Bivariate Data – Scatter Plots and Correlation Coefficient……

Section 3.1 and 3.2

2 Quantitative Variables…… We represent 2 variables that are quantitative

by using a scatter plot.

Scatter Plot – a plot of ordered pairs (x,y) of bivariate data on a coordinate axis system. It is a visual or pictoral way to describe the nature of the relationship between 2 variables.

Input and Output Variables…… X:

a. Input Variable

b. Independent Var

c. Controlled Var

Y:

a. Output Variable

b. Dependent Var

c. Results from the Controlled

variable

Example…… When dealing with

height and weight, which variable would you use as the input variable and why?

Answer:

Height would be used as the input variable because weight is often predicted based on a person’s height.

Constructing a scatter plot…… Do a scatter plot of the

following data:

Independent Dependent

Variable Variable

Age Blood Pressure

43 128

48 120

56 135

61 143

67 141

70 152

What do we look for?...... A. Is it a positive correlation, negative

correlation, or no correlation?

B. Is it a strong or weak correlation?

C. What is the shape of the graph?

Answer……With TI

Age Blood Pressure

43 128

48 120

56 135

61 143

67 141

70 152

Notice…… Notice the following:

A. Strong Positive – as x increases, y

also increases.

B. Linear - it is a graph of a line.

Example 2……By HandIndependent Dependent

Variable Variable

 

# of Absences Final Grade

6 82

2 86

15 43

9 74

12 58

5 90

8 78

Example 2……With TIIndependent Dependent

Variable Variable

 

# of Absences Final Grade

6 82

2 86

15 43

9 74

12 58

5 90

8 78

Notice…… Notice the following:

A. Strong Negative – As x increases, y decreases

B. Linear – it’s the graph of a line.

Example 3……By HandIndependent Dependent

Variable Variable

 

Hrs. of Exercise Amt of Milk

3 48

0 8

2 32

5 64

8 10

5 32

10 56

2 72

1 48

Example 3……With TIIndependent Dependent

Variable Variable

 

Hrs. of Exercise Amt of Milk

3 48

0 8

2 32

5 64

8 10

5 32

10 56

2 72

1 48

Notice…… Notice:

There seems to be no correlation between the hours or exercise a person performs and the amount of milk they drink.

Steps to see on Calculator…… Put x’s in L1 and y’s in L2

Click on “2nd y=“

Set scatter plot to look like the screen to the right.

Press zoom 9 or set your own window and then press graph.

Linear CorrelationSection 3.2

Correlation…… Definition – a

statistical method used to determine whether a relationship exists between variables.

3 Types of Correlation:

A. Positive

B. Negative

C. No Correlation

Positive Correlation: as x increases, y increases or as x decreases, y decreases.

Negative Correlation: as x increases, y decreases.

No Correlation: there is no relationship between the variables.

Linear Correlation Analysis …… Primary Purpose: to measure the strength of

the relationship between the variables.

*This is a test question!!!!

Coefficient of Linear Correlation The numerical measure

of the strength and the direction between 2 variables.

This number is called the correlation coefficient.

The symbol used to represent the correlation coefficient is “r.”

The range of “r” values…… The range of the correlation coefficient is -1

to +1.

The closer to 0 you get, the weaker the correlation.

Range……

Strong

Negative No Linear Relationship Strong

Positive

____________________________________ -1 0

+1

Computational Formula using z-scores of x and y……

1

n

zzr yx

deviationst

meanvaluez

.

Example 1…… Find the correlation

coefficient (r) of the following example.

Use the lists in the calculator.

x y

2 80

5 80

1 70

4 90

2 60

Find mean and st. dev first…… Since you will be using a

formula that uses z-scores, you will need to know the mean and standard deviation of the x and y values.

Put x’s in L1 Put y’s in L2 Run stat calc one var

stats L1 – Write down mean & st. dev.

Run stat calc one var stats L2 – Write down mean & st. dev.

