Author: Walter Vredeveld
Learning Task: How Tall are Our Students?
In this task, students will gather sample data, calculate statistical parameters and draw inferences
about populations. Students will learn to take samples, understand different types of sampling
(random, convenience, self-selected, and systematic) and sampling bias, extrapolate from a sample to a
population and use statistical methodologies to make predictions regarding the full population.
STANDARDS ADDRESSED IN THIS TASK:
Advanced Algebra CCGPS Standards
MCC9-12.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 data
sets.
MCC9-12.S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there are data sets for
which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate
areas under the normal curve.
MCC9-12.S.IC.1 Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
Math III GPS Standards
MM3D1. Students will create probability histograms of discrete random variables, using both
experimental and theoretical probabilities.
MM3D2. Students will solve problems involving probabilities by interpreting a normal
distribution as a probability histogram for a continuous random variable.
a. Determine intervals about the mean that include a given percent of data.
c. Estimate how many items in a population fall within a specified interval.
Advanced Mathematical Decision Making (AMDM) GPS Standards
DATA ANAYLSIS AND PROBABILITY - Students will explore representations of data and
models of data as tools in the decision making process.
MAMDMD2. Students will build the skills and vocabulary necessary to analyze and critique
reported statistical information, summaries, and graphical displays.
MAMDMD3. Students will apply statistical methods to design, conduct, and analyze statistical
studies.
Mathematical Process Standards GPS
MM#P1.Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
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d. Monitor and reflect on the process of mathematical problem solving.
MM#P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
STANDARDS FOR MATHEMATICAL PRACTICE
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
In order for students to be successful, the following skills and concepts need to be maintained:




Students will need to be familiar with histograms, normal distributions and with statistical
measures of central tendency (mean) and dispersal (standard deviation)
Students must understand the use of z-scores to make predictions about a population from
smaller samples.
Students must understand sampling techniques and sampling types, including random,
convenience, self-selected, and systematic and sampling bias
Students must be able to read and create tabular data to be analyzed.
ESSENTIAL QUESTIONS



How do I conduct a sampling operation to gather data about a population?
How do I develop statistical parameters of the data that was gathered?
How do I make predictions about the total population, based on the sample?
MATERIALS:

Scientific calculator.

