You can calculate: Central tendency Variability You could graph the data
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Transcript of You can calculate: Central tendency Variability You could graph the data
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Smile
Jerry 10
Elan 6
George 8
Newman 9
Kramer 7
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Smile
Jerry 10
Elan 6
George 8
Newman 9
Kramer 7
You can calculate:
Central tendency
Variability
You could graph the data
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Talk
Jerry 5
Elan 1
George 3
Newman 4
Kramer 2
You can calculate:
Central tendency
Variability
You could graph the data
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Bivariate Distribution
Smile Talk
Jerry 10 5
Elan 6 1
George 8 3
Newman 9 4
Kramer 7 2
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Positive Correlation
Smile Talk
Jerry 10 5
Elan 6 1
George 8 3
Newman 9 4
Kramer 7 2
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Positive Correlation
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
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Regression Line
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
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Correlation
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
r = 1.00
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Regression Line
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
. . .. .
r = .64
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Regression Line
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
. .. .
r = .64
.
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Practice
Smile Talk
Jerry 9 5
Elan 2 1
George 5 3
Newman 4 4
Kramer 3 2
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Regression Line
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
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Regression Line
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
.
.. ..
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Regression Line
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Smil
e
.
.. ..
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Frown Talk
Jerry 10 2
Elan 6 6
George 8 4
Newman 9 3
Kramer 7 5
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Frown Talk
Jerry 10 2
Elan 6 6
George 8 4
Newman 9 3
Kramer 7 5
Negative Correlation
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Negative Correlation
0
2
4
6
8
10
12
2 3 4 5 6
Talk
Fro
wn
r = - 1.00
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Negative Correlation
0
2
4
6
8
10
12
1 2 3 4 5
Talk
Fro
wn
.
.
. .. r = - .85
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Gas in car Talk
Jerry 10 8
Elan 6 9
George 8 3
Newman 9 4
Kramer 7 3
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Gas in car Talk
Jerry 10 8
Elan 6 9
George 8 3
Newman 9 4
Kramer 7 3
Zero Correlation
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Zero Correlation
0
2
4
6
8
10
12
3 4 5 6 7 8 9
Talk
Gas
in c
ar
.
... .r = .00
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Correlation Coefficient
• The sign of a correlation (+ or -) only tells you the direction of the relationship
• The value of the correlation only tells you about the size of the relationship (i.e., how close the scores are to the regression line)
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Excel Example
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• Which is a bigger effect?
r = .40 or r = -.40
How are they different?
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Interpreting an r value
• What is a “big r”
• Rule of thumb:
Small r = .10
Medium r = .30
Large r = .50
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Practice
• Do you think the following variables are positively, negatively or uncorrelated to each other?
• Alcohol consumption & Driving skills• Miles of running a day & speed in a foot race• Height & GPA• Forearm length & foot length• Test #1 score and Test#2 score
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Statistics Needed
• Need to find the best place to draw the regression line on a scatter plot
• Need to quantify the cluster of scores around this regression line (i.e., the correlation coefficient)
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Covariance• Correlations are based on the statistic called
covariance
• Reflects the degree to which two variables vary together– Expressed in deviations measured in the original
units in which X and Y are measured
1
))((
N
YYXXCOVXY
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• Note how it is similar to a variance– If Ys were changed to Xs it would be s2
• How it works (positive vs. negative vs. zero)
1
))((
N
YYXXCOVXY
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Computational formula
1
NN
YXXY
COVXY
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Smile Talk
Jerry 9 5
Elan 2 1
George 5 3
Newman 4 4
Kramer 3 2
1
NN
YXXY
COVXY
Ingredients:
∑XY
∑X
∑Y
N
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Smile (Y)
Talk (X)
XY
Jerry 9 5
Elan 2 1
George 5 3
Newman 4 4
Kramer 3 2
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Smile (Y)
Talk (X)
XY
Jerry 9 5 45
Elan 2 1 2
George 5 3 15
Newman 4 4 16
Kramer 3 2 6
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Smile (Y)
Talk (X)
XY
Jerry 9 5 45
Elan 2 1 2
George 5 3 15
Newman 4 4 16
Kramer 3 2 6
∑ = 23
∑ = 15
∑ = 84
N = 5
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∑XY = 84
∑Y = 23
∑X = 15
N = 5
1
NN
YXXY
COVXY
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∑XY = 84
∑Y = 23
∑X = 15
N = 5
1
84
NN
YX
COVXY
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∑XY = 84
∑Y = 23
∑X = 15
N = 5
1
)23(1584
NNCOVXY
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∑XY = 84
∑Y = 23
∑X = 15
N = 5
155
)23(1584
XYCOV
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∑XY = 84
∑Y = 23
∑X = 15
N = 5
155
)23(1584
75.3
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Problem!
• The size of the covariance depends on the standard deviation of the variables
• COVXY = 3.75 might occur because– There is a strong correlation between X and
Y, but small standard deviations
– There is a weak correlation between X and Y, but large standard deviations
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Solution
• Need to “standardize” the covariance
• Remember how we standardized single scores
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Correlation
YX
XY
SS
COVr
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Smile (Y)
Talk (X)
XY
Jerry 9 5
Elan 2 1
George 5 3
Newman 4 4
Kramer 3 2
SY =2.70 SX =1.58
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Correlation
YX
XY
SS
COVr
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Correlation
)70.2(58.1
75.388.