X values: Y values:

Write down on your paper……You’ll use them later. X Values:

Mean = 2.8

St. Dev = 1.643167673

Y Values:

Mean = 76

St. Dev = 11.40175425

Calculator Lists……

Set Formula Set Formula Set Formula

L1 L2 L3 = (L1-2.8)/1.643167673 L4 = (L2-76)/11.40175425 L5 = L3 x L4

x y z(of x) z (of y) z (of x) times z(of y)

2 80 -0.4869 0.35082 -0.1708

5 80 1.3389 0.35082 0.46971

1 70 -1.095 -0.5262 0.57646

4 90 0.7303 1.2279 0.89672

2 60 -0.4869 -1.403 0.68321

2.455298358

Calculate “r”…… From the lists….. n = 5

455298395.2 yx zz61.0

4

455298395.2

1

n

zzr yx

What does that mean? Since r = 0.61, the

correlation is a moderate correlation.

Do we want to make predictions from this?

It depends on how precise the answer needs to be.

Example 2…… Find the correlation

coefficient (r) for the following data.

Do you remember what we found from the scatter plot?

Age Blood Pressure

43 128

48 120

56 135

61 143

67 141

70 152

Let’s do this one together…… Remember to use your lists in the calculator. Don’t round numbers until your final answer. Find the mean and st. dev. for x and y. Explain what you found.

X Values: Y Values:

List values you should have……

L1 L2 L3 L4 L5

43 128 -1.368 -0.7458 1.0205

48 120 -0.8965 -1.448 1.2978

n=6 56 135 -0.1415 -0.1316 0.01863

61 143 0.33028 0.57031 0.18836

67 141 0.89647 0.39483 0.35395

70 152 1.1796 1.36 1.6042

    4.483364073

Compute “r”……

897.05

483364073.4

1

n

zzr yx

Describe it…… Since r = 0.897

Strong Positive Correlation

Example 3…… Find the correlation

coefficient for the following data.

Do you remember what we found from the scatter plot?

# of Absences Final Grade

6 82

2 86

15 43

9 74

12 58

5 90

8 78

X Values: Y Values:

List Values you should have……

L1 L2 L3 L4 L5

6 82 -0.4898 0.53626 -0.2626

2 86 -1.404 0.7746 -1.088

15 43 1.5673 -1.788 -2.802

n=7 9 74 0.19591 0.05958 0.01167

12 58 0.88158 -0.8938 -0.7879

5 90 -0.7183 1.0129 -0.7276

8 78 -0.0327 0.29792 -0.0097

-5.66529102

Compute “r”……

944.06

66529102.5

1

n

zzr yx

Describe it…… Since r = -0.944

Strong Negative Correlation

Example 4…… Find the correlation

coefficient of the following data.

Do you remember what we found from the scatter plot?

Hrs of Exercise Amt of Milk

3 48

0 8

2 32

5 64

8 10

5 32

10 56

2 72

1 48

X Values: Y Values:

List Values you should have……Hrs of Exercise Amt of Milk L3 L4 L5

3 48 -0.3015 0.30713 -0.0926

0 8 -1.206 -1.476 1.7804

2 32 -0.603 -0.4062 0.24495

5 64 0.30151 1.0205 0.30768

n=9 8 10 1.206 -1.387 -1.673

5 32 0.30151 -0.4062 -0.1225

10 56 1.8091 0.66379 1.2008

2 72 -0.603 1.3771 -0.8304

1 48 -0.9045 0.30713 -0.2778

0.537689672

Compute “r”……

067.8

5376896717.

1

n

zzr yx

Describe It…… Since r = .067

No Correlation…..No correlation exists

What is It is the coefficient of determination.

It is the percentage of the total variation in y which can be explained by the relationship between x and y.

A way to think of it: The value tells you how much your ability to predict is improved by using the regression line compared with NOT using the regression line.

?2r

For Example…… If it means that 89% of the variation

in y can be explained by the relationship between x and y.

It is a good fit.

89.2 r

Assignment…… Worksheet