS?
 Predict the appearance of the graph for YCS.
 Is a pie chart the best method of representing the data?
 What problem is the author trying to address in the statistical analysis?
 What conclusions can you formulate by examining the sources?
(MP-2, MP-4, L.3, L.8)
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TEACHER NOTES
2. Discuss the types of problems that use statistics to solve and those that do not. Listed below are a few to
use for discussion. A suggestion for an activity is to mix up the two lists and have students decide which
ones need statistics to solve and which ones do not. (S.ID.2, MP.3, L.2, L.4)
NEED STATISTICS:
a) Finding averages such as an average cost, averages in sports, grade point average, average
salary.
b) Look for trends
c) Estimate future values
d) Outliers used in curving test grades
DO NOT NEED STATISTICS
a) Geometric sequence problems
b) Finding angles of a triangle
c) Finding coordinates from a transformation
d) Finding slope of a line
3. When collecting data, students need to be exposed to the appropriateness of the data in relation to the
data collected. To aid students, the following questions need to be discussed: (S.ID.3, MP.1, MP.8, L.2)
a) In collecting data to find free throw averages in NBA, what needs to be collected? (Teacher note:
ask students what are some data that would not apply)
b) How do teachers determine the procedure used to curve test grades? (i.e. don’t like student, one
student scored significantly higher than everyone else in the class)
c) Are the data skewed? Present this problem to the class to discuss the appropriateness of the data:
How can we project students’ choice in the decision making process? We will need a good
random sample, so what would be the best way to do this? Should we survey French class,
survey faculty, or survey a random sample of students and faculty?
4. Note to teacher: You may need to review or re-teach mean, median, mode and outlier before working
on this problem. The following web-sites will aid in the review:
http://learni.st/users/S33572/boards/3202-measures-of-central-tendency-and-distribution-common-corestandard-9-12-s-id-2 rap on finding measures of central tendency
http://regentsprep.org/Regents/math/ALGEBRA/AD2/measure.htm examples on how outliers affect
measures of central tendency
Have students solve the following problem using a line (dot) plot: The number of Atlantic hurricanes that
occurred each year from 2001 – 2012 are listed below in the table. Find the measures of central
tendency: mean, median, and mode and explain their meaning in relation to the problem. Are there
any outliers, what affect does it have on the measures of central tendency? (S.ID.1, S.ID.3, MP.1, MP.2,
MP.4, MP.5, L.2, L.6)
Year
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
No. of
9
4
7
9
15
5
6
8
3
12
7
10
Hurricanes
5. Review with students dot plot, histogram, box plot, median, skew, frequency table, quartiles (lower
and upper), interquartile range, and outlier (S.ID.1, S.ID.2, S.ID.5, MP.2, MP.4, L-2, L-6)
http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/extra_examples/chapter13/lesson13_3.pdf
nice review of histograms
http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/extra_examples/chapter13/lesson13_4.pdf
nice review of range, quartiles, inner quartile range and outliers
http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/extra_examples/chapter13/lesson13_5.pdf
nice review of box and whisker plots
http://glencoe.mcgraw-hill.com/sites/dl/free/0078885159/632857/pahwp.pdf pages 185 through 188 for
box and whisker plot and histograms;
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TEACHER NOTES
http://www2.oakwood.k12.oh.us/~colson_kelly/Colson/DA1_CH1_files/DA1%20CH1%20More%20Practice%20Your
%20Skills.pdf pages 1-8 for dot plots and bar graphs
6. Use the time traveled to school (column M) in the census data to create a problem to use in determining
which type of plot to use (dot plot, histogram, or box plot). Note: be sure there is an outlier in the
data. Then use the class data to create a dot plot and from that a histogram, making sure to include the
following measurement classes (bins), frequency, frequency table. Refer to your textbook in the 2005
edition section 13-3 and in the 2012 edition page 41 and chapter 12 and the National Library of Virtual
Manipulative. Once the dot plot and histogram are completed, have students create a box and
whisker plot, include the terms mean, median, quartile (lower and upper), outliers (multiply 1.5 by the
interquartile range and add to Q3, then multiply 1.5 by the interquartile range and subtract from Q1. Any
number outside these two numbers is an outlier). When using the 2005 edition of the textbook, refer to
page 733 which contains the definition of an outlier. (S.ID.1, SD.3, MP.1, MP.2, MP.4, L.2, L.6)
7. Ask students “For this particular situation, what data representation do they feel is the best? Why?”
(S.ID.2, MP.2, MP.4, L.6)
Suggest: stop for a teacher generated assessment
8. Build on interquartile range from the box and whisker plot, then lead into the MAD (mean absolute
deviation)(S.ID.1, S.ID.2, MP.1, MP.2, MP.4, MP.8, L.2, L.6)
http://www.glencoe.com/sites/pdfs/impact_math/ls9_c1_mean_absolute_deviation.pdf discusses how to
find MAD and contains a number of examples and the Youtube video explains the procedure
http://www.youtube.com/watch?v=PwsXncM2pas ). After students understand MAD, proceed to standard
deviation using the YouTube video (http://www.youtube.com/watch?v=dq_D30kyR1A ) which
explains the concept of standard deviation in relation to a bell shape curve. Watch the video and then
discuss concepts with students.
