Statistics: A Gentle Introduction By Frederick L. Coolidge, Ph.D. Sage Publications

1
Statistics A Gentle Introduction By Frederick
L. Coolidge, Ph.D.Sage Publications

• Chapter 2
• Descriptive Statistics
• Understanding Distributions of Numbers

2
0730 Q1 Results N20

• 15
• 21124456679
• 3001124779

3
0900 Q1 Results N32

• 1249
• 20335567799
• 32224444445566889
• 4001

4
Overview

• Graphs and tables
• Whats the point?
• The nasty tricks of the trade
• Types of distributions
• Grouping data
• Cumulative frequency distributions
• Stem-and-leaf plot

5
Graphs and TablesWhats the point?

• Whats the point?
• Document the sources of statistical data and its
  characteristics.
• Where did you get it?
• What is it measuring?

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Graphs and TablesWhats the point?

• Make appropriate comparisons.
• Compare similar data.
• Make the point more clearly.
• Make data more understandable.
• Eliminate doubt.

7
Frequency Distributions

• A table reporting the number of
  observations falling into each category of the
  variable
• Frequency count for data value is of times
  value occurs in data set
• Ungrouped frequency distribution lists the data
  values w/frequency count with which each value
  occurs
• Relative frequency for any class is obtained by
  dividing frequency for that class by total of
  observations.

8
Cumulative Frequency(CF) and Cumulative Relative
Freq(CRF)

• CF- a specific value in a frequency table is sum
  of frequencies for all values at or below the
  given value
• CRF- the sum of the relative frequencies for all
  values at or below the given value expressed as a
  proportion
• Grouped Frequency distribution is obtained by
  constructing intervals for data and listing
  frequency count in each interval

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MathAnxiety Scores Freq Relative Freq Cumulative Freq Cumulative Relative Freq Cumulative Relative Freq
1 1 0.05 1 0.05
2 2 0.09 3 0.14
3 3 0.14 6 0.28
4 4 0.18 10 0.46
5 5 0.23 15 0.69
6 0 0 15 0.69
7 2 0.09 17 0.78
8 3 0.14 20 0.92
9 1 0.05 21 0.97
10 1 0.05 22 1.02
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MathAnxietyScore730class(Grouped Freq
Distribution
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Histogram Math Anxiety Scores

• .30
• .25
• .20
• .15
• .10
• .5
• .5 2.5 4.5 6.5 8.5 10.5

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Blacks More Pessimistic than whites economic
opportunities
What Govts Role in improving economic position of minorities Non-Hispanic Whites() Blacks() Hispanics
Major Role 32 68 67
Minor Role 51 22 21
No Role 16 9 8

13
Laws Covering Sales of Firearms Increase
Restrictions( 2000)?
More Less Same No opinion
Men(N493) 256 39 193 5
Women(N538) 387 11 129 11
14
Men and Firearm Restrictions Frequency
Distribution(N493)
F CF RF CRF
More 256 256 .52 .52
Less 39 295 .08 .60
Same 193 488 .39 .99
No opinion 5 493 .01 1


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Women and Firearm Restrictions Frequency
Distribution(N538)
F CF RF CRF
More 387 387 .719 .719
Less 11 398 .020 .739
Same 129 527 .239 .978
No opinion 11 538 .020 .998


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Graphs and TablesWhats the point?

• Demonstrate the mechanisms of cause and effect
  and express the mechanisms quantitatively.
• If you vary the cause and the results change in a
  predictable and uniform manner, then you make a
  stronger case for cause and effect.

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Graphs and TablesWhats the point?

• Recognize the inherent multivariate (more than
  one cause) nature of the problem.
• Is there anything with just one cause?
• Temperature of boiling water
• Altitude of water
• What is in the water (salt)?

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Graphs and TablesWhats the point?

• Inspect and evaluate alternative hypotheses.
• Cigarette smoking is related to a lower incidence
  of Alzheimers disease.
• Is it the cigarettes?
• Is it the dying at an earlier age, before
  Alzheimers is diagnosable?

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Graphs and TablesThe nasty tricks of the trade

• The nasty tricks of the trade
• Adjust the scale to make the point
• Show only part of the scale
• Omit the units of measure
• Change the scale along the graph
• Include too much junk
• Not enough to bother graphing

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Graphs and TablesThe nasty tricks of the trade
Is Brand One really any better than the others?
21
Stem-and-leaf plot

• Presents the frequency of data points without
  losing important information.
• Data set 25, 27, 29
• Stem ? 2 579 ? Leaves

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Stem-and-leaf plot

• The first digit is the stem
• The second digit is each leaf
• 25 27 29
• Stem ? 2 579 ? Leaves

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Stem-and-leaf plot

• The first digit is the stem
• The second digit is each leaf
• 25 27 29
• Stem ? 2 579 ? Leaves

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Stem-and-leaf plot

• Lets try it
• Data set 30, 32, 32, 34, 37, 37, 39
• Data set 5, 9, 10, 11, 11, 23, 25, 27

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Types of DistributionsFrequency Distribution

• Frequency distribution
• Showing what you have
• A way to illustrate how many of each thing.

