Download Stats review

Why do we need statistics?
A.
B.
C.
D.
To confuse students
To torture students
To put the fear of the almighty in them
To ruin their GPA, so that they don’t get
into grad school, have to buss tables and
move back in with parents
E. All of the above
F. All of the above (and other tragic
outcomes)
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A positive optimistic view…
 It is a tool that could help you
succeed and move out of your
parents house
 There is nothing to fear but fear itself
 You need a passing grade of C
 Can help to get into grad school
 It’s important to understand so you
don’t get scammed…
2
The Caveat: Remember…
 There are lies
 There are d#$m (darn) lies
and then
 There are statistics
 Magic
3
Statistics
 The science of collecting, displaying
and analyzing data
 Based on quantitative measurements
of samples
 Allow us to objectively evaluate data
 Descriptive
 Inferential
4
Defining variability
 Amount of change or fluctuation
 Some variability is expected
 Is the observed variability due to the
usual variability among subjects from
the population?
 Or is the observed variability greater
than the usual variability
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Highest
Frequency (# of Subjects)
Frequency Distribution
From population
Sample 1
Sample 2
Sample 3
0
Dependent variable
highest
score
6
Frequency (# of Subjects)
Frequency Distribution
Untreated groups of
an experiment
Experimental
Control
Dependent variable
7
Frequency (# of Subjects)
Frequency Distribution
Treated Groups
Control
Experimental
Dependent variable
8
Beginning steps of an Experiment







