Chapter 5Summarizing

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.

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

Does there seem to be a relationship between age and weight of these adults?

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?

Does there seem to be a relationship between height and weight of these adults?

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!

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

--

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

Correlation

measures the direction and the strength of a linear relationship between 2 quantitative variables.

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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

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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:

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

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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

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

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

Some Correlation Pictures

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.

• 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

• 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

• 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

• 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!

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.

Correlation does not imply causation

Correlation does not imply causation

Correlation does not Correlation does not imply causationimply causation