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Types of DistributionsFrequency Distribution
27
Types of DistributionsNormal Distribution

• Normal distribution
• Also known as the bell-shaped curve
• An illustration of the expectation of what most
  types of data will look like
• A few data points at each extreme
• Most data points in the middle area

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Types of DistributionsNormal Distribution
29
Types of DistributionsPositively Skewed
Distribution

• Not all data are created equal
• Positive skew
• Many data points near the origin of the graph

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Types of DistributionsNegatively Skewed
Distribution

• Negative skew
• Many data points away from the origin of the
  graph

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Types of DistributionsBimodal Distribution

• Bimodal
• Two areas under the curve with many data points

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Types of DistributionsNon-normal Distributions

• Nonnormal distributions
• But not abnormal
• Platykurtic flat like a plate

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Bi-Modal Distribution Spring 2010 Quiz Scores
F CF RF CRF
10-16 5 5 .227 .227
17-23 3 8 .136 .363
24-30 2 10 .090 .453
31-37 8 18 .363 .816
38-44 4 22 .181 .997

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Types of DistributionsNon-normal Distributions

• Leptokurtic up down (like leaping)
• Bimodal lumpy

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Grouping data

• A way of organizing data so that they are
  manageable.
• Which is easier to understand?
• 3, 1, 7, 4, 1, 2, 3, 5, 4, 9
• or
• 1, 1, 2, 3, 3, 4, 4, 5, 7, 9

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Grouping dataTips for grouping data

• Tips for grouping lots of data
• Choose interval widths that reduce your data to 5
  to 10 intervals.

5
10
15
20
25
30
35
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Grouping dataTips for grouping data

• Choose meaningful intervals.
• Which is easier to understand at a glance?

5
10
15
20
25
30
35
or
4
7
10
13
16
19
22
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Grouping dataTips for grouping data

• Interval widths must be the same.

5
10
15
20
25
30
35
NOT
5
10
20
22
30
33
35
39
Grouping dataTips for grouping data

• Intervals cannot overlap.

5-10
11-15
16-20
21-25
26-30
31-35
36-40
NOT
5-10
10-15
14-20
20-26
25-30
30-35
35
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Grouping dataAn example

• The data are displayed using
• A frequency table of individual data points
• A frequency table by intervals
• Graph of data by intervals

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Grouping dataAn example
42
Grouping dataAn example
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Grouping dataAn example
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Freq Distribution Using Stated limits
Age Category Freq CF
20-29 7 7
30-39 7 14
40-49 12 26
50-59 3 29
60-69 3 32
70-79 6 38
80-89 2 40

Total 40

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Problem w/ Stated Limits

• Gap of one between adjacent intervals
• Problem for scores with fractional values where
  classify a woman 49.25 years old? Here age would
  actually fall between intervals 40-49 and 50-59!!
• Real limits extend upper and lower limits by .5

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Freq Distribution Using Real Upper and Lower
limits
Age Category Freq CF
19.5-29.5 7 7
29.5-39.5 7 14
39.5-49.5 12 26
49.5-59.5 3 29
59.5-69.5 3 32
69.5-79.5 6 38
79.5-89.5 2 40

Total 40

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Upper/Lower limits Fractional Values

• Scores falling exactly at upper real limit or
  lower real limit are rounded to closest even
  number EX59.5 rounded to 60 and included in
  interval
• 59.5-69.5
• Where would you classify respondent 49.25 years?
  How about 59.4?

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Cumulative Frequency Distribution

• Cumulative frequency distribution
• Shows how many cases (data points) have been
  accounted for out of the total number of cases
  (data points).

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Cumulative Frequency Distribution

• How many data points have accounted for as each
  group is displayed.

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Cumulative Frequency Distribution

• Cumulative frequencies can also be illustrated
  using percentages.

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Cumulative Frequency Distribution

• Cumulative distributions can help give a
  reference point for an individual score.
• Percentile
• What percentage scored above or below the score
  of interest
• Quartile
• Divides the scores into four groups
• 25 1st, 2nd, 3rd, 4th

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Cumulative Frequency Distribution
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Statistics A Gentle Introduction By Frederick
L. Coolidge, Ph.D.Sage Publications

• Chapter 2
• Descriptive Statistics
• Understanding Distributions of Numbers