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Welcome to MM150!
Kirsten Meymaris
Thursday, Mar. 31st
Plan for the hour
 Final Project Reminder – due Tuesday, April 5th
 What is “average”?
(Measures of Central Tendency, 9.1)
 How far apart are the data?
(Measures of Dispersion, 9.2)
 Completing this course
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Final Project
You will need to create 5 slides or pages:
On slide or page 1: provide your name, the project title, and the
course and section number.
On slides or pages 2, 3 and 4: These three slides (or pages) are the
main part of your project. Introduce your chosen profession and
give a brief overview of the math concept (or concepts) you will
apply to the profession. Then go into more detail, and describe how
the concept can apply to your chosen profession. You do not have
to “do the math.” You can simply describe how you would use it.
Make sure you offer an example or examples of situations in which
you would use the concept you have chosen. Provide detail –
actively discuss your example.
On slide or page 5, provide any resources you have used to give credit
to others’ ideas and information. Every student must have a
reference page. If the textbook in your only “source,” that’s fine!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Final Project Reminders
• Choose either Microsoft Word or Microsoft PowerPoint in which to
create your final project. The final project must be submitted via
the Unit 9 drop box - Due April 5th!!
• Students MUST include a reference page! If your examples are
based on your actual real-life day at the job (i.e., you did no
“research”), then list the textbook as your reference. Every
student can list the textbook as a reference.
• Check spelling and grammar and visit the Writing Center if needed.
• Submit your (ungraded) final project into the Unit 10 Math Fair.
Spend some time there, reading your classmates’ projects. I always
learn something new from my students’ projects.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
9.1
Measures of Central Tendency
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Measures of Central Tendency
Measure
Mean
What
Median
middle value of
set of ranked data
Sum of values
divided by number
in set
How
x
x
n
Rank data,
Select middle value
(* in some cases, you’ll calculate
mean of two middle values)
Mode
Midrange
value that occurs
most frequently
in set
value halfway
between lowest (L)
and highest (H)
values in set
Rank data, Select value(s)
that occurs the most
(in some cases, there isn’t one!)
LH
2
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Questions to ask
• When are different measures of central
tendency used?
• How different can these measures be? Can
they ever be the same?
• Does every set always have each measure?
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
What is the average age of
students here at Kaplan?
Last week,
we surveyed
(took a sample)
of students in
M150-20AU (30
students) to
estimate the whole
Kaplan student
body.
22
28
33
55
20
38
27
41
42
34
25
29
33
69
20
36
24
45
39
47
26
30
37
19
50
32
44
21
79
48
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Mean
• The mean, is the sum of the data divided by
the number of pieces of data. The formula for
calculating the mean is
x
x
n
• where x represents the sum of all the data
and n represents the number of pieces of
data.
Note: The arithmetic mean, or simply the mean is symbolized by “x bar” when it is a sample of a
population or by the Greek letter mu, , when it is the entire population.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Mean
What is the mean age of
students in MM150?
x
x
n
22
28
33
55
20
38
27
41
42
34
25
29
33
69
20
36
24
45
39
47
26
30
37
19
50
32
44
21
79
48
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Solution
all ages

X
30 students
(22  25  26  28  29  30
 33  33  37  55  69  19
 20  20  50  38  36  32
 27  24  44  41  45  21
 42  39  79  34  47  48)
X
30
X  36.43
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Median
• The median is the value in the middle of a set
of ranked data.
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Median
• The median is the value in the middle of a set
of ranked data.
19
29
41
Middle value
(median)
In this case
needs to be
averaged
20
20
21
22
24
25
26
27
28
30
32
33
33
34
36
37
38
39
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Median
• The median is the value in the middle of a set
of ranked data.
19
29
41
33  34
Median 
2
Median  33.5
20
20
21
22
24
25
26
27
28
30
32
33
33
34
36
37
38
39
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Mode
• The mode is the piece of data
that occurs most frequently.
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Mode
• The mode is the piece of data
that occurs most frequently.
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Mode = 20 and 33 (bimodal!)
Note: A set does not necessarily have a mode. Also, the set
could have more than one mode!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Midrange
• The midrange is the value
halfway between the lowest
(L) and highest (H) values in
a set of data.
lowest value + highest value
Midrange 
2
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example - Midrange
• The midrange is the value
halfway between the lowest
(L) and highest (H) values in
a set of data.
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
19  79
Midrange 
2
Midrange  49
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example – Review of Measures
• What is the average age of
students in MM150?
Mean = 36.43
Median = 33.5
Mode = 20 and 33 (bimodal!)
Midrange = 49
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Measures of Position
• Measures of position are often used to make
comparisons.
• Two measures of position are percentiles and
quartiles.
• The nth percentile means that you outscored
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Percentiles
• The nth percentile means that you are older
than n% of the students and younger than
(100-nth)% of the students.
• For example, age = 27
• You are in the 45th percentile.
• “45% of the people are younger than you and
65% of the students are older than you!”
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
To Find the Quartiles
of a Set of Data
1.
2.
3.
4.
Order the data from smallest to largest.
Find median of the set, Q2
Find the median of the first (lower) half, Q1
Find the median of the last (upper) half, Q3
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Quartiles
• The ages of Kaplan students in one section of
MM150. Determine Q1, Q2, and Q3.
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Quartiles
Q1 = 26
Q2 = 33.5
Q3 = 44
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
33.5
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Using Excel to Calculate Stats
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
9.2
Measures of Dispersion
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Measures of Dispersion
• Measures of dispersion are used to indicate
the spread of the data.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Measures of Dispersion
Measure
Range
What
How
Total spread of
data in set
Rank the data,
Highest(H) – Lowest(L)
Standard
Deviation
How different
from the mean

