Download Mini Statistics Review

Mini Statistics Review
I. Descriptive statistics
A. Measures of central tendency
1. Mean: average value
2. Median: middle value
3. Mode: most frequent value
B. Measures of variability
1. Range: difference between highest and lowest scores
2. Standard deviation: reflects variability around mean
3. Variance: what you get when you square the standard deviation
C. Skewness: reflects how asymmetrical the distribution is
D. Kurtosis: reflects how flat or peaked the distribution is
E. Correlation: reflects degree of relationship between 2 variables (bivariate r)
F. Pictures (e.g., histograms, polygons, stem & leaf diagrams, scatterplots)
II. Inferential tests
A. One group tests
1. z-test for one group (know population variance)
2. t-test for one group (don't know population variance)
3. Chi-square goodness of fit test (data in categories)
B. Two group tests
1. z-test for two independent groups (know population variance)
2. t-test for two independent groups (don't know population variance)
3. t-test for two correlated (dependent or related) groups
(don't know population variance)
4. Mann-Whitney U-test (for two independent groups; ranked data)
5. Wilcoxon signed-ranks test (for two related groups; ranked data)
6. Chi-square test for independence (data in categories)
C. More than two independent groups
1. One-way analysis of variance (extension of t-test for independent groups)
2. Kruskal-Wallis test (ranked data)
3. Chi-square test for independence (data in categories)
D. More than two related groups: Repeated measures analysis of variance
E. More than one factor (factorial designs): Factorial analysis of variance
2
z-test for one group
Question: Do people who take a prep course score differently on a standardized
test than the general population of test-takers?
What we know:
Population mean = 500
Population standard deviation = 100
Data from those who take the prep course:
600
550
500
400
450
560
sample mean = 510
sample standard deviation = 74.83
N=6
z = .245, p > .05
Answer: Those who take the prep course do not score significantly different
than the general population of test-takers, z = .245, p > .05, two-tailed.
t-test for one group
Question: Does studying for an exam improve performance on the exam?
What we know: Population mean of those who don’t study = 10
Data from those who study:
10
12
14
14
10
14
15
9
10
sample mean = 12
sample standard deviation = 2.29
N=9
t(8) = 2.62, p < .05
Answer: People who study for the exam score significantly higher than people
who don’t study, t(8) = 2.62, p < .05, one-tailed.
3
Chi-square goodness of fit test
Question: Do students show preferences among different types of exam questions?
Data from our sample:
Exam Question Type
Multiple choice
True/false
Short answer
O
15
20
10
-------------------------------------------------E
15
15
15
Total
45
45
2 = 3.33, p > .05
Answer: Students did not show a significant preference among the different types of
exam questions, 2(2, N = 45) = 3.33, p > .05.
t-test for two independent groups
Question: Do people who take statistics exams in the morning perform differently on
the exam than people who take statistics exams in the afternoon?
What we know about the populations: nothing, although we are making
assumptions about it
Data from our samples:
Morning students
80
90
75
85
70
________
M = 80
SD = 7.91
N1 = 5
Afternoon students
75
85
65
80
70
________
M = 75
SD = 7.91
N2 = 5
t(8) = 1, p > .05
Answer: People who take exams in the morning do not perform significantly different
than those who take exams in the afternoon, t(8) = 1, p > .05, two-tailed.
4
Mann-Whitney U-test
Question: Is there a difference between the study times of people who do and do
not pass an exam?
Data from our samples:
Not Passing
2
4
5
11
_________
M = 5.5
SD = 3.87
N1 = 4
Passing
Rank
1
2
3
6
8
9
12
12
15
40
_________
M = 16
SD = 12.02
N2 = 6
Rank
4
5
7.5
7.5
9
10
U = 2, p < .05
Answer: The results indicate a significant difference in study time between those not
passing and passing the exam, U = 2, p < .05, two-tailed.
t-test for two dependent (correlated or related) groups
(repeated measures or matched subjects)
Question: Is a person's anxiety level before a test higher than their anxiety level
after the test?
Data from repeated (before and after) measurements:
Participant
pre-anx.
post-anx.
1
11
8
2
15
13
3
12
9
N = # of pairs = 5
4
8
8
5
9
7
_________
_________
M = 11
M= 9
SD = 2.74
SD = 2.35
t(4) = 3.65, p < .05
Answer: Anxiety level before the exam is significantly higher than
anxiety level after the exam, t(4) = 3.65, p < .05, one-tailed.
