Download Slide 1 - Interpersonal Research Laboratory

Review
• Ways to “see” data
–
–
–
–
–
–
Simple frequency distribution
Group frequency distribution
Histogram
Stem-and-Leaf Display
Describing distributions
Box-Plot
• Measures of central tendency
– Mean
– Median
– Mode
Review
• Measures of variability
– Range
– IQR
– Standard deviation
Compute a standard deviation
with the Raw-Score Method
• Previously learned the deviation formula
– Good to see “what's going on”
• Raw score formula
– Easier to calculate than the deviation formula
– Not as intuitive as the deviation formula
• They are algebraically the same!!
Raw-Score Formula
-1
Step 1: Create a table
Coffee
X
4
10
22
2
6
X2
Step 2: Square each value
Coffee
X
4
10
22
2
6
X
2
16
100
484
4
36
Step 3: Sum
Coffee
X
4
10
22
2
6
X = 44
X
2
16
100
484
4
36
2
 X = 640
Step 4: Plug in values
-1
N= 5
X = 44
 X2 = 640
Step 4: Plug in values
5
5-1
N= 5
X = 44
 X2 = 640
Step 4: Plug in values
44
5
5-1
N= 5
X = 44
 X2 = 640
Step 4: Plug in values
44
640
5
5-1
N= 5
X = 44
 X2 = 640
Step 5: Solve!
1936
44
640
5
5-1
Step 5: Solve!
1936
44
640 387.2
5
4
Step 5: Solve!
1936
44
64063.2387.2
5
5
Answer = 7.95
Practice
• You are interested in how citizens of the
US feel about the president. You asked
8 people to rate the president on a 10
point scale. Describe how the country
feels about the president -- be sure to
report a measure of central tendency
and the standard deviation.
8, 4, 9, 10, 6, 5, 7, 9
Central Tendency
8, 4, 9, 10, 6, 5, 7, 9
4, 5, 6, 7, 8, 9, 9, 10
Mean = 7.25
Median = (4.5) = 7.5
Mode = 9
Standard Deviation
-1
X
8
4
9
10
6
5
7
9
X2
64
16
81
100
36
25
49
81
= 58  = 452
Standard Deviation
452
58
8
8 - 1-1
X
8
4
9
10
6
5
7
9
X2
64
16
81
100
36
25
49
81
= 58  = 452
Standard Deviation
58
452
8
8 - 1-1
X
8
4
9
10
6
5
7
9
X2
64
16
81
100
36
25
49
81
= 58  = 452
Standard Deviation
452
420.5
7
-1
X
8
4
9
10
6
5
7
9
X2
64
16
81
100
36
25
49
81
= 58  = 452
Standard Deviation
2.12
-1
X
8
4
9
10
6
5
7
9
X2
64
16
81
100
36
25
49
81
= 58  = 452
Variance
• The last step in calculating a standard
deviation is to find the square root
• The number you are fining the square root
of is the variance!
Variance
S2=
Variance
S2=
-1
Practice
• Below are the test score of Joe and
Bob. What are their means, medians,
and modes? Who tended to have the
most uniform scores?
• Joe
80, 40, 65, 90, 99, 90, 22, 50
• Bob
50, 50, 40, 26, 85, 78, 12, 50
Practice
• Joe
22, 40, 50, 65, 80, 90, 90, 99
Mean = 67
• Bob
12, 26, 40, 50, 50, 50, 78, 85
Mean = 48.88
Practice
• Joe
22, 40, 50, 65, 80, 90, 90, 99
Median = 72.5
• Bob
12, 26, 40, 50, 50, 50, 78, 85
Median = 50
Practice
• Joe
22, 40, 50, 65, 80, 90, 90, 99
Mode = 90
• Bob
12, 26, 40, 50, 50, 50, 78, 85
Mode = 50
Practice
• Joe
22, 40, 50, 65, 80, 90, 90, 99
S = 27.51; S2 = 756.80
• Bob
12, 26, 40, 50, 50, 50, 78, 85
S = 24.26; S2 = 588.55
Thus, Bob’s scores were the most uniform
Review
• Ways to “see” data
–
–
–
–
–
–
Simple frequency distribution
Group frequency distribution
Histogram
Stem-and-Leaf Display
Describing distributions
Box-Plot
• Measures of central tendency
– Mean
– Median
– Mode
Review
• Measures of variability
– Range
– IQR
– Standard deviation
– Variance
What if. . . .
• You recently finished taking a test that you
received a score of 90 and the test scores
were normally distributed.
•
•
•
•
It was out of 200 points
The highest score was 110
The average score was 95
