Download variance! - Interpersonal Research Laboratory

Formula
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
Note: This is the formula for both  and S
Step 1: Create a table
Coffee
X
4
10
22
2
6
X
2
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
N= 5
X = 44
 X2 = 640
Step 4: Plug in values
5
5
N= 5
X = 44
 X2 = 640
Step 4: Plug in values
44
5
5
N= 5
X = 44
 X2 = 640
Step 4: Plug in values
44
5
640
5
N= 5
X = 44
 X2 = 640
Step 5: Solve!
1936
44
640
5
5
Step 5: Solve!
1936
44
640 387.2
5
5
Step 5: Solve!
1936
44
64050.56
387.2
5
5
Answer = 7.11
Practice
• Use the raw score formula and find the
standard deviation of:
6, 3, 4, 10, 8
X
6
3
4
10
8
X = 31
2
X
36
9
16
100
64
 X2 = 225
31
5
225
5
N= 5
X = 31
 X2 = 225
2.56 =
1936
44
225 192.2
5
5
Ŝ
• What if we want to use a sample standard
deviation to estimate the population ?
• We need to make one small change to the
formula to do this
• You need to make the s an “unbiased
estimator”
Ŝ
• To do that you use Ŝ
– This provides an estimate of the populations
variability
Remember
Just “ - 1”
Ŝ
Remember
S=
Just “ - 1”
Ŝ=
-1
Why?
• The first formula is biased -- its answer
tends to be too small
• Don’t worry about why -- unless you want
too!!
Practice!
• Below is data from 5 people in this class.
What is the estimated standard deviation
of all the students in this class? Use the Ŝ
raw score formula.
• Neuroticism scores
12, 15, 22, 10, 9
X
12
15
22
10
9
X = 68
2
X
144
225
484
100
81
2
 X = 1034
68
1034 5
5-1
N= 5
X = 68
 X2 = 1034
5.22 =
1936
44
1034 924.8
5
4
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!
 2 = population variance
Ŝ 2 = sample variance used to estimate  2
Variance
S 2,  2 =
Ŝ2=
Variance
S 2,  2 =
Ŝ2=
-1
There are 12 different formulas!
• Standard Deviation
– Deviation Formula , S, Ŝ
– Raw Formula , S, Ŝ
• Variance
– Deviation Formula  2, S 2, Ŝ 2
– Raw Formula  2, S 2, Ŝ 2
Review -- Important Formulas
• Standard Deviation -- Deviation Formula
 =
Ŝ=
Review -- Important Formulas
• Standard Deviation -- Deviation Formula
Review -- Important Formulas
• Variance -- Deviation Formula
2 =
Ŝ2 =
Review -- Important Formulas
• Variance -- Deviation Formula
2
Review -- Important Formulas
• Standard Deviation -- Raw Formula
 and S =
Ŝ=
-1
Review -- Important Formulas
• Variance -- Raw Formula
2 and S2 =
Ŝ2 =
-1
How to know which to use
• 1) Does the question want a standard
deviation or a variance (most of the time
standard deviations are used)
• 2) Is the group of scores a sample or
population?
• 3) If it’s a sample, do you want to
generalize the findings to a population?
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
• Want to use Ŝ
Standard Deviation
• Want to use Ŝ
-1
X
8
4
9
10
6
5
7
9
X2
64
16
81
100
36
25
49
81
= 58  = 452
Standard Deviation
• Want to use Ŝ
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
• Want to use Ŝ
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
• Want to use Ŝ
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
• Want to use Ŝ
2.12
-1
X
8
4
9
10
6
5
7
9
X2
64
16
81
100
36
25
49
81
= 58  = 452
Boxplots
• The boxplot graphically displays three
different characteristics of the distribution
– Extreme scores
– Interquartile range
– Median
Boxplot
40
30
20
10
0
N=
100
NEUR
Boxplot
40
30
Interquartile range
25th - 75th percentile
20
10
0
N=
100
NEUR
Boxplot
40
30
Extreme Scores
20
10
0
N=
100
NEUR
Boxplot
40
30
Median
20
10
0
N=
100
NEUR
Boxplot
40
Skew -- Look at the
“whiskers” to
determine if the
distribution is
skewed
30
20
10
0
N=
100
NEUR
Create a boxplot
• Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Create a boxplot
• Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median =
25th =
75th =
Lowest =
Highest =
Create a boxplot
• Create a boxplot with this data set
2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99
Median = 22.5
25th = 6
75th = 62
Lowest = 2
Highest = 99
120
100
80
60
40
20
0
-20
N=
12
VAR00004
Neuroticism
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
N=
130
130
CNEUR
MNEUR
Extraversion
5.0
4.5
72
152
4.0
3.5
3.0
2.5
106
120
2.0
N=
130
130
CEXTRA
MEXTRA
Conscientiousness
5
4
3
2
1
0
N=
128
128
CCON
MCON
Which distribution has a positive skew?
120
C
100
E
A
80
B
60
D
40
20
0
-20
N=
9
9
9
9
9
VAR00002
VAR00003
VAR00004
VAR00005
VAR00006
Which distribution has a negative skew?
120
C
100
E
A
80
B
60
D
40
20
0
-20
N=
9
9
9
9
9
VAR00002
VAR00003
VAR00004
VAR00005
VAR00006
Which distribution is most compact?
120
C
100
E
A
80
B
60
D
40
20
0
-20
N=
9
9
9
9
9
VAR00002
VAR00003
VAR00004
VAR00005
VAR00006
Which distribution has a median close to 25?
120
C
100
E
A
80
B
60
D
40
20
0
-20
N=
9
9
9
9
9
VAR00002
VAR00003
VAR00004
VAR00005
VAR00006
Which distribution is most symmetrical?
120
C
100
E
A
80
B
60
D
40
20
0
-20
N=
9
9
9
9
9
VAR00002
VAR00003
VAR00004
VAR00005
VAR00006
Which distribution has has the largest range?
120
C
100
E
A
80
B
60
D
40
20
0
-20
N=
9
9
9
9
9
VAR00002
VAR00003
VAR00004
VAR00005
VAR00006
Effect Size Index
• On average, males are taller than females.
• On average, females are less extraverted
than males.
• On average, middle children tend to be
more rebellious than first born children.
Effect Size Estimate
• Gives a mathematical way to answer the
question: How much more?
Effect Size Estimate
• Ingredients:
Two different means

an estimated SD of both samples
Effect Size Estimate

* Note: This is used to examine differences between MEANS
Effect Size Estimate

** When calculating put the bigger mean first (so d will
always be positive)
• d can range from 0 upward
• Small effect
• Medium effect
• Large effect
d = .20
d = .50
d = .80
Heights
• Do you think the difference in males and
females heights is small, medium, or
large?
•
Women = 64.6 inches
Men
= 69.8 inches
 = 2.8 inches

69.8 - 64.6
2.8
69.8 - 64.6
2.8
= 1.86
Rebelliousness
• Do you think the difference in
rebelliousness between middle and first
born children is large?
•
Middle = 19.45
First =
9.88
 = 26.68
19.45 - 9.88
26.68
= .36
The mean difference for rebelliousness is 9.57
The mean difference for height is 5.2
Why does Height have a bigger effect size, but smaller mean
difference?
Rebelliousness
Heights
19.45 - 9.88
= .36
26.68
69.8 - 64.6
2.8
= 1.86
Practice
• Make a box plot using the simple
frequency distribution of Satisfaction with
Life scores on page 27
Practice
• 4.15
• 4.18