Tape Measure or other measurement device (“height pipes” can be created from thin
1
wall 12 inch PVC pipe).
GROUPING
Teams of two to four students.
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TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
I.
Purpose
To give students the experience of gathering sample data, calculating statistical
parameters and drawing inferences about populations.
II.
Data Sampling (Height Measurement)
a. Each team will utilize a measuring tape or “height pipe” and the “Height Sampling
Data” worksheet. They will measure other students, recording Grade, Gender and
Height (to the nearest inch), and record their observations on the worksheet. (This
works best when a wide range of students are available, e.g. during lunch)
b. Teams should NOT duplicate students measured, i.e. ask the subject if they have
already been measured by another team. If they answer yes, then find another student
to measure.
III.
Develop Statistical Parameters of the Sample
a. Draw a Histogram Chart for your sample data.
Height Sample Histogram
16
14
Number of Students
12
10
8
6
4
2
0
45
50
55
60
65
70
Height in Inches
75
80
85
b. Calculate the mean, standard deviation and % of the total population (total number of
students in the population*) for your sample. (Use the Sample Standard Deviation S on
your calculator).
Sample Size (n)
% Total Population*
* Need to obtain total population numbers for comparison.
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Mean ( x )
Standard Deviation (S)
c. Draw a Normal Distribution Chart for your sample data.
Normal Distribution - Student Height
25
# of Students
20
15
10
5
0
21
22
-3ϭ
IV.
-2ϭ
23
x
-1ϭ
Height - Inches
24
+1ϭ
25
+2ϭ
+3ϭ
In-depth Sample Analysis
a. “Disaggregate” your data into gender and grade and compare your sample size to the
total school population* breakdown.
Grade
Female
Sample
Mean
Total
School*
Male
%
Sample
9th
10th
11th
12th
TOTAL
* Need to obtain total population numbers for comparison.
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Mean
Total
School*
%
Total
School*
b. Draw a Histogram of your Means by Gender and Grade.
Height
74
72
70
68
66
64
62
60
58
V.
9th
9th
10th
10th
11th
11th
12th
12th
Female
Male
Female
Male
Female
Male
Female
Male
Draw Inferences about the Entire Population
Using your normal distribution and the z-score table
determine what percent of students are predicted to be:
Less than 5 feet
VI.
Between 5 feet and 6 feet
𝑥 −𝑥
𝑧=
𝑆
Greater than 6 feet
Representative Sample?
a. How would you describe your sampling methods?
__ Self-Selected
__ Convenience
__ Systematic
__ Random
Describe your method:
b. Do you think that your sample is representative of the total population? Why?
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Height Sampling Data Worksheet
# Grade Gender
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
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Height (inches)
# Grade Gender
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Height (inches)
Z-Score Table
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Answers will vary depending on the sample
data. This answer key is representative of how
the response should be structured.
Data Sample - Answer Key
(For Sample below)
II.
Data Sampling (Height Measurement)
8|Page
III.
Develop Statistical Parameters of the Sample
a. Draw a Histogram Chart for your sample data.
Height Sample Histogram
8
Number of Students
7
6
5
4
3
2
1
0
60
61
62
63
64
65
66
68
69
71
72
73
74
75
Height in Inches
b. Calculate the mean, standard deviation and % of the total population (total number of
students in the population*) for your sample. (Use the Sample Standard Deviation S on
your calculator).
Sample Size (n)
% Total Population*
Mean ( x )
Standard Deviation (S)
47
3.4%
69.0
4.0
* Need to obtain total population numbers for comparison.
Total School population of 1,374
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c. Draw a Normal Distribution Chart for your sample data.
Normal Distribution - Student Height
90
80
# of Students
70
60
50
40
30
20
10
0
IV.
1
57
-3ϭ
2
61
-2ϭ
3
65
-1ϭ
4
69
x
Height - Inches
5
73
+1ϭ
77
+2ϭ
6
81
+3ϭ
7
In-depth Sample Analysis
a. “Disaggregate” your data into gender and grade and compare your sample size to the
total school population* breakdown.
Grade
9th
10th
11th
12th
TOTAL
Female
Sample
Mean
5
3
5
2
15
65.0
65.7
63.8
65.0
64.7
Total
School*
189
151
141
160
641
Male
%
Sample
Mean
2.6%
2.0%
3.5%
1.3%
3
3
17
9
32
70.0
73.7
69.9
69.8
70.3
2.3%
* Need to obtain total population numbers for comparison.
Total School population of 1,374
10 | P a g e
Total
School*
230
184
174
145
733
%
1.3%
1.6%
9.8%
6.2%
4.4%
Total
School*
419
335
315
305
1374
b. Draw a Histogram of your Means by Gender and Grade.
Height by Gender/Grade
76
74
72
70
68
66
64
62
60
9th
65
70
Female
Male
10th
65.7
73.7
Female
V.
11th
63.8
69.9
12th
65
69.8
Male
Draw Inferences about the Entire Population
Using your normal distribution and the z-score table determine
what percent of students are predicted to be:
𝑧=
60 −69
4
= 2.25
100% minus % less than 5
feet + % greater than 6 feet
𝑧=
𝑧=
𝑥 −𝑥
𝑆
72 − 69
= 0.75
4
Less than 5 feet (60 inches)
Between 5 feet and 6 feet
Greater than 6 feet (72 inches)
1.2%
76.1%
22.7%
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VI.
Representative Sample?
b. How would you describe your sampling methods?
__ Self-Selected
__ Convenience
__ Systematic
__ Random
Describe your method:
The samples were obtained by measuring student heights during 1st lunch (one lunch out
of three). Students were asked if we could measure their height and we collected the data.
c. Do you think that your sample is representative of the total population? Why?
NO.
Students were only measured during one lunch. Any possible sample data from the other two
lunches was missed.
Looking at the data, juniors and seniors are over-represented and freshmen and sophomores are
under-represented. Also more male than female measurements were taken.



