Aron chpt 1 ed

Statistics for the Behavioral and Statistics for the Behavioral and Social Sciences: Social Sciences: A Brief CourseA Brief Course Fifth EditionFifth Edition

Arthur Aron, Elaine N. Aron, Elliot CoupsArthur Aron, Elaine N. Aron, Elliot Coups

Copyright © 2011 by Pearson Education, Inc. All rights reserved.

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Prepared by: Genna Hymowitz

Stony Brook University

Displaying the Order in a Group of Displaying the Order in a Group of Numbers Using Tables and GraphsNumbers Using Tables and Graphs

Chapter 1


Chapter OutlineChapter OutlineThe Two Branches of Statistical MethodsSome Basic ConceptsKinds of VariablesFrequency TablesHistogramsShapes of Frequency DistributionsFrequency Tables and Histograms in

Research Articles


Why Learn Statistics?Why Learn Statistics?Increase your understanding of

research articles Help you learn how to do your

own researchCan improve your reasoning and

intuition◦Help you make sound decisions


What is Statistics?What is Statistics?A branch of mathematics that focuses

on the organization, analysis, and interpretation of a group of numbers

Two Main Branches of Statistics◦ descriptive statistics:

used to summarize and describe a group of numbers from a research study

◦ inferential statistics: procedures for drawing conclusions based on the

scores collected in a research study but going beyond them


Basic ConceptsBasic ConceptsVariable

◦ characteristic or condition that can have different values e.g., level of stress age Gender

Value ◦ possible number or category a score

can have e.g., 0–10 35 Male

Score ◦ particular person’s value

e.g., a study participant rates her current level of stress as a 5 on a scale of 0–10


Kinds of VariablesKinds of VariablesNumeric (Quantitative)Variable

◦variable that has values that are numbers

Nominal (Categorical)Variable◦variable that has values that are

names or categories e.g., gender, religion, ethnicity


Level of MeasurementLevel of MeasurementType of underlying numerical

information provided by a measure◦ equal-interval

numeric variable in which differences between values correspond to differences in the underlying thing being measured interval – a scale in which the units of

measurement (intervals) between the numbers are all equal in size but there is no absolute zero. e.g. , intelligence, temperature

ratio – in addition to order and equal units of measurement, there is an absolute zero that indicates an absence of the variable being measured. e.g. , height, weight


Not in text


information provided by a measure◦rank-order (ordinal)

numeric variable in which values correspond to the relative position of things measured

e.g., class standing, birth order, position in a race



information provided by a measure◦ nominal

variable in which values are categories e.g., gender, religion, ethnicity


How Are You Doing?How Are You Doing?You are conducting a study to evaluate

how happy people are in their job. • For this study, you ask people to indicate

their job title.1)What is your variable of interest?2)Is your variable

a) numericb) nominal

3)What level of measurement are you using?

a) ratiob) intervalc) rank-order variabled) nominal



how happy people are in their job. . • For this study, you ask people to rate their

level of happiness on a scale of 0–10.1)What is your variable of interest?2)Is your variable


3)What level of measurement are you using?

a) ratiob) intervalc) rank-orderedd) nominal



how happy people are in their job. • For this study, you ask people to rate their

level of happiness as “very happy”, “happy”, “unhappy”, “very unhappy”.

1)What is your variable of interest?2)Is your variable


3)What level of measurement are you using?a) ratiob) intervalc) rank-orderedd) nominal


http://www.edugamer.net/app/playGame.aspx?userGameId=4213

FrequencyFrequencyGiven a set of numbers, how can

we make sense of them?

Scores on a Job Happiness Survey

8, 2, 3, 1, 2, 9, 1, 5, 6, 9, 4, 4, 2, 3, 3, 5, 4, 7, 5, 3


FrequencyFrequencyGiven a set of numbers, how can

we make sense of them?◦frequency

number of scores with a particular value If 5 students reported that their level of

happiness on the job was a 2 on a 0–10 scale, the frequency for a rating of 2 would be 5.

◦frequency table a table displaying the pattern of

frequencies over different values


Steps for Making a Steps for Making a Frequency TableFrequency TableStep 1:

◦ Make a list down the page of each possible value, from lowest to highest.

Step 2: ◦ Go one by one through the scores, making

a mark for each next to its value on the list. Step 3:

◦ Make a table showing how many times each value on your list was used.

Step 4: ◦ Figure the percentages of scores for each

value.


Frequency Table Step 1Frequency Table Step 1 Your research study used a happiness scale that

ranges from 0 (not at all happy) to 10 (extremely

happy).


Happiness Rating

0

1

2

3

4

5

6

7

8

9

10

Frequency Table Step 2Frequency Table Step 2• Your study resulted in the following scores:

• 8 2 3 1 2 9 1 5 6 1 9 4 4 2 3 3 5 4 7 5 3


Happiness Rating

Frequency Tally

0

1 II

2 III

3 IIII

4 III

5 III

6 I

7 I

8 I

9 II

10

Frequency Table Step 3Frequency Table Step 3


Happiness Rating

Frequency Tally

Frequency

0 0

1 II 2

2 III 3

3 IIII 4

4 III 3

5 III 3

6 I 1

7 I 1

8 I 1

9 II 2

10 0

Frequency Table Step 4Frequency Table Step 4

Figure the percentage of scores for each value.◦Take the frequency of the value,

divide it by the total number of scores, and multiply by 100.