The lowest score was 90
Z-score
• A mathematical way to modify an
individual raw score so that the result
conveys the score’s relationship to the
mean and standard deviation of the other
scores
• Transforms a distribution of scores so they
have a mean of 0 and a SD of 1
Z-score
• Ingredients:
X
Raw score
Mean of scores
S
The standard deviation of scores
Z-score
What it does
• xTells you how far from the mean
you are and if you are > or < the mean
• S Tells you the “size” of this difference
Example
• Sample 1:
X=8
=6
S =5
Example
• Sample 1:
X=8
=6
S =5
Z score = .4
Example
• Sample 1:
X=8
=6
S = 1.25
Example
• Sample 1:
X=8
=6
S = 1.25
Z-score = 1.6
Example
• Sample 1:
X=8
=6
S = 1.25
Z-score = 1.6
Note: A Z-score tells you
how many SD above or
below a mean a specific
score falls!
Practice
• The history teacher Mr. Hand announced
that the lowest test score for each student
would be dropped. Jeff scored a 85 on his
first test. The mean was 74 and the SD
was 4. On the second exam, he made
150. The class mean was 140 and the SD
was 15. On the third exam, the mean was
35 and the SD was 5. Jeff got 40. Which
test should be dropped?
Practice
• Test #1
Z = (85 - 74) / 4 = 2.75
• Test #2
Z = (150 - 140) / 15 = .67
• Test #3
Z = (40 - 35) / 5 = 1.00
Practice
Time
(sec)
30
Distance
(feet)
6
Joey
40
8
Ross
25
4
Monica
45
10
Chandler
33
9
Rachel
Which challenge did Ross do best? Which did
Monica do best?
Time
(sec)
30
Distance
(feet)
6
Joey
40
8
Ross
25
4
Monica
45
10
Chandler
33
9
Rachel
Practice
Time
(sec)
30
Distance
(feet)
6
Joey
40
8
Ross
25
4
Monica
45
10
Chandler
33
9
Rachel
= 34.6
S = 7.96
= 7.4
S = 2.41
Practice
Rachel
Time
(sec)
30
Distance
(feet)
-.58
6
Joey
40
.68
8
Ross
25
-1.21
4
Monica
45
1.31
10
Chandler
33
-.20
9
= 34.6
S = 7.96
= 7.4
S = 2.41
Practice
Rachel
Time
(sec)
30
Distance
(feet)
-.58
6
-.58
Joey
40
.68
8
.25
Ross
25
-1.21
4
-1.66
Monica
45
1.31
10
1.08
Chandler
33
-.20
9
.66
= 34.6
S = 7.96
= 7.4
S = 2.41
Ross did worse in the throwing challenge than the
endurance and Monica did better in the endurance
than the throwing challenge.
Rachel
Time
(sec)
30
Distance
(feet)
-.58
6
-.58
Joey
40
.68
8
.25
Ross
25
-1.21
4
-1.66
Monica
45
1.31
10
1.08
Chandler
33
-.20
9
.66
= 34.6
S = 7.96
= 7.4
S = 2.41
Shifting Gears
Question
• A random sample of 100 students found:
– 56 were psychology majors
– 32 were undecided
– 8 were math majors
– 4 were biology majors
• What proportion were psychology majors?
• .56
Question
• A random sample of 100 students found:
– 56 were psychology majors
– 32 were undecided
– 8 were math majors
– 4 were biology majors
• What is the probability of randomly
selecting a psychology major?
Question
• A random sample of 100 students found:
– 56 were psychology majors
– 32 were undecided
– 8 were math majors
– 4 were biology majors
• What is the probability of randomly
selecting a psychology major?
• .56
Probabilities
• The likelihood that something will occur
• Easy to do with nominal data!
• What if the variable was quantitative?
Extraversion
50
40
30
20
Count
10
0
1.13
1.63
1.38
BFISUR
2.13
1.88
2.63
2.38
3.13
2.88
3.63
3.38
4.13
3.88
4.63
4.38
4.88
Openness to Experience
40
30
20
Count
10
0
1.60
2.20
2.00
BFIOPN
2.60
2.40
3.00
2.80
3.40
3.20
3.80
3.60
4.20
4.00
4.60
4.40
5.00
4.80
Neuroticism
40
30
20
Count
10
0
1.25
1.75
1.50
BFISTB
2.25
2.00
2.75
2.50
3.25
3.00
3.75
3.50
4.25
4.00
4.88
4.50
Probabilities