http://www.youtube.com/watch?v=klP8ElkVxJk also shows the calculations for finding the standard
deviation in a different way. When using the 2012 edition of the textbook, refer to pages 758 and 759
and the PDF file version of Chapter 12
http://www.glencoe.com/sites/common_assets/mathematics/TN_2012/Alg1_se/Chapter_12_8952705.pdf
http://www.tutorcircle.com/interpreting-and-understanding-standard-deviation-worksheet-wst-5GR.html
contains ten questions on finding the MAD and standard deviation.
9. Using the example from step #6, compare the data from 2 different classes, including measures center
of central tendency (mean and median) and spread (interquartile range and standard deviation). In
looking at the dot plots and box and whisker plot, examine overlapping data and discuss inferences
that could be made. Informally examine the histograms and look for similarities and differences.
(S.ID.1, S.ID.2, MP.2, MP.4, L-2, L-6)
10. Show students the histograms and their standard deviation and how standard deviation affects the
shape of the histogram. (S.ID.3, MP.2, MP.4, MP.5, MP.6, MP.7, L.2, L.6) - - (To help show how
standard deviation affects the shape of the histogram, use the exploring data web site:
http://math.cowpi.com/geogebra/histogram_mean_median.html) and also a matching exercise matching
histograms with
summary statistic (http://exploringdata.net/histsum2.htm )
Administer School City Test on single variable statistics at this time
11. Teacher explains joint, marginal, and conditional relative frequencies; (see attached for full detail on
pages 9-11 of Unit Plan) (S.ID.5, MP.4, MP.5, MP.7, MP.8, L.2, L.6)
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TEACHER NOTES
https://commoncorealgebra1.wikispaces.hcpss.org/file/detail/S.ID.5+Lesson+Two+Way+Tables.docx
After completing tasks1 through11 on pages 9 – 11, to reinforce these concepts use the web
site, http://www.pbs.org/greatwar/resources/casdeath_pop.html, choose 2 countries and
choose two of the columns: killed, wounded and/or prisoners or missing in action, to create a 2
way frequency table. Have students answer the following questions similar to the ones in task
5, 7, 8, and 9.
12. Use the following examples to discuss line of best fit as a straight line that goes in the same
direction as the points and has the same number of points above and below the line; discuss the
purpose for line of best fit is to make predictions; same number of points above the line as below the
line. Address correlation compared to causation as you do these examples (ice cream cones; sea
level rise; yearly cash production per person versus birth rate)
http://www.mathsisfun.com/data/scatter-xy-plots.html (Question #2 on this website is good to
use as there is no correlation or cause. Also look for examples that are not linear and some that have
negative correlation.) Additional examples:
a. number of ice cream cones sold at $4.00 each and temperature
# of ice
cream cones
Temperature
in C
20
60
140
190
250
280
320
7
10
13
16
19
22
25
b. sea level rise and year
Year
Rise in
sea level
in mm.
1990
1994
1998
2002
2006
2010
3.2 mm
3.4 mm
2.9 mm
3.2 mm
3.0 mm
3.3 mm
c. yearly cash production per person versus birth rate
Year
1950
1960
1970
1980
1990
2000
2010
Birth
rate
24.1
23.7
18.4
15.9
16.7
14.7
13.8
d. Value of an Escalade over a 5 year period, applying depreciation.
Year
Value of
Car
2012
$65,000
2013
2014
2015
2016
2017
$51,750
$45,439
$39,478
$34,251
$29,656
Using the examples above:





Have discussion about which variable should be the dependent variable and which
variable should be the independent variable
Plot points on graph paper; and draw line of best fit
Compare two students’ lines and discuss which is a better line of best fit; what is a
better estimate(e.g., for 20°C)
Discuss contextual meaning of slope and y intercept
Ask questions about specific temperatures (e.g., 20°C, how many ice cream cones are
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YCS ALGEBRA I: Unit 3: Descriptive Statistics 2013-2014 5
TEACHING-LEARNING
sold? - - When revenue of sales is $1,000, - - What is the temperature?