5
Wilcoxon signed ranks test
Question: Is there a difference between pretest and posttest in how many items
participants answer incorrectly?
Data from our sample:
Pair
A
B
C
D
E
F
G
Pretest
3
14
2
3
6
2
4
________
M = 4.86
SD = 4.26
Posttest
D
3
0
1
13
1
1
2
1
4
2
3
-1
3
1
________
M = 2.43
SD = 1.13
N = # of pairs = 7
Rank
6
2.5
2.5
5
2.5
2.5
Signed Rank
6
2.5
2.5
5
-2.5
2.5
# pairs with nonzero D = 6
R+ = 18.5
R- = 2.5
T = smaller of R+ and R- = 2.5
Answer: The results indicate no significant difference between pretest and posttest
scores, T = 2.5, p > .05, two-tailed.
Chi-square test for independence
Question: Is cereal preference related to gender? (Is the distribution of cereal
preferences the same for men and women?)
Data from our samples:
Cereal Preference
Oat squares
Fruit loops
Corn flakes
----------------------------------------------------Men O
5
10
5
20
E
6.67
6.67
6.67
---------------------------------------------------Women O
10
5
10
25
E
8.33
8.33
8.33
---------------------------------------------------15
15
15
n = 45
2 = 4.5, p > .05
Answer: Men and women do not differ significantly in their cereal preferences,
2(2, N = 45) = 4.5, p > .05
6
One-way analysis of variance
(for between subjects designs)
Question: Does the number of hours of sleep a person gets before an exam influence
the number of errors he/she makes on the exam?
Data from our samples:
no hours
1
2
0
1
3
2
M1 = 1.5
SD1 = 1.05
n1 = 6
HOURS OF SLEEP
4 hours
2
3
1
2
1
3
M2 = 2
SD2 = .89
n2 = 6
8 hours
0
1
0
2
0
0
M3 = .5
SD3 = .84
n3 = 6
Summary table for analysis:
Source
Between groups
Within groups
Total
df
2
15
17
SS
7
13
20
MS
3.5
.867
F__
3.5/.867 = 4.04
Answer: There is a significant difference among the mean errors made by people
getting different amounts of sleep before an exam, F(2, 15) = 4.04,
p < .05.
The magnitude of the effect (e.g., value of eta-squared) is .35, which means
that 35% of the variability in errors (the dependent variable) can be
explained by the number of hours of sleep (the independent variable).
Tukey HSD post hoc tests revealed that the mean difference between the 4
and 8 hours of sleep groups is significant, HSD = 1.4, p < .05.
7
Kruskal-Wallis test
Question: Is there a difference in the number of errors made on a task by people
working in different sized groups (small, medium, and large)?
Data from our samples:
Small (rank)
0
1.5
1
3.5
2
6
3
8.5
3
8.5
4
10
________ ______
M1 = 2.17
T1 = 38
SD1 = 1.47
n1 = 6
H=
GROUP SIZE
Medium (rank)
0
1.5
2
6
2
6
7
11
8
12.5
9
14
_______ _______
M2 = 4.67 T2 = 51
SD2 = 3.78
n2 = 6
Large
1
8
10
12
13
44
_________
M3 = 14.67
SD3 = 14.99
n3 = 6
(rank)
3.5
12.5
15
16
17
18
______
T3 = 82
2 = 5.98
Answer: The results indicate no significant difference among the groups,
2(2, N = 18) = 5.98, p > .05. (Had the value of the test statistic been
significant, the Mann-Whitney U-test could have been used to make pairwise
comparisons.)
8
Repeated measures analysis of variance
(for within subjects designs)
Question: Are there changes in performance on a quiz over time (i.e., over sessions)?
Data from our sample:
SESSIONS
Session 1 Session 2 Session 3
3
3
6
2
2
2
1
1
4
2
4
6
______
______
______
M1 = 2
M2 = 2.5
M3 = 4.5
SD1 = .82
SD2 = 1.29 SD3 = 1.91
Ss
1
2
3
4
n = # participants = 4
Summary table for analysis (2 versions):
Source
Between Ss
Within Ss
Sessions
Error
Total
Source
Between groups
Within groups
Between subjects
Error
Total
df
3
2
6
11
df
2
9
3
6
11
SS
12
MS
F
14
6
32
7
1
7
MS
7
F
SS
14
18
12
6
p__
< .05
7
p__
< .05
1
32
Answer: There is a significant change in quiz performance across sessions,
F(2, 6)= 7, p < .05.
