Aron chpt 1 ed (1)

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

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

Prepared by:
Genna Hymowitz
Stony Brook University
y y
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l l l di f h

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

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

Chapter 1


Chapter Outline
The Two Branches of Statistical Methods
Some Basic Concepts
Kinds of Variables
Frequency Tables
Histograms
Shapes of Frequency Distributions
p q y
Frequency Tables and Histograms in Research
Articles


Why Learn Statistics?
Increase your understanding of research
y g
articles
Help you learn how to do your own
ep ea ow o you ow
research
Can improve your reasoning and intuition
◦ Help you make sound decisions


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 Concepts
p
Variable
◦ 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
0 10


Kinds of Variables
Numeric (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
g g g y


Level of Measurement
Type 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
measurement,
Not in text there is an absolute zero that indicates an absence of the
variable being measured.
e.g. , height, weight


◦ rank-order (ordinal)
numeric variable in which values correspond to the relative
p
position of things measured
e.g., class standing, birth order, position in a race


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


How Are You Doing?
You are conducting a study to evaluate how happy
p p
people are in their j
job.
• For this study, you ask people to indicate their job
title.
1) What is your variable of interest?
2) Is your variable
a) numeric
b) nominal
i l
3) What level of measurement are you using?
a) ratio
b) interval
c) rank-order variable
d) nominal


How Are You Doing?
p p
people are in their j .
job.
• For this study, you ask people to rate their level of
happiness on a scale of 0–10.
2) Is your variable
a) numeric
b) nominal
i l
a) ratio
b) interval
c) rank-ordered
d) nominal


How Are You Doing?
p p
people are in their job.
j
• For this study, you ask people to rate their level of
happiness as “very happy”, “happy”, “unhappy”, “very
unhappy”.
2) Is your variable
a)
) numeric
b) nominal
a)
) ratio
b) interval
c) rank-ordered
d) nominal


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

Frequency
Given 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


Frequency
Given 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 b 5.
ld be 5
◦ frequency table
a table displaying the pattern of frequencies over
different values


Steps for Making a Frequency Table
Step 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
used.
Step 4:
◦ Figure the percentages of scores for each value.


Frequency Table Step 1
Your research study used a happiness scale that ranges from 0 (not
at all happy) to 10 (extremely happy)
happy).
Happiness Rating

0
1
2
3
4
5
6
7
8
9
10


q y p
• Your study resulted in the following scores:
• 823129156194423354753

Happiness Frequency
Rating Tally
0
1 II
2 III
3 IIII
4 III
5 III
6 I
7 I
8 I
9 II
10


Happiness Frequency Frequency
Rating Tally
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


Figure the percentage of scores for each
value.
◦ Take the frequency of the value divide it by the
value,
total number of scores, and multiply by 100.


Completed Frequency Table
Happiness Frequency Percent
Rating
0 0 0%
1 2 10%
2 3 15%
5%
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:

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
Variables
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 Percent
Psychology 5 25

Sociology 8 40

Anthropology 3 15

Political Science
P liti l S i 4 20


Another Example:
Psychology majors?
y gy j
Sociology majors?
CJ majors?
Other majors?

Grouped Frequency Table
A frequency table that uses intervals of values
Lists the number of participants for each
interval of values
If the list of possible values ranges from 0 10 a
ossible al es ran es 0–10,
possible set of intervals is:
0–1
2–3
4–5
6–7
8–9
10 11
10–11


Histogram
Graph of the information on a frequency
table
◦ The height of each bar is the frequency of
each value in the frequency table
table.


Histogram
Graph of the information on a frequency
p q y
table
◦ The height of each bar is the frequency of
g q y
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 Histogram
Step 1 Happiness Frequency
Rating
◦ Make a frequency table 0 0
or grouped frequency
1 2
table.
2 3
Scores on a Job Happiness Survey 3 4
4 3
8, 2 3 1, 2 9, 1, 5 6, 9,
8 2, 3, 1 2, 9 1 5, 6 9
5 3
4, 4, 2, 3, 3, 5, 4, 7, 5, 3
6 1
7 1
8 1
9 2
10 0


Step 2
◦ Put the values at the bottom of the page going from left to right,
from l
f lowest to hi h
highest
Step 3
◦ Make a scale of frequencies along the left edge of the page (0 will
be
b at the bottom and the highest value will be at the top).
h b d h hi h l ill b h )

6

5

4

3

2

1

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


Step 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 Distributions
Show 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


8

7

6

5

4

3

2

1

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


q y
Show 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

◦ 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


◦ 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


◦ rectangular distribution
when all values h
h ll l have approximately th same f
i t l the frequency

3

2

1

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


Symmetrical and Skewed Distributions
y
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 Distributions
y
Skewed distributions are distributions where the
scores pile up on one side of the middle.
◦ characterized b the side of th distribution where scores
h t i d by th id f the di t ib ti h
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 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 Effects
Floor 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).
(e.g., children)

8

7

6

5

4

3

2

1

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


Floor and Ceiling Effects
Ceiling Effect
g
◦ 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 t t)
t ti ti 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 Light-Tailed
Heavy- Light-
Distributions
Normal Curve
◦ bell-shaped, unimodal, and symmetrical
Light-Tailed Distribution
◦ There are fe sc res in the tails (the tails are thin)
few scores thin).
Heavy-Tailed Distribution
◦ There are many scores in the tails (the tails are thick).


Key Points
y
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.
Af frequency table organizes the scores i
bl i h into a table that li each possible
bl h lists h ibl
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
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.


Aron chpt 1 ed (1)

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