Are the data correlated - - e.g., ice cream sales and temperature?

Show “r” value computed by calculator - - the closer to 1 or -1, the stronger the
correlation.

Examine additional scatter plots at the website listed above. (S.ID.6.a, c, S.ID.7,
S.ID.8, S.ID.9, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, L.2, L.6)
TEACHER NOTES
13. Next address correlation compared to causation for the data in the following problems; (S.ID.9, MP.2,
L.2, L.6)
a. number of house fires versus temperature in the United States (-)
# of house
fires
Temperatur
e F0
40,000
42,000
44,000
47,000
51,000
56,000
62,000
850 F
650 F
450 F
250 F
50 F
00 F
-150 F
b. education versus yearly income (+)
Yearly income
$87,334
$67,600
$55,432
$40,820
$33,904
$24,492
Education beyond
high school
8 years
6 years
4 years
2 years
0 years
-2 years
c. number of pairs of underwear versus math scores (no correlation)
# of pairs of
underwear
SAT math
scores
10
25
12
6
15
4
11
740
680
640
600
540
460
420
d. battery life of cell phone and usage of cell phone for an Apple I phone(causation)
Battery life
100%
80%
65%
50%
38%
25%
10%
Usage in hours
8.48
6.68
5.06
3.45
2.05
1.27
0.5
Have students work in small groups and discuss: no correlation, positive correlation, negative
correlation, causation. Show that rituals are not causation (e.g., doing certain things for your sports
team - - wearing a certain color; doing a certain thing, wearing jewelry, etc.) (Extra examples are
attached on page 12 of the unit)
14. Give students more practice on graphing data and drawing line of best fit and correlation. (S.ID.6, a, b,
c, S.ID.7, S.ID.8, MP.1, MP.2, MP.4, MP.5, L.2, L.6) The following activities from NCTM Illuminations
are excellent:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=146 has a graphing tool where points can be plotted
and a line of best fit drawn with correlation coefficient, it is excellent.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L663 is an activity on investigating fire environments.
It is comprised of 5 lessons which can be done independently. Lesson 2 deals with line of best fit and
lesson 4 deals with slope.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L298 is an activity on exploring linear data and deals
with oil changes vs. cost of repair (negative correlation), bike weight vs. jump height (negative correlation)
and dosage of a medication vs. weight.
15. Using the Census At School website, each student chooses 6 data points using left foot length and left
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TEACHING-LEARNING
TEACHER NOTES
index finger length. Students plot points, draw line of best fit, find the equation of the line in y intercept
form. Students explain the contextual meaning of slope and y intercept, discuss appropriateness of the
answer, and find the correlation coefficient. As students explain, ask students if what they are saying
makes sense and the reasonableness of their answers. (S.ID.6a, c, S.ID.7, S.ID.8, MP.1, MP.2, MP.4,
MP.5, L.2, L.6)
16. Teacher asks students if they know or have ideas about what the word - - residual might mean. Relate
the word to something they know (e.g., resident and where you live and the bus stop and how far you
have to go to get to your residence. Below is the definition and an example of bus stops along a street.
(S.ID.6b, MP.2, MP.4, MP.5, L.2, L.6)
17. Teacher explains residual as the vertical distance between a data point and the graph of a regression
equation. The residual is positive if the data point is above the graph. The residual is negative if the data
point is below the graph. The residual is 0 only when the graph passes through the data point.
Use the following example to explain the term residual:
Below is a line representing a street that a school bus travels to pick up students. The dots represent the
students’ houses. The bus stops at the end of every student’s driveway which is the vertical length from the
house to the street. The residual is the vertical distance from the residence to the street. Using the
graph, have students approximate the residuals and determine if this is an acceptable method of picking up
students.
Upon completion of this exercise, refer to the examples in T-L #12 and approximate residuals and ask
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TEACHING-LEARNING
TEACHER NOTES
students if this is an appropriate line of best fit based on the residuals. (The closer the residual is to zero the
better the fit)
Administer School City Test on 2-variable statistics at this time.
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Unit Test with Multiple Choice Questions
TEACHER CLASSROOM ASSESSMENT
TEACHER NOTES
1. 2- and 4-point questions
2. Other in class assessments
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Students are to state a problem or pose a question that can be solved by using 2 variable statistics, and then
develop a plan to answer the question or solve the problem. They will find and analyze the data and predict
a future event. The following steps will be included.
