Dispersion

© Relay Graduate School of Education. All rights reserved. 1
DISPERSION

AGENDA OBJECTIVES
Agenda and Objectives
• Descriptive statistics
• Dispersion
• Aggregate data
• The right questions and graphics
• Closing
 Compare basic descriptive statistics
and identify their limitations
 Describe common mistakes associated
with analyzing "on average" data
 Explain the purpose of the Data
Narrative analyses
 Evaluate research questions against
criteria for quality
2 2

In the last activity, we
realized our need for
more descriptors of
the data beyond just
mean, median, mode,
range and n-count

© 2014 Relay Graduate School of Education. All rights reserved. 4
Increase quality control testing
Customer Satisfaction Improvement Plan
Tree of Statistical Terminology
BASIC
DESCRIPTIVE
STATISTICS
Dataset
contents
Measures of
central
tendency
Dispersion
Variables
Missing data
Mean
Median
Range & Outliers
Standard Deviation
Distributions
Frequency/Quantiles
Mode
Raw data

Frequency/Quantiles
• Quantiles: ranges of scores, partitioned into bins of the same
size
• Can be divided up in quintiles, quartiles, deciles, etc
• Allows us to determine the frequency of scores in each range

Let’s create a frequency
table of our own

• How were the intervals partitioned (binned) for this sample?
• What did we learn about the samples by looking at the frequencies?
• What is another way we could have partitioned (binned) the intervals?
Practice With Frequency Tables
1

Click ahead when
you’ve completed the
appropriate section
of your Handout

Check Your Work

• The intervals were binned into quintiles, in ranges of 20%
• When we partitioned the data in this manner, we could see that the
distribution of performance in Class #1 was indeed different from Class
#2.
– Class #2 has 3 students who are “passing” and Class #1 has only 1. But,
class #2 also has 3 students who are FAR below passing in the 0-20%
range.
• We could have partitioned the intervals by 10% range. We could have
seen more clearly how many students reached the Proficient Goal. But we
only had 10 students in each class, so that wouldn’t have worked well.
– When deciding what range to use for creating the bins, there is no
better rule of thumb than to use your noodle!
Check Your Work

Let’s try another
frequency table

Frequency Tables: Changing the Bin Size
• What did we learn about the two samples by looking at the
frequencies?
• What is a helpful rule of thumb we can use for partitioning
frequencies into appropriate intervals?

Check your work

Check your work
5
5
5
5

Check your work
• This makes it look like Class #1 and Class #2 performed equally!
• When deciding what range to use for creating the bins, there is no
better rule of thumb than to use your noodle!
5
5
5
5

“Remember, 71.3% of all
statistics…

statistics…
are made up on
the spot!”

statistics…
are made up on
the spot!”
Oops. This
should say
“81.3%”, not
71.3%.

BASIC
DESCRIPTIVE
STATISTICS
Dataset
contents
Measures of
central
tendency
Dispersion
Variables
Missing data
Mean
Median
Range & Outliers
Standard Deviation
Distributions
Frequency/Quantiles
Mode
Raw data

Standard Deviation: Technical Definition
• The average distance of the data from the mean
(average)

Standard Deviation: Technical Definition
• The average distance of the data from the mean
(average)
• Formula is:
• You do not need to calculate with the formula, but…
• You should understand the conceptual meaning

Standard Deviation: Conceptual Meaning
A “low” standard deviation - data are clustered close to the
mean
A “high” standard deviation - data are spread across the range
of values

Standard Deviation: Conceptual Meaning
A “low” standard deviation - data are clustered close to the
mean
A “high” standard deviation - data are spread across the range
of values
What constitutes “low” or “high”? Let’s look at Class 1 and
Class 2.

Standard Deviation: Class 1 vs Class 2
• What class’ scores have greater average distance from the mean?
(remember that the mean is 50 for both classes)

Check Your work

Check Your Work
• Class #1 has a lower standard deviation. Its data points have a
lower average distance from the average.

Check Your Work
• Class #1 has a lower standard deviation. Its data points have a
lower average distance from the average.
We won’t focus on
what this number
24.2 means, we’ll
just notice that it’s
lower than Class #2

Check Your Work
• Here’s a graphical way of seeing that Class #1 has data points
that are more clustered around the average score
0
20
40
60
80
100
120
Class #1
0
20
40
60
80
100
120
Class #2

Standard Deviation: Follow up questions
• The standard deviation would be even lower if the points were
clustered even closer to the mean
• If everybody scored the same score, the standard deviation
would be zero

“Remember…Even if you are a
one-in-a-million kind of guy,

“If you are a one-in-a-
million kind of guy,
in China and India

in China and India
there are thousands
more just like
you!!”

in China and India
there are thousands
more just like
you!!”
Numbers only have
meaning in context!
“High” or “Low”
standard deviation
means nothing without
additional info!