Sample from population
Hypothesis
Define variables
Assign subjects to conditions
Measure performance
Calculate means
Calculate variability
Heading Error: Calculating Variance
Deviation from the mean
for each subject
Sex
SUBJE
CT$
Female
rat1
-4.4
4.4
-4.68
21.86
Female
rat3
11
11
1.93
3.71
Female
rat5
2.3
2.3
-6.78
45.90
Female
rat7
8.5
8.5
-0.58
0.33
Female
rat9
6.9
6.9
-2.18
4.73
Female
rat11
-10.8
10.8
1.73
2.98
Female
rat13
-10.9
10.9
1.83
3.33
Female
rat15
17.8
17.8
8.73
76.13
Male
rat2
29.6
29.6
6.86
47.09
Male
rat4
-18.5
18.5
-4.24
17.96
Male
rat6
14.5
14.5
-8.24
67.86
Male
rat8
58.2
58.2
35.46
1257.59
Male
rat10
-18.7
18.7
-4.04
16.30
Male
rat12
-17.3
17.3
-5.44
29.57
Male
rat14
-14.8
14.8
-7.94
63.00
Male
rat16
10.3
10.3
-12.44
154.69
HEADGdeg
(Xi – X)
Square the deviation from
the mean for each subject
(Xi – X)2
Add the squared deviations
together
(Xi – X)2
Female = 158.96
Male = 1654.06
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Heading Error: Calculating Variance
Sex
SUBJE
CT$
Female
rat1
-4.4
4.4
-4.68
21.86
Female
rat3
11
11
1.93
3.71
Female
rat5
2.3
2.3
-6.78
45.90
Female
rat7
8.5
8.5
-0.58
0.33
Female
rat9
6.9
6.9
-2.18
4.73
Female
rat11
-10.8
10.8
1.73
2.98
Female
rat13
-10.9
10.9
1.83
3.33
Female
rat15
17.8
17.8
8.73
76.13
Male
rat2
29.6
29.6
6.86
47.09
Male
rat4
-18.5
18.5
-4.24
17.96
Male
rat6
14.5
14.5
-8.24
67.86
Male
rat8
58.2
58.2
35.46
1257.59
Male
rat10
-18.7
18.7
-4.04
16.30
Male
rat12
-17.3
17.3
-5.44
29.57
Male
rat14
-14.8
14.8
-7.94
63.00
Male
rat16
10.3
10.3
-12.44
154.69
HEADGdeg
Compute the Variance
s2
=
(Xi – X)2
n-1
s2Female = 22.71
s2Male = 236.29
11
Heading Error: Calculating standard deviation
Sex
SUBJE
CT$
Female
rat1
-4.4
4.4
-4.68
21.86
Female
rat3
11
11
1.93
3.71
Female
rat5
2.3
2.3
-6.78
45.90
Female
rat7
8.5
8.5
-0.58
0.33
Female
rat9
6.9
6.9
-2.18
4.73
Female
rat11
-10.8
10.8
1.73
2.98
Female
rat13
-10.9
10.9
1.83
3.33
Female
rat15
17.8
17.8
8.73
76.13
Male
rat2
29.6
29.6
6.86
47.09
Male
rat4
-18.5
18.5
-4.24
17.96
Male
rat6
14.5
14.5
-8.24
67.86
Male
rat8
58.2
58.2
35.46
1257.59
Male
rat10
-18.7
18.7
-4.04
16.30
Male
rat12
-17.3
17.3
-5.44
29.57
Male
rat14
-14.8
14.8
-7.94
63.00
Male
rat16
10.3
10.3
-12.44
154.69
HEADGdeg
Standard deviation
s or SD = s2
sFemale = 4.77
sMale = 15.37
12
Heading Error: Calculating standard error of
the Mean
Sex
SUBJE
CT$
Female
rat1
-4.4
4.4
-4.68
21.86
Female
rat3
11
11
1.93
3.71
Female
rat5
2.3
2.3
-6.78
45.90
Female
rat7
8.5
8.5
-0.58
0.33
Female
rat9
6.9
6.9
-2.18
4.73
Female
rat11
-10.8
10.8
1.73
2.98
Female
rat13
-10.9
10.9
1.83
3.33
Female
rat15
17.8
17.8
8.73
76.13
Male
rat2
29.6
29.6
6.86
47.09
Male
rat4
-18.5
18.5
-4.24
17.96
Male
rat6
14.5
14.5
-8.24
67.86
Male
rat8
58.2
58.2
35.46
1257.59
Male
rat10
-18.7
18.7
-4.04
16.30
Male
rat12
-17.3
17.3
-5.44
29.57
Male
rat14
-14.8
14.8
-7.94
63.00
Male
rat16
10.3
10.3
-12.44
154.69
HEADGdeg
Standard error of the Mean
SEM =
SD
n
SEMFemale = 1.68
SEMMale =
5.43
13
Heading Error: Group Means with SEM
Absolute Heading Error (deg)
30
25
20
Male
15
Female
10
5
0
Group
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Heading Error: Group Means with
95% Confidence Interval
Confidence intervals (CI) represent a range of values above and
below our sample mean that is likely to contain the population
mean; i.e., the true mean of the population is likely (we’re 95%
confident) to fall somewhere within the CI range.
( )
SD
n
35
Heading Error
CI = X± tcrit
40
30
25
Male
20
Female
15
10
5
0
Group
15
Heading Error: Group Means with SEM
variance (s2)- average squared deviation of scores from their mean
standard deviation (SD)- average deviation of scores about the mean
standard error of the mean (SEM)- dispersion of the distribution of
sample means
=
(Xi –
n-1
SD = s2
SEM =
SD
n
30
Absolute Heading Error (deg)
s2
X)2
25
20
Male
15
Female
10
5
0
Group
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Choosing a significance level
Significance level - A criterion for deciding
whether to reject the null hypothesis or not.
• What is the convention?
p < .05 ( level)
• A stricter criterion may be required if the risk
of making a wrong decision (a Type I error) is
greater than usual. p < .01 or p < .001.
• But there is a trade off in using a stricter
criterion.
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Choosing a significance level
Type II error – Failure to reject the null
hypothesis when it is really false ().
• Concluding that the difference is due to chance
variation when it is really due to the
independent variable
• Power of the statistical test (1 - )
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Summary Chart
Reality
Check
Decision based on Statisical Results
Fail to reject Ho
Reject Ho
Ho is true
Correct
p=1-
Type I error
p=
Ho is False
Type II error
P=
Correct
p=1-
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Time estimation experiment
Time will go faster for people having fun
than for those not having fun.
Two group design: Fun - views cartoons with the captions for 10 min.
No Fun – views cartons without captions for 10 min.
Ho = the time estimates of the two groups will be the same.
H1 = the fun group will have shorter estimates than the control group.
Table 13-2 possible errors in the time estimation experiment (p.381, 6th ed.)
What type of errors were made in the two descriptions?
Type I
Type II
We conclude that there was no difference in the time estimates made by the
“fun” and “no fun” groups even though the treatments did produce an effect.
Type 1 = Reporting an effect that doesn’t really exist
Type I
Type II
We conclude that there was a difference in the time estimates made by the
“fun” and “no fun” groups even though the treatments produced little or no
effect at all.
Type 2 = Missing an effect that does really exist