The students taking the measurements were juniors and seniors. They were more
easily able to approach fellow juniors and seniors to be sampled.
Males may have been more willing to volunteer than females.
The lunch period sampled MAY have contained more males and/or more upper
classmen.
This was definitely a CONVENIENCE sample.
12 | P a g e
Answers will vary depending on the sample
data. This answer key is representative of how
the response should be structured.
Aggregate Data - Answer Key
(For ALL Sample Data Collected)
III.
Develop Statistical Parameters of the Sample
a. Draw a Histogram Chart for your sample data.
Aggregate of All Samples
35
# of Students
30
25
20
15
10
5
0
52 55 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79
Height (inches)
b. Calculate the mean, standard deviation and % of the total population (total number of
students in the population*) for your sample.
Team #
Sample Size
(n)
% Total
Population
Mean ( x )
Std Deviation
S
1
40
2.9
66.0
3.8
2
3
40
31
2.9
2.2
67.9
65.0
3.6
3.0
4
5
6
7
8
38
33
40
29
47
2.8
2.4
2.9
2.0
3.4
65.4
65.8
68.4
67.4
69.0
4.1
4.0
5.4
4.2
4.0
TOTAL
297
21.6
67.0
4.3
* Need to obtain total population numbers for comparison.
Total School population of 1,374
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c. Draw a Normal Distribution Chart for your sample data.
Aggregate Normal Distribution - Student Height
90
80
# of Students
70
60
50
40
30
20
10
0
1
54.1
2
-3ϭ
IV.
58.4
-2ϭ
3
62.7
4
67.0
5
71.3
75.6
-1ϭ
x
+1ϭ
+2ϭ
Height - Inches
6
79.9
7
+3ϭ
In-depth Sample Analysis
a. “Disaggregate” your data into gender and grade and compare your sample size to the
total school population* breakdown.
Grade
9th
10th
11th
12th
TOTAL
Female
Sample
Mean
19
23
49
36
127
63.2
62.9
64.3
63.6
63.7
Total
School*
189
151
141
160
641
Male
%
Sample
Mean
10.0%
15.2%
34.8%
22.5%
22
25
59
64
169
69.0
68.8
69.4
69.9
69.4
19.8%
* Need to obtain total population numbers for comparison.
Total School population of 1,374
14 | P a g e
Total
School*
230
184
174
145
733
%
9.6%
13.6%
33.9%
44.1%
23.1%
Total
School*
419
335
315
305
1374
b. Draw a Histogram of your Means by Gender and Grade.
Aggregate Height by Gender Grade
72
70
68
66
64
62
60
58
9th
63.2
69.0
Female
Male
10th
62.9
68.8
Female
V.
11th
64.3
69.4
12th
63.6
69.9
Male
Draw Inferences about the Entire Population
Using your normal distribution and the z-score table determine
what percent of students are predicted to be:
𝑧=
60 −67
4.3
= 1.63
100% minus % less than 5
feet + % greater than 6 feet
𝑥 −𝑥
𝑧=
𝑆
𝑧=
72 − 67
= 1.16
4.3
Less than 5 feet (60 inches)
Between 5 feet and 6 feet
Greater than 6 feet (72 inches)
5.3%
82.5%
12.2%
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VI.
Representative Sample?
c. How would you describe your sampling methods?
__ Self-Selected
__ Convenience
__ Systematic
__ Random
Describe your method:
The samples were obtained by measuring student heights during 1st lunch (one lunch out
of three). Students were asked if we could measure their height and we collected the data.
d. Do you think that your sample is representative of the total population? Why?
NO.
Students were only measured during one lunch. Any possible sample data from the other two
lunches was missed.
Looking at the data, juniors and seniors are over-represented and freshmen and sophomores are
under-represented. Also more male than female measurements were taken.



The students taking the measurements were juniors and seniors. They were more
easily able to approach fellow juniors and seniors to be sampled.
Males may have been more willing to volunteer than females.
The lunch period sampled MAY have contained more males and/or more upper
classmen.
This was definitely a CONVENIENCE sample.
16 | P a g e