1. State the problem or question.
2. Look up data on the internet and state the web site
3. Present the data in a table
4. Graph the data and analyze by finding the line of best fit and stating its equation.
5. Explain the appropriateness of the line of best fit using the correlation coefficient as calculated by the
calculator and residuals using an approximation from the graph.
6. Interpret the data based on the context of the original question.
7. Predict a future event using your graph
(S.ID.1, S.ID.2, S.ID.3, MP.1, MP.2, MP.4, L.2, L.6, L.7)
2. Students evaluate goals for the unit.
AUTHENTIC ASSESSMENT RUBRIC
PROJECT CRITERIA
0
2
State problem
Did not attempt
Stated problem
Denote web site
Did not attempt
Identified Web site
Table of data
Did not attempt
Created table of data
Graph of data
Did not attempt
Plotted points to graph the data
Graph of line of best fit
Did not attempt
Showed line of best fit on graph
Equation of line of best fit
Did not attempt
Stated equation of line of best fit
State correlation coefficient
Did not attempt
Stated correlation coefficient
Explanation of appropriateness of Did not attempt
Explained the line using
line using correlation coefficient
correlation coefficient
Explanation of appropriateness of Did not attempt
Explained line using residuals
line using residuals
Explanation of interpretation of
Did not attempt
Interpreted the data in context of
data in context of the problem
the identified problem
Prediction of a future event
Did not attempt
Made logical prediction of future
event
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YCS ALGEBRA I: Unit 3: Descriptive Statistics 2013-2014 8
T-L #11: (S.ID. 5) Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in
the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and
trends in the data.
Lesson Tasks, Problems, and Activities
1) Students write down their answer to the question about what extracurricular activities they do.
2) Collect and organize their answers to the question about the activities they do. Allow students to suggest methods of
organizing the data. Then create a blank table similar to the one below, filling the total number of students in the class. Tell
them that each student can only have one favorite then go around to each student and fill in the chart. Use tick marks as you
collect the data and then write a final number in each cell. Use this data to refer to as you progress through the lesson.
Favorite Extra-Curricular Activities for (your number) Students
Athletics
Music/Arts
SGA
Other
Male
Female
3) Talk about how this table of data is called a two-way frequency table. Each of the cells (values) in the table are called
counts and they measure the frequency of the choice selected. These are also called the joint frequencies. Ask the students
how we could graphically represent this data. Acceptable answers are bar chart or pie graph. Talk about how this data is
called categorical (since it can only be measured in counts) and cannot be represented by something like a histogram, which
represents quantitative data only.
4) Tell the students that they can do some analysis on this data but we need to find a few totals first. Add a column and a row to
the table labeled “Total.” Have the students calculate the row, column and table totals. The totals are called the marginal
frequencies. Make sure that students recognize that the table total should sum to the number of students in the class.
Male
Female
Total
Athletics
Music/Arts
SGA
Other
Athletics total
Music/arts total
SGA total
Other total
Total
Male total
Female total
Table total
5) Although the joint and marginal frequencies are good to know, more likely we want to know how it relates to the rest of the
data. The first thing we can find is something called relative marginal frequencies. Relative marginal frequencies are
calculated by taking the row or column totals and dividing it by the table total. Do an example or two from your class data.
Note that either fractions or percents are acceptable answers here. Here is a sample chart with solutions:
Sample completed chart:
Favorite Extra-Curricular Activities for 30 Students
Athletics
Music/Arts
SGA
Other
Total
Male
10
5
1
2
18
Female
5
6
0
1
12
Total
15
11
1
3
30
What is the relative marginal frequency of students who favor athletics? 15/30 or 50%
What is the relative marginal frequency of students who favor SGA? 1/30 or ≈ 3.3%
What is the relative marginal frequency of males? 18/30 or 3/5 or 60%
6) Mention to the students that sometimes the question that asks for relative marginal frequencies may be asked differently, such
as “What is the distribution of favorite extra-curricular activities?” In this case they want you to list each extra-curricular activity
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YCS ALGEBRA I: Unit 3: Descriptive Statistics 2013-2014 9
with its relative marginal frequency. For our data that answer would look like: Athletics: 15/30 or ½ or 50%; Music/Arts: 11/30
≈ 36.7%; SGA: 1/30 or ≈ 3.3%; Other: 3/30 or 1/10 or 10%.
7) Another important type of frequencies compares the data in respect to one specific condition and is called relative conditional
frequencies. These are calculated by dividing a cell by its respective row or column total. For example, the relative
conditional frequency of SGA for males is 1/18 or ≈ 5.6% because there is 1 male out of the 18 males total that prefers SGA.
