Download Z-scores and Standardized Distributions

End of Year Review
PSY 211
12-3-07
I suggest prioritizing your studying to focus on the
concepts presented in this review and in the study
guide; however, an “A” performance requires greater
depth of knowledge.
A. Unit 1: Basic Concepts
Distribution Shapes
 Positive skew
o Most scores low, tail at the high end
 Negative skew
o Most scores high, tail at the low end
 Symmetrical
What would be the shape of a distribution for…
Attractiveness scores?
Annual number of sexual partners?
Depressive symptoms?
Central Tendency
 Mean:
o M for sample
o μ for population
o Sum of scores divided by n (sample size)
o ΣX / n
 Median:
o Middle number in a distribution
o If two middle numbers, average them
 Mode:
o Most popular or most common choice
o Can have multiple modes
o If all have the same frequency, no mode
Some quiz scores:
1 1 3 4 4 5
Mean?
Median?
Mode?
Variability
 Standard deviation: average distance scores
vary from the mean
o s for sample
o σ for population
o SD for population or sample
 Variance: standard deviation squared,
commonly used in various calculations
o s2 for sample
o σ2 for population
Would the Lions rather have a football punter with
high or low variability in kicking distance?
What’s the approximate standard deviation for a
7-point “happiness” scale?
B. Unit 2: Correlation and Regression
 Statistics used for describing the relationship
between continuous variables
Correlation
 Pearson’s r, or just plain “r”
 Sign indicates direction
o Positive or direct
o Negative or inverse
 Number indicates magnitude
Indicate the direction and magnitude of these
correlations:
r = 0.12
r = -0.68
r = 1.37
r = 0.08
r = -0.49
Correlation Guide:
impossible!
large
medium
small
zero / near-zero
small
medium
large
impossible!
more extreme than +1.00
+.50 to +1.00
+.30 to +.49
+.10 to +.29
0 ish
-.10 to -.29
-.30 to -.49
-.50 to -1.00
more extreme than -1.00
positive
(direct)
negative
(inverse)
a = +0.8ish, b = -0.3ish, c = -1.0, d = 0.0
stick = 1.0, hot dog = 0.8, sub sandwich = 0.5, egg = 0.3, circle = 0.0
 We spent a lot of time on correlation, so make
sure to review the notes in detail
Regression (R)
 Statistical technique for finding the best fitting
straight line for a set of data
Model Summary
Model
1
R
.293a
R Square
.086
Adjusted
R Square
.083
Std. Error of
the Estimate
1.1061
a. Predictors: (Constant), 59. Agreeableness
Coeffi cientsa
Model
1
(Const ant)
59. Agreeableness
Unstandardized
Coeffic ient s
B
St d. Error
3.325
.278
.288
.052
St andardiz ed
Coeffic ient s
Beta
.293
t
11.964
5.522
Sig.
.000
.000
a. Dependent Variable: 37. Helpfulness
predicted Helpfulness = .288(Agreeableness) + 3.325
What if my agreeableness score is 1?
 When using one predictor, r and R are the same
 Multiple R: Examine how well multiple
independent variables combine to predict the
dependent variable
From Homework #7…
Model Summary
Model
1
R
.314a
R Square
.099
Adjusted
R Square
.093
Std. Error of
the Estimate
1.5667
a. Predictors: (Constant), 78. Life Satisfaction, 55.
Self-reported Intelligence
ANOVAb
Model
1
Regres sion
Residual
Total
Sum of
Squares
86.966
792.838
879.804
df
2
323
325
Mean Square
43.483
2.455
F
17.715
Sig.
.000a
a. Predic tors : (Const ant), 78. Life Satisfaction, 55. Self-report ed Intelligence
b. Dependent Variable: 75. Overwhelmed
Coefficientsa
Model
1
(Constant)
55. Self-reported
Intelligence
78. Life Satisfaction
Unstandardized
Coefficients
B
Std. Error
6.708
.537
Standardized
Coefficients
Beta
t
12.495
Sig.
.000
-.170
.083
-.109
-2.047
.041
-.340
.063
-.286
-5.393
.000
a. Dependent Variable: 75. Overwhelmed
C. Unit 3: Probability and Standard Scores
Probability
 Calculations are de-emphasized on the exam,
and no appendices from the book will be used
 Do know the 68-95-99 rule
T scores 20
IQ scores 55
30
70
40
85
50
100
60
115
70
130
80
145
 I suggest re-reading probability in detail if you
are struggling with issues, such as “p < .05” or
determining whether a result is significant or
reliable
 Probability is the basis for inferential statistics. A
result is defined as statistically significant or
reliable when there is less than a 5% probability
that it would have occurred due to sampling
error or “chance”
 p values can be thought of as the probability of
incorrectly rejecting the null hypothesis, the
probability of making a Type I error, or the
probability that results might occur by chance
 p values are like golf scores; low is good
What proportion of the population has an IQ between
100 and 115?