Type III – failure to accurately identify a type 1 or 2 error
Note - The error has been corrected in the 7th ed., p. 390.
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Questions to ask when
selecting a test statistic
Table 14-1 The parameters of data analysis
___________________________________________________
1. How many independent variables are there?
2. How many treatment conditions are there?
3. Is the experiment run between or within subjects?
4. Are the subjects matched?
5. What is the level of measurement of the dependent variable?
___________________________________________________
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Answers based on the water maze study
Table 13-1 The parameters of data analysis
___________________________________________________
1. How many independent variables are there? one
2. How many treatment conditions are there? one
3. Is the experiment run between or within subjects? between
4. Are the subjects matched? no
5. What is the level of measurement of the dependent variable? ratio
___________________________________________________
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Levels of Measurement
Ratio – a measure of magnitude having equal
intervals between values and having an
absolute zero point.
Interval – same as ratio except that there is no
true zero point.
Ordinal – a measure of magnitude in the form of
ranks (not sure of equal intervals and no
absolute zero).
Nominal – items are classified into categories
that have no quantitative relationship to one
another.
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Choosing a test statistic
TABLE 14-2 Selecting a possible statistical test by number of independent variables and level of measurement
One Independent Variable
Two Treatments
Level of
measurement
of dependent
variable
Two
Independent
Groups
Two matched
groups (or
within
subjects)
Two Independent Variables
More Than Two Treatments
Multiple
independent
groups
Multiple
matched
groups (or
within
subjects)
Interval or
ratio
t test for
independent
groups
t test for
matched groups
One-way
ANOVA
One-way
ANOVA
(repeated
measures)
ordinal
MannWhitney U
test
Wilcoxon test
KruskalWallis test
Friedman
test
Nominal
Chi square
test
Chi square
test
Factorial Designs
Independent
groups
Matched
groups (or
within
sujects)
Independent
groups and
matched groups
(or between
subjects and
within subjects
Two-way
ANOVA
Two-way
ANOVA
(repeated
measures)
Two-way
ANOVA (mixed)
Chi square
test
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Heading Error: Statistical Analysis
t test for Independent Groups
1) Lay out Formula
tobs =
(
X 1 – X2
)( )
(n1 – 1) s21 + (n2 – 1) s22
(n1 + n2 – 2)
1 1
+
n1 n2
2) Plug in Values
tobs =
22.74 – 9.08
(
)( )
(8–1)236.29+(8–1)22.71
(8 + 8 – 2)
1 1
+
8 8
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Heading Error: Statistical Analysis
t test for Independent Groups
8) Divide the numerator by the denominator.
tobs =
13.66
5.69
Formula
tobs =
tobs = 2.40
X1 – X2
(
)( )
(n1 – 1) s21 + (n2 – 1) s22
(n1 + n2 – 2)
1 1
+
n1 n2
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Determining significance
1.Was the hypothesis directional or
nondirectional?
2.What was the significance level?
3.How many degrees of freedom do we
have?
Degrees of freedom (df)– the number of members in a set of
data that can vary or change value without changing the
value of a known statistic for those data.
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Answers to the questions
1.Was the hypothesis directional or
nondirectional? Nondirectional, so twotailed.
2.What was the significance level? p < .05
3.How many degrees of freedom do we
have? 14
Look on page 531 of Myers & Hansen to find
the critical value of t…
28
Answers to the questions
Or you could just go on-line… e.g.,
http://www.psychstat.missouristate.edu/introbook/tdist.htm
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Heading Error: Statistical Analysis
t test for Independent Groups
8) Divide the numerator by the denominator.
tobs =
13.66
5.69
tobs = 2.40
p < .05, two-tailed
tcrit = 2.145
Formula
tobs =
X1 – X2
(
)( )
(n1 – 1) s21 + (n2 – 1) s22
(n1 + n2 – 2)
1 1
+
n1 n2
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Compare to our computer output from SPSS
8) Divide the numerator by the denominator.
tobs =
13.66
5.69
tobs = 2.40
p < .05, two-tailed
tcrit = 2.145
Formula
tobs =
X1 – X2
(
)( )
(n1 – 1) s21 + (n2 – 1) s22
(n1 + n2 – 2)
1 1
+
n1 n2
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Conclusion
Decision: Reject the null hypothesis
Are we done?
- How much importance should we attach to
this finding?
- Was the effect just barely significant
(p<.05)?
- What if the sig level was, p<.0001? Would
this be a larger effect?
32
Answers to the questions
Assess the quality of the Experiment
1)Were control procedures adequate?
2)Were variables defined appropriately?
3)Is a Type I error likely?
The t test is a robust statistic…
Means that assumptions can be violated without changing the rate
of type I or type II error.
33
Effect size
Convert t to a correlation coefficient
r=

t2
t2 +df
r=

(2.40)2
(2.40)2 +14
r = .54
r2 = .15
According to Cohen (1988), r ≥ .50 is considered a large effect
(.30 is a moderate effect and below .30 is a small effect).
The r2 of .15 indicates that the IV accounts for 15% of the variability
observed in the DV.
Online site for effect size calculator:
http://web.uccs.edu/lbecker/Psy590/escalc3.htm
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Effect size
Convert t to a correlation coefficient
r=

t2
t2 +df
r=

(2.40)2
(2.40)2 +14
r = .54
r2 = .15
Online site for effect size calculator:
http://web.uccs.edu/lbecker/Psy590/escalc3.htm
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