Aron chpt 1 ed (1)

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

Arthur Arthur AronAron, Elaine N. , Elaine N. AronAron, Elliot Coups, Elliot Coups

Prepared by:Genna Hymowitz

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

Chapter 1


Chapter OutlineChapter OutlineChapter OutlineChapter Outline

The Two Branches of Statistical MethodsSome Basic ConceptsKinds of VariablesFrequency TablesFrequency TablesHistogramsShapes of Frequency Distributionsp q yFrequency Tables and Histograms in Research Articles


Why Learn Statistics?Why Learn Statistics?Why Learn Statistics?Why Learn Statistics?

Increase your understanding of research y garticles Help you learn how to do your own e p you ea ow to o you ow researchCan improve your reasoning and intuitionCan improve your reasoning and intuition◦ Help you make sound decisions


What is Statistics?What is Statistics?What is Statistics?What is Statistics?A branch of mathematics that focuses on the organization, analysis, and interpretation of a group of numbersTwo Main Branches of Statistics◦ descriptive statistics:

used to summarize and describe a group of numbers from a 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 ConceptsppVariable ◦ characteristic or condition that can have different

valuese.g., level of stressageGenderGender

Value ◦ possible number or category a score can have

e.g., 0–1035Male

Score Score ◦ particular person’s value

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


Kinds of VariablesKinds of VariablesKinds of VariablesKinds of Variables

Numeric (Quantitative)Variable( )◦ variable that has values that are numbers

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

categoriese.g., gender, religion, ethnicityg g g y


Level of MeasurementLevel of MeasurementLevel of MeasurementLevel of MeasurementType of underlying numerical information provided by a measureprovided 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 (intervals) between the numbers are all equal in size but there is no absolute zero.

e.g. , intelligence, temperatureratio – in addition to order and equal units of measurement 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


Level of MeasurementLevel of MeasurementLevel of MeasurementLevel of MeasurementType of underlying numerical information provided by a measureprovided by a measure◦ rank-order (ordinal)

numeric variable in which values correspond to the relative pposition of things measurede.g., class standing, birth order, position in a race


Level of MeasurementLevel of MeasurementLevel of MeasurementLevel of MeasurementType of underlying numerical information provided by a measureprovided by a measure◦ nominal

variable in which values are categoriese.g., gender, religion, ethnicity


How Are You Doing?How Are You Doing?How Are You Doing?How Are You Doing?You are conducting a study to evaluate how happy people are in their job. p p j

• For this study, you ask people to indicate their job title.

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

a) numericb) i lb) nominal

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


How Are You Doing?How Are You Doing?How Are You Doing?How Are You Doing?You are conducting a study to evaluate how happy people are in their job. . p p j

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

a) numericb) i lb) nominal

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


How Are You Doing?How Are You Doing?How Are You Doing?How Are You Doing?You are conducting a study to evaluate how happy people are in their job. p p j

• 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

a) numeric)b) nominal

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


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FrequencyFrequencyFrequencyFrequencyGiven 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


FrequencyFrequencyFrequencyFrequencyGiven 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

ld b 5would be 5.

◦ frequency table a table displaying the pattern of frequencies over a table displaying the pattern of frequencies over different values


Steps for Making a Frequency TableSteps for Making a Frequency TableSteps for Making a Frequency TableSteps for Making a Frequency Table

Step 1: ◦ Make a list down the page of each possible value,

from lowest to highest. Step 2: 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 your list was used. Step 4: ◦ Figure the percentages of scores for each value.


Frequency Table Step 1Frequency Table Step 1Frequency Table Step 1Frequency Table Step 1Your research study used a happiness scale that ranges from 0 (not

at all happy) to 10 (extremely happy)at all happy) to 10 (extremely happy).

Happiness Rating

00

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Frequency Table Step 2Frequency Table Step 2q y pq y p• 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

01 II1 II2 III3 IIII4 III5 III6 I7 I8 I9 II


10

Frequency Table Step 3Frequency Table Step 3Frequency Table Step 3Frequency Table Step 3Happiness Rating

Frequency Tally

Frequency

0 01 II 22 III 333 IIII 44 III 35 III 35 III 36 I 17 I 18 I 19 II 210 0


Frequency Table Step 4Frequency Table Step 4Frequency Table Step 4Frequency Table Step 4

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

total number of scores, and multiply by 100.


Completed Frequency TableCompleted Frequency TableCompleted Frequency TableCompleted Frequency TableHappiness Rating

Frequency Percent

0 0 0%1 2 10%2 3 15%3 5%3 4 20%4 3 15%5 3 15%5 3 15%6 1 5%7 1 5%8 1 5%9 2 10%10 0 0%


Another Example:Another Example:Another Example:Another Example:

4 3 10

5 4 2

9 6 8

3 1 7

5 5 6

2 5 4

6 86 7 8

7 3 5

Frequency Tables for Nominal Frequency Tables for Nominal VariablesVariables

Follow the same four steps that you would Follow the same four steps that you would for a numeric variable.◦ Remember that the values in which you are y

interested are names or categories rather than numbers.

