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Alliance Stat Class
Understanding Measures of Center and
Spread
What is the median?
How do you define and how do you find the value?
What happens if you have an odd number of data
values?
What happens if you have an even number of data
values?
Example: find the median for the following sets of
values:
12 14 15 15 16 18 21 25 27 30 30 30
30 30 26 12 15 9 27 9 13 13 13 18
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Practice finding the Median
Using the back to back stemplot of the
battery data from your assignment
Find the median for each battery type.
What is the mean?
Can you define or tell
me what it means
without saying the word
average?
How do you determine the mean?
Versus
How do you interpret the mean?
Level A Activity
Length of First Name
A Conceptual Activity for:
•
Developing an Understanding of the Mean as
the “Fair Share” value
•
Developing a Measure of Variation from “Fair
Share”
A Statistical Question
How long are the first names of
students in class?
• Nine students were asked what was the
length of their first name.
• Each student represented her/his name
length with a collection snap cubes.
Snap Cube Representation for Nine Name Lengths
How might we examine the data on
the name length for these nine
children?
2
3
3
4
4
5
6
7
Ordered Snap Cube & Numerical Representations
of Nine Name Lengths
9
Notice that the name lengths vary.
What if we used all our name
lengths and tried to make all names
the same length, in which case
there is no variability.
How many people would be in each
name length?
How can we go about
creating these new groups?
We might start by separating
all the names into one large
group.
All 43 letters in the students’ names
Create Nine “New” Groups
Create Nine “New” Groups
Original Question:
What if we used all our name lengths
and tried to make all names the same
length, in which case there is no
variability.
How many people would be in each
name length?
This is the value of the mean number of
letters in the 9 students names.
What is the formula to find the mean?
∑x
= ____
n
How does this formula compare with how we
found the mean through the idea of fair
share?
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A New Problem
What if the fair share value or the
mean number of letters for nine
children is 6?
What are some different snap cube
representations that might produce a
fair share value of 6?
How many total number of cubes do you need?
Snap Cube Representation of Nine Families,
Each of Size 6
Two Examples with Fair
Share Value of 6.
Which group is “closer”
to being “fair?”
How might we quantity “how
close” a group of name lengths
is to being fair?
Steps to Fair
•
One step occurs when a snap cube is
removed from a stack higher than the
fair
share value and placed on a stack
lower
than the fair share value .
•
A measure of the degree of fairness in a
snap cube distribution is the “number of
steps” required to make it fair.
Note -- Fewer steps indicates closer to fair
Number of Steps to
Make Fair: 8
Number of Steps to
Make Fair: 9
Students completing Level A understand:
•
the notion of “fair share” for a set of
numeric data
•
the fair share value is also called the
mean value
•
the algorithm for finding the mean
•
the notion of “number of steps” to make fair as a
measure of variability about the mean
•
the fair share/mean value provides a basis for
comparison between two groups of numerical data
with different sizes (thus can’t use total)
Level B Activity
The Name Length Problem
•
How long are the first names of
students in class?
A Conceptual Activity for:
•
Developing an Understanding of the Mean as
the “Balance Point” of a Distribution
•
Developing Measures of Variation about the
Mean
Level B Activity
How long is your first name?
Nine children were asked this question.
The following dot plot is one possible
result for the nine children:
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-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
In which group do the data
(name length) vary (differ) more
from the mean value of 6?
1
4
2
1
2
0
1
2
3
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0
0
4
3
2
0
2
3
4
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In Distribution 1, the Total Distance
from the Mean is 16.
In Distribution 2, the Total Distance
from the Mean is 18.
Consequently, the data in Distribution
2 differ more from the mean than the
data in Distribution 1.
The SAD is defined to be:
The Sum of the
Absolute Deviations
Note the relationship between SAD and
Number of Steps to Fair from Level A:
SAD = 2xNumber of Steps
1
4
2
1
2
0
1
2
3
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Note the total distance for the values below the mean
of 6 is 8, the same as the total distance
for the values above the mean.
Hence the distribution will “balance” at 6 (the mean)
An Illustration where the SAD
doesn’t work!
4
4
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1
1
1
1
1
1
1
1
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Finding SAD
Since both points are 4 from the mean
SAD = 8
Since all 8 are 1 from the mean
SAD = 8
Find the “average” distance from mean
2 data points: SAD/ 2 = 4
8 data points: SAD / 8 = 1
We now have found MAD = Mean Absolute Deviation
Adjusting the SAD for group sizes
yields the:
MAD = Mean Absolute Deviation
Summary of Level B
•
Mean as the balance point of a
distribution
•
Mean as a “central” point
•
Various measures of variation about
the mean.
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X
X – X bar
Absolute
Deviation
2
2-6 = -4
4
4
4-6= -2
2
5
5 – 6 = -1
1
5
5 – 6 = -1
1
6
6–6=0
0
7
7–6=1
1
8
8–6=2
2
9
9–6=3
3
10
10 – 6 = 4
4
Sum of Absolute Deviations = 18
Mean Absolute Deviation (MAD) = 18/9 = 2
Interpret MAD
On “average” how far is the data from the mean.
Example: Find MAD
3035 60 63
1.Find the mean
2.Complete table
X
X – x bar
Absolute
deviations
3. Find the mean of the absolute deviations
X
X – x bar
Absolute
deviations
30
30 – 47 = -17
17
35
35 – 47 = -12
12
60
60 – 47 = 13
13
63
63 – 47= 16
16
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Another Measure of Spread:
Standard Deviation
This is the most common measure of
Variability
X
X – x bar
Squared
Deviations
30
30 – 47 = -17
289
35
35 – 47 = -12
144
60
60 – 47 = 13
169
63
63 – 47= 16
256
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1.
2.
X
2.
3.
4.
X – x bar
Squared
Deviations
Ages (x)
8
7
5
8
Ages (x)
X – x bar
Squared
deviations
14
14– 11.25 = 2.75
7.5625
9
9 – 11.25 = -2.25
5.0625
10
10 – 11.25 = -1.25
1.5625
12
12 – 11.25 = .75
.5625