s
 xx

2
n 1
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Range
• What is the range of ages of
students at Kaplan?
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Range
• What is the range of ages of
students at Kaplan?
Range = 79 - 19 = 60
19
20
20
21
22
24
25
26
27
28
29
30
32
33
33
34
36
37
38
39
41
42
44
45
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Standard Deviation
• The standard deviation measures how much
the data differ from the mean. It is symbolized
with s when it is calculated for a sample, and
with  (Greek letter sigma) when it is
calculated for a population.
s

 xx

2
n 1
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
What does Standard Deviation
mean?
Consider two data sets with mean = 9.5
{5, 8, 9, 10, 12, 13}
{8, 9, 9,10, 10, 11}
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
What does Standard Deviation
mean?
• The standard deviation measures how much
the data differ from the mean.
• Sometimes a small standard deviation is good
For example – A production line wanting all
products of roughly the same size.
• Sometimes a large standard deviation is good
For example – Ages of students at Kaplan – a
wide spread of ages can be a good thing!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
To Find the Standard
Deviation of a Set
1. Find the mean
2. Setup table
Data
Data-Mean
(Data-Mean)2
3. Sum last column (Data-Mean)2
4. Divide by (n-1) ( n=number of data in set)
5. Take the square root – Voila!
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example
• Find the standard deviation of the following
prices of selected washing machines:
$280, $217, $665, $684, $939, $299
Find the mean.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example
• Find the standard deviation of the following
prices of selected washing machines:
$280, $217, $665, $684, $939, $299
Find the mean.
280  217  665  684  939  299 3084
x

 514
6
6
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example continued,
mean = 514
Data
217
280
299
665
684
939
Data  Mean
(Data  Mean)2
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example continued,
mean = 514
Data
217
280
299
665
684
939
Data  Mean
297
234
215
151
170
425
0
(Data  Mean)2
(297)2 = 88,209
54,756
46,225
22,801
28,900
180,625
421,516
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example continued,
mean = 514
4. Divide by (n-1) , n=6
5. Take square root
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example continued,
mean = 514
s

 xx
n 1

2
421,516

 84303.2  290.35
5
• The standard deviation is $290.35.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example – Standard Deviation
• Find the standard deviation of ages of Kaplan
students in MM150.
19
29
41
20
30
42
20
32
44
Mean = 36.43
21
33
45
22
24
25
26
27
28
33
34
36
37
38
39
47
48
50
55
69
79
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Data
Data-Mean
(Data-Mean)2
19
-17.43
999.058886
20
-16.43
269.9449
20
-16.43
269.9449
21
-15.43
238.0849
22
-14.43
208.2249
24
-12.43
154.5049
25
-11.43
130.6449
26
-10.43
108.7849
27
-9.43
88.9249
28
-8.43
71.0649
29
-7.43
55.2049
30
-6.43
41.3449
32
-4.43
19.6249
33
-3.43
11.7649
33
-3.43
11.7649
34
-2.43
5.9049
36
-0.43
0.1849
37
0.57
0.3249
38
1.57
2.4649
39
2.57
6.6049
41
4.57
20.8849
42
5.57
31.0249
44
7.57
57.3049
45
8.57
73.4449
47
10.57
111.7249
48
11.57
133.8649
50
13.57
184.1449
55
18.57
344.8449
69
32.57
1060.8049
79
42.57
1812.2049
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example – Standard Deviation
• Find the standard deviation of ages of Kaplan
students in MM150.
19
29
41
20
30
42
20
32
44
1. Mean = 36.43
21
33
45
22
33
47
2. Make table
24
34
48
25
36
50
26
37
55
3. Sum (Data-Mean)2
27
38
69
28
39
79
= 6524.62
4. Divide by (n-1) = 29 students
= 6524.62/29
5.
=
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example – Standard Deviation
• Find the standard deviation of ages of Kaplan
students in MM150.
19
29
41
20
30
42
20
32
44
1. Mean = 36.43
21
33
45
22
33
47
2. Make table
24
34
48
25
36
50
26
37
55
3. Sum (Data-Mean)2
27
38
69
28
39
79
= 6524.62
4. Divide by (n-1) = 29 students
= 6524.62/29 = 224.99
5.
224.99 = 14.99 ~ 15
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Thank You!
Ask, Ask, Ask any
questions you have.
I hope to “see” you again in
another math class!
kmeymaris@kaplan.edu
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.