Major Frequency PercentPsychology 5 25

Sociology 8 40Sociology 8 40

Anthropology 3 15

P liti l S i 4 20


Political Science 4 20

Another Example:Another Example:Another Example:Another Example:

Psychology majors?y gy jSociology majors?CJ majors?CJ majors?Other majors?

Grouped Frequency TableGrouped Frequency TableGrouped Frequency TableGrouped Frequency TableA frequency table that uses intervals of valuesLists the number of participants for each interval of valuesIf the list of ossible al es ran es from 0 10 a If the list of possible values ranges from 0–10, a possible set of intervals is: 0–12–34–56–78–910–1110 11


HistogramHistogramHistogramHistogramGraph of the information on a frequency table table ◦ The height of each bar is the frequency of

each value in the frequency tableeach value in the frequency table.


HistogramHistogramHistogramHistogram

Graph of the information on a frequency p q ytable ◦ The height of each bar is the frequency of g q y

each value in the frequency table.8

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1


00 1 2 3 4 5 6 7 8 9 10

How to Make a HistogramHow to Make a HistogramHow to Make a HistogramHow to Make a Histogram

Step 1 Happiness Rating

Frequency

◦ Make a frequency table or grouped frequency table.

Rating0 01 2table.

Scores on a Job Happiness Survey

8 2 3 1 2 9 1 5 6 9

2 33 44 38, 2, 3, 1, 2, 9, 1, 5, 6, 9,

4, 4, 2, 3, 3, 5, 4, 7, 5, 3 5 36 17 17 18 19 2


10 0

How to Make a HistogramHow to Make a HistogramHow to Make a HistogramHow to Make a HistogramStep 2◦ Put the values at the bottom of the page going from left to right,

f l hi h from lowest to highest Step 3 ◦ Make a scale of frequencies along the left edge of the page (0 will

b h b d h hi h l ill b h )be at the bottom and the highest value will be at the top).

66

5

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2

1


00 1 2 3 4 5 6 7 8 9 10

How to Make a HistogramHow to Make a HistogramHow to Make a HistogramHow to Make a Histogram

Step 4◦ Make a bar for each value.

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22

1

0


0 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency DistributionsFrequency DistributionsFrequency Distributions

Show the pattern of frequencies over the various values (how the frequencies are spread out).8

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00 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency DistributionsFrequency DistributionsFrequency Distributions

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Frequency DistributionsFrequency Distributionsq yq yShow the pattern of frequencies over the various values (how the frequencies are spread out).◦ unimodal distribution a histogram with one very high ◦ unimodal distribution - a histogram with one very high

area

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00 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency DistributionsFrequency DistributionsFrequency Distributions◦ bimodal distribution

a distribution with two fairly equal high pointsa distribution with two fairly equal high points

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00 1 2 3 4 5 6 7 8 9 10

Frequency DistributionsFrequency DistributionsFrequency DistributionsFrequency Distributions◦ multimodal distribution

a distribution with two or more high pointsa distribution with two or more high points

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00 1 2 3 4 5 6 7 8 9 10


Frequency DistributionsFrequency DistributionsFrequency DistributionsFrequency Distributions◦ rectangular distribution

h ll l h i t l th fwhen all values have approximately the same frequency

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2

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00 1 2 3 4 5 6 7 8 9 10


Symmetrical and Skewed DistributionsSymmetrical and Skewed DistributionsyyIn the social and behavioral sciences, most scores are symmetrically distributed.◦ They have approximately the same number of scores on both

sides of the distribution.

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Symmetrical and Skewed DistributionsSymmetrical and Skewed DistributionsyySkewed distributions are distributions where the scores pile up on one side of the middle.

h t i d b th id f th di t ib ti h ◦ characterized by the side of the distribution where scores are more spread out (tail)

◦ negatively skewed distributiontail is to the lefttail is to the left

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Skewed DistributionsSkewed DistributionsSkewed DistributionsSkewed Distributions

positively skewed distribution◦ tail is to the right

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Floor and Ceiling EffectsFloor and Ceiling EffectsFloor 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)a lower score (e.g., number of children).

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00 1 2 3 4 5 6 7 8 9 10

Floor and Ceiling EffectsFloor and Ceiling EffectsFloor and Ceiling EffectsFloor and Ceiling Effects

Ceiling Effectg◦ Scores pile toward the upper end of the distribution

because it is not possible to have a higher score (e.g., t ti ti t t)scores on a very easy statistics test).

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00 1 2 3 4 5 6 7 8 9 10

Normal, HeavyNormal, Heavy--Tailed, and LightTailed, and Light--Tailed Tailed DistributionsDistributions

Normal Curve◦ bell-shaped, unimodal, and symmetricalLight-Tailed Distribution

There are fe sc res in the tails (the tails are thin)◦ 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).There are many scores in the tails (the tails are thick).


Key PointsKey PointsyyDescriptive 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 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 f bl i h i bl h li h ibl 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 leftskewed 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.


Date post:	19-Oct-2014
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Aron chpt 1 ed (1)

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