Completed Frequency TableCompleted Frequency Table


Happiness Rating

Frequency Percent

0 0 0%

1 2 10%

2 3 15%

3 4 20%

4 3 15%

5 3 15%

6 1 5%

7 1 5%

8 1 5%

9 2 10%

10 0 0%

Another Example:Another Example:

4 3 10

5 4 2

9 6 8

3 1 7

5 5 6

2 5 4

6 7 8

7 3 5

Frequency Tables for Nominal Frequency Tables for Nominal VariablesVariablesFollow the same four steps that

you would for a numeric variable.◦Remember that the values in which

you are interested are names or categories rather than numbers.


Major Frequency Percent

Psychology 5 25

Sociology 8 40

Anthropology 3 15

Political Science 4 20

Another Example:Another Example:Psychology majors?Sociology majors?CJ majors?Other majors?

Grouped Frequency TableGrouped Frequency TableA frequency table that uses intervals of

valuesLists the number of participants for

each interval of valuesIf the list of possible values ranges from

0–10, a possible set of intervals is: 0–12–34–56–78–910–11


HistogramHistogramGraph of the information on a

frequency table ◦The height of each bar is the

frequency of each value in the frequency table.


HistogramHistogramGraph of the information on a

frequency table ◦The height of each bar is the

frequency of each value in the frequency table.


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

How to Make a HistogramHow to Make a HistogramStep 1

◦ Make a frequency table or grouped frequency table.


Scores on a Job Happiness Survey

8, 2, 3, 1, 2, 9, 1, 5, 6, 9, 4, 4, 2, 3, 3,

5, 4, 7, 5, 3

Happiness Rating

Frequency

0 0

1 2

2 3

3 4

4 3

5 3

6 1

7 1

8 1

9 2

10 0

How to Make a HistogramHow to Make a Histogram Step 2

◦ Put the values at the bottom of the page going from left to right, from lowest to highest

Step 3 ◦ Make a scale of frequencies along the left edge of the

page (0 will be at the bottom and the highest value will be at the top).


6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

How to Make a HistogramHow to Make a HistogramStep 4

◦ Make a bar for each value.


6

5

4

3

2

1

0

0 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency DistributionsShow the pattern of frequencies over

the various values (how the frequencies are spread out).


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency Distributions


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency DistributionsShow the pattern of frequencies over the various

values (how the frequencies are spread out).◦unimodal distribution - a histogram with one

very high area


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency Distributions◦bimodal distribution

a distribution with two fairly equal high points


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency Distributions◦multimodal distribution

a distribution with two or more high points


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency Distributions◦rectangular distribution

when all values have approximately the same frequency


3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Symmetrical and Skewed Symmetrical and Skewed DistributionsDistributions In the social and behavioral sciences, most

scores are symmetrically distributed.◦ They have approximately the same number of

scores on both sides of the distribution.


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Symmetrical and Skewed Symmetrical and Skewed DistributionsDistributions

Skewed distributions are distributions where the scores pile up on one side of the middle. ◦ characterized by the side of the distribution

where scores are more spread out (tail) ◦ negatively skewed distribution

tail is to the left


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Skewed DistributionsSkewed Distributions

positively skewed distribution◦ tail is to the right


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Floor and Ceiling EffectsFloor and Ceiling EffectsFloor Effect

◦Scores pile toward the lower end of the distribution because it is not possible to have a lower score (e.g., number of children).


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Floor and Ceiling EffectsFloor and Ceiling EffectsCeiling Effect

◦ Scores pile toward the upper end of the distribution because it is not possible to have a higher score (e.g., scores on a very easy statistics test).


8

7

6

5

4

3

2

1

0 0 1 2 3 4 5 6 7 8 9 10

Normal, Heavy-Tailed, and Normal, Heavy-Tailed, and Light-Tailed DistributionsLight-Tailed Distributions

Normal Curve◦ bell-shaped, unimodal, and symmetrical

Light-Tailed Distribution◦ There are few scores in the tails (the tails

are thin).Heavy-Tailed Distribution

◦ There are many scores in the tails (the tails are thick).


Key PointsKey Points Descriptive statistics are used to describe and summarize a group of

numbers from a research study. A value is a number or category; a variable is a characteristic that

can have different values; a score is a particular person’s value on the variable.

Some numeric variables are rank-ordered and some variables are names or categories and not numbers.

A frequency table organizes the scores into a table that lists each possible value from lowest to highest along with the frequency of each value.

A grouped frequency table is used when there are many different values. Intervals are given for a range of values.

A histogram visually displays the information in a frequency table. The general shape of a histogram can be unimodal, bimodal,

multimodal, or rectangular, and the distribution can be symmetrical, skewed to the right, or skewed to the left.

Frequency tables, when used in research articles, are used to summarize the characteristics of study participants. Histograms almost never appear in articles, but the shapes of the distribution are sometimes described in words.


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