Normality frequently occurs in many situations of psychology,
and other sciences
COMPUTER PROG
• http://www.jcu.edu/math/isep/Quincunx/Qu
incunx.html
• http://webphysics.davidson.edu/Applets/G
alton/BallDrop.html
• http://www.ms.uky.edu/~mai/java/stat/Galt
onMachine.html
Next step
• Z scores allow us to modify a raw score so that it
conveys the score’s relationship to the mean
and standard deviation of the other scores.
• Normality of scores frequently occurs in many
situations of psychology, and other sciences
• Is it possible to apply Z score to the normal
distribution to compute a probability?
Theoretical Normal Curve
-3
-2
-1

1
2
3
Theoretical Normal Curve
-3
-2
-1

1
2
3
Theoretical Normal Curve
-3
-2
-1

1
2
3
Theoretical Normal Curve
-3
-2
-1

1
2
3
Note: A Z-score tells you how many SD above
or below a mean a specific score falls!
Theoretical Normal Curve
Z-scores
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
We can use the theoretical normal distribution to
determine the probability of an event.
For example, do you know the probability of getting a Z
score of 0 or less?
.50
Z-scores
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
We can use the theoretical normal distribution to
determine the probability of an event.
For example, you know the probability of getting a Z
score of 0 or less.
.50
Z-scores
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
With the theoretical normal distribution we know the
probabilities associated with every z score! The
probability of getting a score between a 0 and a 1 is
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
What is the probability of getting a score of 1 or
higher?
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
These values are given in Appendix Z
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
Mean to Z
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
Smaller Portion
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
Larger Portion
.84
.1587
Z-scores
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
• Above z = 2.25?
• Above z = -1.45
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
.2123
• Above z = 2.25?
• Above z = -1.45
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
.2123
• Above z = 2.25?
.0122
• Above z = -1.45
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
.2123
• Above z = 2.25?
.0122
• Above z = -1.45
.9265
Practice
• What proportion of this class would have
received an A on the last test if I gave A’s
to anyone with a z score of 1.25 or higher?
• .1056
Example: IQ
• Mean IQ = 100
• Standard deviation = 15
• What proportion of people have an IQ of
120 or higher?
Step 1: Sketch out question
-3
-2
-1

1
2
3
Step 1: Sketch out question
120
-3
-2
-1

1
2
3
Step 2: Calculate Z score
(120 - 100) / 15 = 1.33
120
-3
-2
-1

1
2
3
Step 3: Look up Z score in
Table
Z = 1.33
120
.0918
-3
-2
-1

1
2
3
Example: IQ
• A proportion of .0918 or 9.18 percent of
the population have an IQ above 120.
• What proportion of the population have an
IQ below 80?
Step 1: Sketch out question
-3
-2
-1

1
2
3
Step 1: Sketch out question
80
-3
-2
-1

1
2
3
Step 2: Calculate Z score
(80 - 100) / 15 = -1.33
80
-3
-2
-1

1
2
3
Step 3: Look up Z score in
Table
Z = -1.33
80
.0918
-3
-2
-1

1
2
3
Example: IQ
• Mean IQ = 100
• SD = 15
• What proportion of the population have an
IQ below 110?
Step 1: Sketch out question
-3
-2
-1

1
2
3
Step 1: Sketch out question
110
-3
-2
-1

1
2
3
Step 2: Calculate Z score
(110 - 100) / 15 = .67
110
-3
-2
-1

1
2
3
Step 3: Look up Z score in
Table
Z = .67
110
.7486
-3
-2
-1

1
2
3
Example: IQ
• A proportion of .7486 or 74.86 percent of
the population have an IQ below 110.
Finding the Proportion of the
Population Between Two
Scores
• What proportion of the population have IQ
scores between 90 and 110?
Step 1: Sketch out question
90
110
?
-3
-2
-1