Which of the following correlations are statistically
significant?
r = 0.39, p = 0.23
r = 0.10, p = 0.02
r = 0.16, p = 0.06
Standard Scores
Z score = X – Mean
SD
X = Meandesired + (Z score)*(SDdesired)
 Z scale: M = 0, SD = 1
 T scale: M = 50, SD = 10
 IQ scale: M = 100, SD = 115
Which of the following scores is more extreme?
Z = 1.5
T = 30
IQ = 125
And these?
Z = -2.1
T = 70
IQ = 70
D. Unit 4: Introduction to Hypothesis Testing
 Use sample data to determine whether the
findings are reliably – likely to occur at the
population level
 p value indicates the probability that the findings
are due to sampling error or “chance”
 Low p value, means low probability that findings
are due to chance, so findings must be reliable,
likely to generalize to the population
Z statistic
 Can be used to compare an individual score or a
sample mean to the population mean, when the
population standard deviation is know
 Critical Z value (where p < .05) always occurrs
at a Z of ±1.96
Single-sample t-test
 Can be used to compare a sample mean to the
population mean, when only the sample
standard deviation is know (population standard
deviation is unknown)
 Critical t value (where p < .05) depends on the
degrees of freedom (df; n – 1) because the
shape of the t distribution changes, depending
on sample size; smaller samples require a
higher t value to obtain a significant or reliable
result
Between-group t-test
 Compare two samples (two groups within a
sample) to determine whether group differences
are reliable enough to likely generalize to the
population
 Critical t depends on df, as for all t-tests
Within-subject t-test
 a.k.a. Repeated-measures t-test
 Compare the same group of people across two
different time points to determine whether
scores reliably differ over time
 Participants act as own control, so power is a bit
higher than with other t-tests
 Sometimes also used for matched samples,
such as twin or Alzheimer’s studies
 Critical t depends on df, as for all t-tests
Effect Size
 The p value describes whether a result is
reliable and likely to generalize to the
population, but it is highly dependent on sample
size and does not describe how big or important
a difference is
 r, r2, and Cohen’s d are measures of effect size,
indicating the magnitude of an effect
 p values describe whether a result is reliable,
whereas effect sizes indicate how big or small
the group differences may be
Cohen’s d =
(M 1  M 2 )
s
What type of test would be used to compare gender
differences in neuroticism?
What type of test would be used to compare CMU
students to the overall population in terms of IQ?
Happiness scores improve from 4.7 to 6.2 during the
course of therapy (SD = 1.5). What is Cohen’s d?
large
medium
small
zero / near-zero
small
medium
large
+.80 to infinity
+.50 to +.79
+.20 to +.49
0 ish
-.20 to -.49
-.50 to -.79
-.80 to -infinity
E. Unit 5: Advanced Topics in Hypothesis Testing
 t-tests are often unsatisfying because
researchers want to examine more than two
groups or time points
ANOVA
 Similar to the between-group t-test but can
examine differences in scores across 2 or more
groups
 Factor vs. level
Repeated-measures ANOVA
 Similar to the repeated-measures t-test but can
examine differences in scores across 2 or more
time points
F statistic
 Both types of ANOVA rely
on the F statistic, which is
somewhat similar to the
other distributions learned
about, but F is always
positive
increase
(higher)
decrease
(lower)
 F = total variability
chance variability
 In other words, the F statistics indicates whether
the variability in scores is more than would be
expected to occur due to chance
o Also, recall that the repeated-measures
ANOVA controls for chance variability due
to individual differences, so this test is
more powerful
 Why would variability be greater than chance?
o Treatment or experimental manipulation
causes scores to change, increasing
variability
 Critical F value (where p < .05) depends on two
df values, one related to sample size and one
related to number of groups
o Recall: the critical t value was always
greater than ±1.96 (and usually
somewhere in the 2’s)
o The critical F value is always greater than
1.57 (and usually somewhere between 2
and 10)
 When F is near 1, the results are likely due to
chance, and when F is greater than the critical
value, the results are unlikely due to chance,
probably due to treatment
Are the following F values statistically significant?
F = 0.93
F = 1.28
F = 4.32
Post hoc Tests
 F tests only tell whether a result is significant;
they signal that some reliable difference has
occurred
 Post hoc tests help clarify which groups (or time
points) differ; they tell where the differences
occur
 Least Significant Difference (LSD), Bonferroni,
and others
Chi-square Test for Goodness of Fit
 Involves a single categorical variable
 Used to determine whether proportions
(frequencies or percentages) obtained in the
sample are similar to those that are
hypothesized or expected to occur in the
population
Chi-square Test for Independence
 Independent (unrelated) vs. dependent (related)
 Determines whether one categorical variable
can be used to predict another categorical
variable
What test would be used to determine whether…
A sample is overly White?
Favorite color relates to happiness?
Treatment is effective at several follow-ups?
Gender is related to political party?
Attractiveness is related to intelligence?