Note that this is different than the relative conditional frequency of males for SGA which is 1/1 or 100% because all students
that prefer SGA are males. Have the students answer the following questions using their class data. Sample answer from the
sample data given are listed as well.
What percent of males prefer athletics? 10/18 ≈ 55.6%
What percent of students who prefer music/arts are females? 6/11≈54.5%
What is the relative conditional frequency of other for males? 2/18 or 1/9 ≈ 11.1%
8) Let’s say that we wanted to compare our class’ favorite extra-curricular activities to other classes. Suppose a survey of a
freshman class at a local high school was performed and the results are listed below. Since we have different sample sizes
we need a way to compare the data without just the counts. We can calculate the relative joint frequencies to do this. Walk
through the calculations from the sample of 200 with the students and then have the students do the same for their class data.
Male
Female
Total
Favorite Extra-Curricular Activities for 200 Students
Athletics
Music/Arts
SGA
Other
28
31
12
19
34
40
14
22
62
71
26
41
Total
90
110
200
Relative Joint Frequencies of the Sample
Athletics
Music/Arts
SGA
Other
Total
Male
28/200=14%
31/200=15.5%
12/200=6%
19/200=9.5%
90/200=45%
Female
34/200=17%
40/200=20%
14/200=7%
22/200=11%
110/200=55%
Total
62/200=31%
71/200=35.5%
26/200=13% 41/200=20.5% 200/200=100%
Note that the bold values are relative marginal frequencies and that the values in the table are the relative joint frequencies.
This means, for example, that 14% of the students are male and prefer athletics.
From our sample class data here are the solutions:
Male
Female
Total
Favorite Extra-Curricular Activities for 30 Students
Athletics
Music/Arts
SGA
Other
10/30≈33.3%
5/30≈16.7%
1/30≈3.3%
2/30≈6.6%
5/30≈16.7%
6/30=20%
0/30=0%
1/30≈3.3%
15/30=50%
11/30≈36.7%
1/30≈3.3%
3/30=10%
Total
18 /30=60%
12/30=40%
30/30=100%
9) Discuss with the class the similarities and differences of their class data with the sample from the other school. Discuss any
advantages and disadvantages of comparing this data. Also ask them if they notice any associations or trends.
Based on the sample data, some possible answers are:
There is a similar percentage of students who prefer music/arts: us – 35.5%; them – 36.7%
Our class prefers athletics about 19% more than the other school: us – 50%; them – 31%
Our class has a larger percentage of females than males as opposed to the other school: us – 45% male and 55% female;
them- 60% male and 40% female.
Our class only represents a small portion of the freshman of the school and may not be representative of the rest of the
freshman class so these results may not be accurate.
We can use these data to see if the funding for these activities match the popularity of it in our school.
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In our class it appears that males who prefer athletics (33.3%) is more than males who prefer the other choices combined
(16.7%+3.3%+6.6% = 26.6%). Females in our class appear to highly prefer athletics and music/arts.
10) Exit slip – have students complete the lesson closure questions.
Evidence of Success:
After this lesson, students should be able to calculate various types of relative frequencies. They also should be able to make
comparisons about the data using these values. Successful completion of the practice resource sheet as well as a possible quiz
would give a good indication of a deepened understanding.
New vocabulary – two way frequency table, cells, marginal frequency, conditional frequency, joint frequency relative marginal
frequency, relative conditional frequency, relative joint frequency, categorical data, quantitative data
Previous vocabulary – counts, frequency, bar chart, pie chart, histogram
Students often confuse the three types of relative frequencies.
Relative conditional frequencies can sometimes be confusing for students to know which cell to divide by which total. Careful
attention of the wording of the question is important to decipher which is requested. Be sure to take the time to ensure that every
student understands the examples presented before moving on.
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T-L # 13
OTHER EXAMPLES FOR CORRELATION AND CAUSATION
1. Students who eat breakfast are more successful in school.
2. People who smoke cigarettes get lung cancer.
3. The manager of a toy store in the mall hires a new worker in December. The worker is a college
student named Stacy. After Stacy is hired, the store’s sales shoot up by 300%. The store
manager says that Stacy is a fantastic sales worker! I haven’t hired anyone else but Stacy. Since
we hired her, our sales have tripled! I’d better give her a raise! Is the manager’s conclusion
logical? Is it true that Stacy must be fantastic at her job?
4. Lucy ate the white berry, and then became sick. First I hit Bob’s foot with a hammer, then his foot
swelled with a purple bruise. I conclude that eating the white berry is what actually make Lucy sick
later. I conclude that being hit with a hammer is what later caused Bob’s foot to swell
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