1
2
3
Step 2: Calculate Z scores for
both values
• Z = (X -  ) / 
• Z = (90 - 100) / 15 = -.67
• Z = (110 - 100) / 15 = .67
Step 3: Look up Z scores
-.67
.2486
-3
-2
-1
.67
.2486

1
2
3
Step 4: Add together the two
values
-.67
.67
.4972
-3
-2
-1

1
2
3
• A proportion of .4972 or 49.72 percent of
the population have an IQ between 90 and
110.
• What proportion of the population have an
IQ between 110 and 130?
Step 1: Sketch out question
110
130
?
-3
-2
-1

1
2
3
Step 2: Calculate Z scores for
both values
• Z = (X -  ) / 
• Z = (110 - 100) / 15 = .67
• Z = (130 - 100) / 15 = 2.0
Step 3: Look up Z score
.67
2.0
.4772
-3
-2
-1

1
2
3
Step 3: Look up Z score
.67
2.0
.4772
.2486
-3
-2
-1

1
2
3
Step 4: Subtract
.4772 - .2486 = .2286
.67
2.0
.2286
-3
-2
-1

1
2
3
• A proportion of .2286 or 22.86 percent of
the population have an IQ between 110
and 130.
Finding a score when given a
probability
• IQ scores – what is the range of IQ scores
we expect 95% of the population to fall?
• “If I draw a person at random from this
population, 95% of the time his or her
score will lie between ___ and ___”
• Mean = 100
• SD = 15
Step 1: Sketch out question
95%
?
100
?
Step 1: Sketch out question
95%
2.5%
?
100
2.5%
?
Step 1: Sketch out question
Z = -1.96
Z = 1.96
95%
2.5%
?
100
2.5%
?
Step 3: Find the X score that
goes with the Z score
• Z score = 1.96
• Z = (X -  ) / 
• 1.96 = (X - 100) / 15
• Must solve for X
• X =  + (z)()
• X = 100 + (1.96)(15)
Step 3: Find the X score that
goes with the Z score
• Z score = 1.96
• Z = (X -  ) / 
• 1.96 = (X - 100) / 15
•
•
•
•
Must solve for X
X =  + (z)()
X = 100 + (1.96)(15)
Upper IQ score = 129.4
Step 3: Find the X score that
goes with the Z score
•
•
•
•
Must solve for X
X =  + (z)()
X = 100 + (-1.96)(15)
Lower IQ score = 70.6
Step 1: Sketch out question
Z = -1.96
Z = 1.96
95%
2.5%
70.6
100
2.5%
129.4
Finding a score when given a
probability
• “If I draw a person at random from this
population, 95% of the time his or her
score will lie between 70.6 and 129.4”
Practice
• GRE Score – what is the range of GRE
scores we expect 90% of the population to
fall?
• Mean = 500
• SD = 100
Step 1: Sketch out question
Z = -1.64
Z = 1.64
90%
5%
?
500
5%
?
Step 3: Find the X score that
goes with the Z score
• X =  + (z)()
• X = 500 + (1.64)(100)
• Upper score = 664
• X =  + (z)()
• X = 500 + (-1.64)(100)
• Lower score = 336
Finding a score when given a
probability
• “If I draw a person at random from this
population, 90% of the time his or her
score will lie between 336 and 664”
Practice
Practice
• The Neuroticism Measure
= 23.32
S = 6.24
n = 54
How many people likely have a neuroticism
score between 18 and 26?
Practice
• (18-23.32) /6.24 = -.85
• area = .3023
•
•
•
•
( 26-23.32)/6.26 = .43
area = .1664
.3023 + .1664 = .4687
.4687*54 = 25.31 or 25 people
SPSS
PROGRAM:
https://citrixweb.villanova.edu/Citrix/XenApp/auth/l
ogin.aspx
BASIC “HOW TO”
http://www.psychology.ilstu.edu/jccutti/138web/sps
s.html
SPSS “HELP” is also good
SPSS PROBLEM #1
• Page 65
• Data 2.1
• Turn in the SPSS output for
•
•
•
•
1) Mean, median, mode
2) Standard deviation
3) Frequency Distribution
4) Histogram