BASIC
DESCRIPTIVE
STATISTICS
Dataset
contents
Measures of
central
tendency
Dispersion
Variables
Missing data
Mean
Median
Range & Outliers
Standard Deviation
Distributions
Frequency/Quantiles
Mode
Raw data

How Does This Relate To The Data Narrative Requirements?
This is in
your
Handout

Here’s what is meant in
the rubric by…
All students’ academic
achievement, displayed
relative to the Proficient
and Ambitious Goal

Displayed Relative to the Proficient and Ambitious Goal
This is for
Class #1.

Displayed Relative to the Proficient and Ambitious Goal
10 students in total.
9 scored under 70%.
1 scored above 80%.

Histogram: The Columns Represent Numerical Ranges
We call this a
histogram because
the columns
represent
numerical ranges

Here’s what is meant in
the rubric by…
Distribution of academic
performance for all
students

Distribution of Academic Performance for All Students
0
10
20
30
40
50
60
70
80
90
100
CS CT RL SG LL S LJ AC MB SW
Score
Individual Students
Class #1 Standards Mastery, All Students
This is for
Class #1.
10 students
in total. 10
columns.

Bar Graphs: The Columns Represent Individuals
0
10
20
30
40
50
60
70
80
90
100
CS CT RL SG LL S LJ AC MB SW
Score
Individual Students
Class #1 Standards Mastery, All Students
We call this
a bar graph
because
each column
represents
an individual

Bar Graphs: The Columns Represent Individuals
0
10
20
30
40
50
60
70
80
90
100
CS PH BC TB CT RL JT SG LL B S LJ AC TF MM MB HC JD SW MA
Score
Individual Students
Class #1 & #2 Standards Mastery, All Students
Class 1
Class 2

Your turn

Histograms: Your Turn
0
1
2
3
4
5
6
7
8
9
10
0% - 20% <20% - 40% <40% - 60% <60% - 80% <80% - 100%
NumberofStudents
Score Range
Class 1
Class 2
• Does this histogram do a good job displaying the results from Class #1 and
Class #2? Why or why not?
• Given that 70% is an important threshold for how we measure Standards
Mastery (it’s the Proficient Goal), how else could we have binned the
frequency table and histogram?

Check Your Work

Check Your Work
0
1
2
3
4
5
6
7
8
9
10
0% - 20% <20% - 40% <40% - 60% <60% - 80% <80% - 100%
NumberofStudents
Score Range
Class 1
Class 2

Histograms: Your Turn
0
2
4
6
8
10
0% -
20%
<20% -
40%
<40% -
60%
<60% -
80%
<80% -
100%
NumberofStudents
Score Range
Class #1 & #2 Standards
Mastery, All Students
Class 1
Class 2
• This histogram does a decent job showing the results, but it doesn’t
account show performance relative to the Proficient Goal at 70%.
• We could have binned the data with ranges of 10%...but there aren’t
enough data points to do that effectively
• We never use different size bins on the same histogram! That’s confusing.

“Before you create a
graphical display…

Figure out what you want
to say.”

Figure out what you want
to say.”
Like “Figure 1.1” –
get it?!?

Distributions
• A graphical display of individual data points or quantile bins
• A way for us to visualize dispersion
You need to know:
• how to create and interpret a graph (plot)
• some basic distribution shapes (normal, skewed, and bimodal)
You do not need to know:
• the theoretical implications of distributions for statistical models

Normal Distribution
• Most commonly discussed distribution
• Aka Bell Curve
• We don’t want this kind of distribution of score results
57

Skewed Distribution
• Looks exactly as it sounds – “skewed”.
• “Skewed left” means that the tail points left, i.e. that we have
very few data points on the lower end of performance
• We do want this kind of distribution of score results!
58
Skewed right
Skewed left

Skewed Left Distribution
• Looks exactly as it sounds – “skewed”
• Do we want this kind of distribution of score results?
59
Skewed right
Skewed left

Bimodal Distribution
• Bimodal = two modes (two most common values)
• We don’t want this kind of distribution of score results
60

BASIC
DESCRIPTIVE
STATISTICS
Dataset
contents
Measures of
central
tendency
Dispersion
Variables
N-count and missing data
Mean
Median
Range & Outliers
Standard Deviation
Distributions
Frequency/Quantiles
Basic Descriptive Statistics Tree
Mode

We’ve reviewed everything
in our tree of statistical
terminology.
Now it’s time to further
analyze and interpret these
descriptive statistics!

Dispersion

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