chapter 5 summarizing bivariate data
DESCRIPTION
Chapter 5 Summarizing Bivariate Data. Correlation. Variables : Response variable (y) measures an outcome (dependent) Explanatory variable (x) helps explain or influence changes in a response variable (independent). Suppose we found the age and weight of a sample of 10 adults. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/1.jpg)
Chapter 5Summarizing
Bivariate Data
Correlation
![Page 2: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/2.jpg)
Variables:
Response variable (y) measures an outcome (dependent)
Explanatory variable (x) helps explain or influence changes in
a response variable (independent)
![Page 3: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/3.jpg)
Suppose we found the age and weight of a sample of 10 adults.
Create a scatterplot of the data below.
Is there any relationship between the age and weight of these adults?
Age 24 30 41 28 50 46 49 35 20 39
Wt 256 124 320 185 158 129 103 196 110 130
![Page 4: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/4.jpg)
Does there seem to be a relationship between age and weight of these adults?
![Page 5: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/5.jpg)
Suppose we found the height and weight of a sample of 10 adults.
Create a scatterplot of the data below.
Is there any relationship between the height and weight of these adults?
Ht 74 65 77 72 68 60 62 73 61 64
Wt 256 124 320 185 158 129 103 196 110 130
Is it positive or negative? Weak or strong?
![Page 6: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/6.jpg)
Does there seem to be a relationship between height and weight of these adults?
![Page 7: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/7.jpg)
When describing relationships between two variables, you
should address:
• Direction (positive, negative, or neither)
• Strength of the relationship (how much scattering?)
• Form ( linear or some other pattern)
• Unusual features (outliers or influential points)
And ALWAYS in context of the problem!
![Page 8: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/8.jpg)
Identify as having a positivepositive association, a negativenegative association, or nono association.
1. Heights of mothers & heights of their adult daughters
++
2. Age of a car in years and its current value
3. Weight of a person and calories consumed
4. Height of a person and the person’s birth month
5. Number of hours spent in safety training and the number of accidents that occur
--++NONO
--
![Page 9: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/9.jpg)
The closer the points in a scatterplot are to a straight
line - the stronger the relationship.
The farther away from a straight line – the weaker the relationship
![Page 10: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/10.jpg)
Correlation
measures the direction and the strength of a linear relationship between 2 quantitative variables.
![Page 11: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/11.jpg)
x
i
s
xxi
y
y y
s
y
i
x
i
s
yy
s
xx
y
i
x
i
s
yy
s
xx
nr
1
1
Height (in)
Weight (lb)
74 256
65 124
77 320
72 185
68 158
60 129
62 103
73 196
61 110
64 130
=
Find the mean and standard deviation of the heights and weights of the 10 students:
67.6x 6.06xs
171.1y
70.25ys
![Page 12: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/12.jpg)
x
i
s
xxi
y
y y
s
y
i
x
i
s
yy
s
xx
y
i
x
i
s
yy
s
xx
nr
1
1
Height (in)
Weight (lb)
74 256 1.06 1.21 1.2865 124 -.43 -.67 .2977 320 1.55 2.12 3.2972 185 .73 .20 .1468 158 .07 -.19 -.0160 129 -1.25 -.60 .7562 103 -.92 -.97 .9073 196 .89 .35 .3261 110 -1.09 -.87 .9564 130 -.59 -.59 .35
=8.24/9=
.9157
Find the mean and standard deviation of the heights and weights of the 10 students:
![Page 13: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/13.jpg)
Correlation Coefficient (r)-• A quantitativequantitative assessment of the strength
& direction of the linear relationship between bivariate, quantitative data
• Pearson’s sample correlation is used most
• parameter - rho)
• statistic - r
y
i
x
i
s
yy
s
xx
nr
1
1
![Page 14: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/14.jpg)
Calculate r. Interpret r in context.
Speed Limit (mph) 55 50 45 40 30 20
Avg. # of accidents (weekly)
28 25 21 17 11 6
There is a strong, positive, linear relationship between speed limit and average number of accidents per week.
r = .9964
![Page 15: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/15.jpg)
Moderate CorrelationStrong correlation
Properties of r(correlation coefficient)
• legitimate values of r is [-1,1]
0 .5 .8 1-1 -.8 -.5
No Correlation
Weak correlation
![Page 16: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/16.jpg)
Some Correlation Pictures
![Page 17: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/17.jpg)
Some Correlation Pictures
![Page 18: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/18.jpg)
Some Correlation Pictures
![Page 19: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/19.jpg)
Some Correlation Pictures
![Page 20: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/20.jpg)
Some Correlation Pictures
![Page 21: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/21.jpg)
Some Correlation Pictures
![Page 22: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/22.jpg)
x (in mm) 12 15 21 32 26 19 24y 4 7 10 14 9 8 12
Find r.Interpret r in context.
.9181
There is a strong, positive, linear relationship between speed limit and the number of weekly accidents.
![Page 23: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/23.jpg)
• value of r is not changed by any transformationstransformations
x (in mm) 12 15 21 32 26 19 24y 4 7 10 14 9 8 12
Find r.Change to cm & find r.
The correlations are the same.
.9181
.9181
Do the following transformations & calculate r
1) 5(x + 14)2) (y + 30) ÷ 4
STILL = .9181
![Page 24: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/24.jpg)
• value of r does not depend on which of the two variables is labeled
x
Switch x & y & find r.
The correlations are the same.
Type: LinReg L2, L1
![Page 25: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/25.jpg)
• value of r is non-resistantnon-resistant
x 12 15 21 32 26 19 24y 4 7 10 14 9 8 22
Find r.
Outliers affect the correlation coefficient
![Page 26: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/26.jpg)
• value of r is a measure of the extent to which x & y are linearlylinearly related
Find the correlation for these points:x -3 -1 1 3 5 7 9Y 40 20 8 4 8 20 40
What does this correlation mean?Sketch the scatterplot
r = 0, but has a definite definite relationship!
![Page 27: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/27.jpg)
1) Correlation makes no distinction between explanatory and response variable. It is
unitless.
2) Correlation does not change when we change the units of measurement of x, y, or both.
3) Correlation requires both variables to be quantitative.
4) Correlation does not describe curved relationship between variables, no matter how
strong. Only the linear relationship between variables.
5) Like the mean and standard deviation, the correlation is not resistant: r is strongly
affected by a fewoutlying observations.
![Page 28: Chapter 5 Summarizing Bivariate Data](https://reader033.vdocuments.site/reader033/viewer/2022061509/56813709550346895d9e9402/html5/thumbnails/28.jpg)
Correlation does not imply causation
Correlation does not imply causation
Correlation does not Correlation does not imply causationimply causation