Post hoc tests can be done to determine which sessions are significantly
different. In addition, one can calculate the magnitude of the effect.
9
Two-way analysis of variance
Questions:
(1) Is there a significant difference between the levels of factor A?
Is there a difference between the mean number of channel changes for men vs.
women? (main effect question)
(2) Is there a significant difference between the levels of factor B?
Is there a difference between the mean number of channel changes in the three
“length of wait” conditions? (main effect question)
(3) Is there an interaction between factors A and B?
Is the effect of length of wait on channel changing different for males and
females? (Does the effect of length of wait on channel changing depend on
gender?)
Data from our samples (borrowed fictitious data):
A1
male
B1
10 minutes
1
6
1
1
1
M =2
SD = 2.24
Factor B (length of wait)
B2
B3
20 minutes
30 minutes_
7
3
7
1
11
1
4
6
6
4
M =7
SD = 2.55
M =3
SD = 2.12
marginal
M = 4
SD = 3.09
Factor A
(gender)
----------------------------------------------------------0
0
0
A2
3
0
2
female
7
0
0
5
5
0
5
0
3
M =4
M =1
M=1
SD = 2.65
SD = 2.24
SD = 1.41
_______________________________________
marginal
marginal
marginal
M =3
M= 4
M =2
SD = 2.54
SD = 3.89
SD = 2.00
Mean
# of
Channel
Changes
7
6
5
4
3
2
1
marginal
M = 2
SD = 2.48
*
+
* A1 (male)
*
+
+ A2 (female)
-----------------------------------------------------------------10 min.
20 min.
30 min.
Factor B (length of wait)
10
Independent variables: gender (2 levels), length of wait (3 levels)
Dependent variable: # of channel changes within the allotted time period
Summary table for analysis:
Source
df
Between groups
5
Gender
1
Length of wait
2
Gender X Length of wait
2
Within groups
24
Total
29
SS
130
30
20
80
120
250
MS
30
10
40
5
F
30/5 = 6
10/5 = 2
40/5 = 8
p___
< .05
> .05
< .05
Answers:
The analysis revealed a significant main effect of gender, F(1, 24) = 6, p < .05, and
a significant interaction between gender and length of wait, F(2, 24) = 8, p < .05. The
effect of length of wait was not significant, F(2, 24) = 2, p > .05. The magnitude of
the effects (eta-squared values) for gender, length of wait, and the interaction were .12,
.08, and .32, respectively.
To further explore the interaction between gender and length of wait, two simple effects
tests were done, one for males and one for females. The simple effects test for males
revealed a significant difference among the length of wait means, F(2, 24) = 7, p < .05.
The test for the females revealed no significant difference, F(2, 24) = 3, p > .05.
To further explore the significant simple effect for males, Tukey HSD tests were used to
make pairwise comparisons between the length of wait means. Both comparisons
involving the mean for 20 minutes (i.e., 10 vs. 20 minutes and 20 vs. 30 minutes)
were significant (HSD = 3.53, p < .05).
11
Choosing the Appropriate Statistical Test
SCALE OF MEASUREMENT
NUMBER & TYPE OF
SAMPLES (GROUPS)
1 sample
Interval/ratio
t-test for 1 group
Ordinal
---
z-test for 1 group
Nominal
Chi-square
goodness of
fit test
Binomial test
2 independent samples
2 correlated samples
t-test for 2
indep. groups
Mann-Whitney
U-test
z-test for 2
indep. groups
Median test
t-test for 2
correlated groups
(related)
Wilcoxon signed
ranks test
Chi-square
test for
independ.
McNemar’s
test
Sign test
More than 2
independent
samples
one-way analysis of
variance
Kruskal-Wallis
test
More than one factor
factorial analysis of
variance
More than 2 correlated
samples
repeated measures
Friedman's rank
analysis of variance
test
Cochran Q
test
Relationship between
2 variables
Pearson correlation
Spearman
correlation
phi-coefficient
Kendall's Tau
Cramer
coefficient
---
Chi-square
test for
independ.
---
Chi-square
test for
independ.
Note: This table contains only a small number of the available tests.