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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
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Chapter OutlineChapter OutlineThe Two Branches of Statistical MethodsSome Basic ConceptsKinds of VariablesFrequency TablesHistogramsShapes of Frequency DistributionsFrequency Tables and Histograms in
Research Articles
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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
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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
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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
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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
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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
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Not in text
Level of MeasurementLevel of MeasurementType of underlying numerical
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
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Level of MeasurementLevel of MeasurementType of underlying numerical
information provided by a measure◦ nominal
variable in which values are categories e.g., gender, religion, ethnicity
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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
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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 rate their
level of happiness on a scale of 0–10.1)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-orderedd) nominal
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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 rate their
level of happiness as “very happy”, “happy”, “unhappy”, “very unhappy”.
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-orderedd) nominal
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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
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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
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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.
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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).
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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
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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
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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.
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Completed Frequency TableCompleted Frequency Table
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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.
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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
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HistogramHistogramGraph of the information on a
frequency table ◦The height of each bar is the
frequency of each value in the frequency table.
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HistogramHistogramGraph of the information on a
frequency table ◦The height of each bar is the
frequency of each value in the frequency table.
Copyright © 2011 by Pearson Education, Inc. All rights reserved.
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.
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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).
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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.
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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).
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8
7
6
5
4
3
2
1
0 0 1 2 3 4 5 6 7 8 9 10
Frequency DistributionsFrequency Distributions
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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
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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
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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
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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
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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.
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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
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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
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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).
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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).
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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).
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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.
Copyright © 2011 by Pearson Education, Inc. All rights reserved.