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Practice
• You are interested in if candy intake is related to childhood depression. You collect data from 5 children.
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Practice
Candy Depression
Charlie 5 55
Augustus 7 43
Veruca 4 59
Mike 3 108
Violet 4 65
Scandy = 1.52 Sdepression = 24.82
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Practice
Candy
(X)
Depression
(Y)
XY
Charlie 5 55 275
Augustus 7 43 301
Veruca 4 59 236
Mike 3 108 324
Violet 4 65 260
∑
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Practice
Candy
(X)
Depression
(Y)
XY
Charlie 5 55 275
Augustus 7 43 301
Veruca 4 59 236
Mike 3 108 324
Violet 4 65 260
∑ 23 330 1396
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∑XY = 1396
∑Y = 330
∑X = 23
N = 5
1
NN
YXXY
COVXY
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∑XY = 1396
∑Y = 330
∑X = 23
N = 5
155
)330(231396
5.30
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Correlation
YX
XY
SS
COVr
COV = -30.5
Sx = 1.52
Sy = 24.82
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Correlation
)82.24(52.1
5.3081.
COV = -30.5
Sx = 1.52
Sy = 24.82
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Hypothesis testing of r
• Is there a significant relationship between X and Y (or are they independent)– Like the X2
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Steps for testing r value
• 1) State the hypothesis
• 2) Find t-critical
• 3) Calculate r value
• 4) Calculate t-observed
• 5) Decision
• 6) Put answer into words
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Practice
• Determine if candy consumption is significantly related to depression.– Test at alpha = .05
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Practice
Candy Depression
Charlie 5 55
Augustus 7 43
Veruca 4 59
Mike 3 108
Violet 4 65
Scandy = 1.52 Sdepression = 24.82
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Step 1
• H1: r is not equal to 0
– The two variables are related to each other
• H0: r is equal to zero
– The two variables are not related to each other
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Step 2
• Calculate df = N - 2
• Page 747– First Column are df– Look at an alpha of .05 with two-tails
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t distributiondf = 3
0
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t distribution
tcrit = 3.182tcrit = -3.182
0
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t distribution
tcrit = 3.182tcrit = -3.182
0
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Step 3
)82.24(52.1
5.3081.
COV = -30.5
Sx = 1.52
Sy = 24.82
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Step 4
• Calculate t-observed
21
2
r
Nrt
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Step 4
• Calculate t-observed
2)81.(1
2581.
t
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Step 4
• Calculate t-observed
2)81.(1
2581.39.2
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Step 5
• If tobs falls in the critical region:
– Reject H0, and accept H1
• If tobs does not fall in the critical region:
– Fail to reject H0
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t distribution
tcrit = 3.182tcrit = -3.182
0
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t distribution
tcrit = 3.182tcrit = -3.182
0
-2.39
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Step 5
• If tobs falls in the critical region:
– Reject H0, and accept H1
• If tIf tobsobs does not fall in the critical region: does not fall in the critical region:
– Fail to reject HFail to reject H00
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Step 6
• Determine if candy consumption is significantly related to depression.– Test at alpha = .05
• Candy consumption is not significantly related to depression– Note: this finding is due to the small sample
size
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Practice
• Is there a significant (.05) relationship between aggression and happiness?
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Aggression Happiness
Mr. Blond 10 9
Mr. Blue 20 4
Mr. Brown 12 5
Mr. Pink 16 6
Mean aggression = 14.50; S2aggression = 19.63
Mean happiness = 6.00; S2happiness = 4.67
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Answer
• Cov = -7.33• r = -.76
• t crit = 4.303
• Thus, fail to reject Ho
• Aggression was not significantly related to happiness
)76.(1
2476.65.1
2
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Practice
• Situation 1• Based on a sample of 100 subjects you find the
correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.
• Situation 2• Based on a sample of 600 subjects you find the
correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.
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Step 1• Situation 1
• H1: r is not equal to 0– The two variables are related to each other
• H0: r is equal to zero– The two variables are not related to each other
• Situation 2
• H1: r is not equal to 0– The two variables are related to each other
• H0: r is equal to zero– The two variables are not related to each other
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Step 2
• Situation 1
• df = 98
• t crit = +1.985 and -1.984
• Situation 2
• df = 598
• t crit = +1.96 and -1.96
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Step 3
• Situation 1
• r = .15
• Situation 2
• r = .15
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Step 4
• Situation 1
• Situation 2
2)15(.100
210015.5.1
215.1
260015.71.3
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Step 5
• Situation 1• If tobs falls in the critical region:
– Reject H0, and accept H1
• If tobs does not fall in the critical region:– Fail to reject H0
• Situation 2• If tobs falls in the critical region:
– Reject H0, and accept H1
• If tobs does not fall in the critical region:– Fail to reject H0
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Step 6
• Situation 1• Based on a sample of 100 subjects you find the
correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.
• There is not a significant relationship between extraversion and happiness
• Situation 2• Based on a sample of 600 subjects you find the
correlation between extraversion is happiness is r=.15. Determine if this value is significantly different than zero.
• There is a significant relationship between extraversion and happiness.