Post on 31-Dec-2015
Chapter 2 1
Statistics: A Gentle Introduction
By Frederick L. Coolidge, Ph.D.Sage Publications
Chapter 2Descriptive Statistics:
Understanding Distributions of Numbers
0730 Q1 Results N=20
1|5 2|1124456679 3|001124779
Chapter 2 2
0900 Q1 Results N=32
1|249 2|0335567799 3|2224444445566889 4|001
Chapter 2 3
Chapter 2 4
Overview
Graphs and tables What’s the point? The nasty tricks of the trade
Types of distributions Grouping data Cumulative frequency distributions Stem-and-leaf plot
Chapter 2 5
Graphs and TablesWhat’s the point?
What’s the point?
Document the sources of statistical data and its characteristics.
Where did you get it? What is it measuring?
Chapter 2 6
Graphs and TablesWhat’s the point?
Make appropriate comparisons.
Compare similar data. Make the point more clearly. Make data more understandable. Eliminate doubt.
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.
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
MathAnxiety Scores Freq
Relative Freq
Cumulative 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
MathAnxietyScore7:30class(Grouped Freq Distribution
Class Intervals F CF RF CRF.5-2.5 3 3 0.136 0.13642.5-4.5 7 10 0.318 0.45464.5-6.5 5 15 0.227 0.68196.5-8.5 5 20 0.227 0.90928.5-10.5 2 22 0.091 1.0002
Histogram Math Anxiety Scores
.30
.25
.20
.15
.10
.5 .5 2.5 4.5 6.5 8.5 10.5
Chapter 2 11
“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 67Minor Role 51 22 21No Role 16 9 8
Laws Covering Sales of Firearms: Increase Restrictions( 2000)?
More Less Same No opinion
Men(N=493) 256 39 193 5Women(N=538)
387 11 129 11
Men and Firearm Restrictions: Frequency Distribution(N=493)
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
Women and Firearm Restrictions: Frequency Distribution(N=538)
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
Chapter 2 16
Graphs and TablesWhat’s 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.
Chapter 2 17
Graphs and TablesWhat’s 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)?
Chapter 2 18
Graphs and TablesWhat’s the point?
Inspect and evaluate alternative hypotheses.
Cigarette smoking is related to a lower incidence of Alzheimer’s disease.
Is it the cigarettes? Is it the dying at an earlier age, before
Alzheimer’s is diagnosable?
Chapter 2 19
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
Chapter 2 20
Graphs and TablesThe nasty tricks of the trade
Is Brand One really any better than the others?
Chapter 2 21
Stem-and-leaf plot
Presents the frequency of data points without losing important information.
Data set: 25, 27, 29
Stem 2 579 Leaves
Chapter 2 22
Stem-and-leaf plot
The first digit is the stem The second digit is each leaf
25 27 29
Stem 2 579 Leaves
Chapter 2 23
Stem-and-leaf plot
The first digit is the stem The second digit is each leaf
25 27 29
Stem 2 579 Leaves
Chapter 2 24
Stem-and-leaf plot
Let’s try itData set: 30, 32, 32, 34, 37, 37, 39Data set: 5, 9, 10, 11, 11, 23, 25, 27
Chapter 2 25
Types of DistributionsFrequency Distribution
Frequency distribution
Showing what you have
A way to illustrate how many of each thing.
Chapter 2 26
Types of DistributionsFrequency Distribution
Chapter 2 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
Chapter 2 28
Types of DistributionsNormal Distribution
Chapter 2 29
Types of DistributionsPositively Skewed Distribution
Not all data are created equal
Positive skew Many data points near the origin of the
graph
Chapter 2 30
Types of DistributionsNegatively Skewed Distribution
Negative skew Many data points away from the origin of
the graph
Chapter 2 31
Types of DistributionsBimodal Distribution
Bimodal Two areas under the curve with many
data points
Chapter 2 32
Types of DistributionsNon-normal Distributions
Nonnormal distributions But not abnormal
Platykurtic: flat like a plate
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
Chapter 2 34
Types of DistributionsNon-normal Distributions
Leptokurtic: up & down (like leaping)
Bimodal: lumpy
Chapter 2 35
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
or1, 1, 2, 3, 3, 4, 4, 5, 7, 9
Chapter 2 36
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
Chapter 2 37
Grouping dataTips for grouping data
Choose meaningful intervals. Which is easier to understand at a
glance?
5 10
15
20
25
30
35
4 7 10
13
16
19
22
or
Chapter 2 38
Grouping dataTips for grouping data
Interval widths must be the same.
5 10
15
20
25
30
35
5 10
20
22
30
33
35
NOT
Chapter 2 39
Grouping dataTips for grouping data
Intervals cannot overlap.
5-10 11-15 16-20
21-25 26-30 31-35
36-40
5-10
10-15 14-20 20-26 25-30 30-35
35
NOT
Chapter 2 40
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
Chapter 2 41
Grouping dataAn example
Chapter 2 42
Grouping dataAn example
Chapter 2 43
Grouping dataAn example
Freq Distribution Using Stated limitsAge 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
Chapter 2 44
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
Chapter 2 45
Freq Distribution Using Real Upper and Lower limitsAge 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
Chapter 2 46
Upper/Lower limits &Fractional Values
Scores falling exactly at upper real limit or lower real limit are rounded to closest even number; EX=59.5 rounded to 60 and included in interval
59.5-69.5 Where would you classify
respondent 49.25 years? How about 59.4?
Chapter 2 47
Chapter 2 48
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).
Chapter 2 49
Cumulative Frequency Distribution
How many data points have accounted for as each group is displayed.
Chapter 2 50
Cumulative Frequency Distribution
Cumulative frequencies can also be illustrated using percentages.
Chapter 2 51
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
Chapter 2 52
Cumulative Frequency Distribution
Chapter 2 53
Statistics: A Gentle Introduction
By Frederick L. Coolidge, Ph.D.Sage Publications
Chapter 2Descriptive Statistics:
Understanding Distributions of Numbers