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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


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 0.052 2 0.09 3 0.143 3 0.14 6 0.284 4 0.18 10 0.465 5 0.23 15 0.696 0 0 15 0.697 2 0.09 17 0.788 3 0.14 20 0.929 1 0.05 21 0.9710 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 opinionMen(N=493) 256 39 193 5Women(N=538) 387 11 129 11

Men and Firearm Restrictions: Frequency Distribution(N=493)

F CF RF CRFMore 256 256 .52 .52Less 39 295 .08 .60Same 193 488 .39 .99No opinion 5 493 .01 1

Women and Firearm Restrictions: Frequency Distribution(N=538)

F CF RF CRFMore 387 387 .719 .719Less 11 398 .020 .739Same 129 527 .239 .978No opinion 11 538 .020 .998

Chapter 2 16


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


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


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 CRF10-16 5 5 .227 .22717-23 3 8 .136 .36324-30 2 10 .090 .45331-37 8 18 .363 .81638-44 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


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


Interval widths must be the same.

5 10

15

20

25

30

35

5 10

20

22

30

33

35

NOT

Chapter 2 39


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


Chapter 2 42


Chapter 2 43


Freq Distribution Using Stated limitsAge Category Freq CF20-29 7 730-39 7 1440-49 12 2650-59 3 2960-69 3 3270-79 6 3880-89 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 CF19.5-29.5 7 729.5-39.5 7 1439.5-49.5 12 2649.5-59.5 3 2959.5-69.5 3 3269.5-79.5 6 3879.5-89.5 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 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


Chapter 2 53

Statistics: A Gentle Introduction

By Frederick L. Coolidge, Ph.D.Sage Publications

Chapter 2Descriptive Statistics:

Understanding Distributions of Numbers

02slidesfreq[feb17]

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