F. SPSS Output
 6 of the 30 multiple choice questions on both the
next exam and on the make-up exam involve
SPSS Output, so know it well
ANOVA
 Most people answered questions correctly on
the ANOVA assignment, suggesting that this is
well-understood
I was curious whether these groups of people differ
on quality of parent relationship:
 Parent as hero (mom or dad)
 Family member, non-parent as hero (sibling or
other family member)
 Non-family member as hero (all others)
Descriptives
83. Parent Relationship
N
Parent
Other Family
Non-family
Total
179
59
88
326
Mean
5.994
5.407
5.295
5.699
Std. Deviation
1.2384
1.6307
1.5841
1.4471
Std. Error
.0926
.2123
.1689
.0801
95% Confidence Interval for
Mean
Lower Bound Upper Bound
5.812
6.177
4.982
5.832
4.960
5.631
5.542
5.857
ANOVA
83. Parent Relationship
Between Groups
W ithin Groups
Total
Sum of
Squares
34.990
645.550
680.540
df
2
323
325
Mean Square
17.495
1.999
F
8.754
Sig.
.000
Minimum
2.0
1.0
1.0
1.0
Maximum
7.0
7.0
7.0
7.0
Post Hoc Tests
Multiple Comparisons
Dependent Variable: 83. Parent Relationship
LSD
(I) hero1
Parent
Other Family
Non-family
(J) hero1
Other Family
Non-family
Parent
Non-family
Parent
Other Family
Mean
Difference
(I-J)
.5876*
.6990*
-.5876*
.1113
-.6990*
-.1113
Std. Error
.2122
.1841
.2122
.2379
.1841
.2379
Sig.
.006
.000
.006
.640
.000
.640
95% Confidence Interval
Lower Bound Upper Bound
.170
1.005
.337
1.061
-1.005
-.170
-.357
.579
-1.061
-.337
-.579
.357
*. The mean difference is significant at the .05 level.
Was the F test significant?
Are people who describe a parent as their hero
reliably more likely to have better parent
relationships than…
people who described a different family member as
a hero?
people who described a non-family member as a
hero?
Do people who choose a non-parent family member
and a non-family member differ in terms of parental
relationships?
Repeated-measures ANOVA
Does treatment for anxiety work?
At 6-month and 1-year follow-up?
Descriptive Statistics
Mean
8.7000
7.5000
7.8667
8.0000
Intake
Discharge
6-month
12-month
Std. Deviation
1.23596
1.10641
1.22428
1.11417
N
30
30
30
30
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
factor1
Error(factor1)
Sphericity Assumed
Greenhous e-Geiss er
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhous e-Geiss er
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
22.700
22.700
22.700
22.700
116.300
116.300
116.300
116.300
df
3
2.559
2.827
1.000
87
74.200
81.993
29.000
Mean Square
7.567
8.872
8.029
22.700
1.337
1.567
1.418
4.010
F
5.660
5.660
5.660
5.660
Pairwise Comparisons
Measure: MEASURE_1
(I) factor1
1
2
3
4
(J) factor1
2
3
4
1
3
4
1
2
4
1
2
3
Mean
Difference
(I-J)
1.200*
.833*
.700*
-1.200*
-.367
-.500
-.833*
.367
-.133
-.700*
.500
.133
Std. Error
.273
.353
.304
.273
.320
.287
.353
.320
.243
.304
.287
.243
a
Sig.
.000
.025
.029
.000
.261
.092
.025
.261
.588
.029
.092
.588
95% Confidence Interval for
a
Difference
Lower Bound Upper Bound
.642
1.758
.112
1.554
.079
1.321
-1.758
-.642
-1.021
.287
-1.086
.086
-1.554
-.112
-.287
1.021
-.631
.364
-1.321
-.079
-.086
1.086
-.364
.631
Based on estimated marginal means
*. The mean difference is significant at the .05 level.
a. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no
adjustments).
Sig.
.001
.003
.002
.024
Chi-square Test for Independence
Does sleeping position (back, side, stomach, fetal)
relate to statistics enjoyment (Yes/No)?
9. Enjoy Stats * 30. Sleeping Position Crosstabulation
9. Enjoy
Stats
No
Yes
Total
Back
20
16.5
2
5.5
22
22.0
Count
Expected Count
Count
Expected Count
Count
Expected Count
30. Sleeping Position
Side
Stomach
Fetal Position
90
57
78
95.4
61.6
71.4
37
25
17
31.6
20.4
23.6
127
82
95
127.0
82.0
95.0
Chi-Square Te sts
Pearson Chi-Square
Lik elihood Ratio
Linear-by-Linear
As soc iation
N of Valid Cases
Value
8.031a
8.775
.478
3
3
As ymp. Sig.
(2-sided)
.045
.032
1
.489
df
326
a. 0 c ells (.0% ) have expected count less than 5. The
minimum expected count is 5.47.
Total
245
245.0
81
81.0
326
326.0