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: IndependentDependent Variable AgeBlood Pressure

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 AgeBlood Pressure

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 Hand IndependentDependent Variable # of AbsencesFinal Grade

Example 2……With TI IndependentDependent Variable # of AbsencesFinal Grade

Notice……  Notice the following: A. Strong Negative – As x increases, y decreases B. Linear – it’s the graph of a line.

Example 3……By Hand IndependentDependent Variable Hrs. of ExerciseAmt of Milk

Example 3……With TI IndependentDependent Variable Hrs. of ExerciseAmt of Milk

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 “2 nd 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 Correlation Section 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 RelationshipStrong Positive ____________________________________

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

Example 1……  Find the correlation coefficient (r) of the following example.  Use the lists in the calculator. xy

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 =  Y Values: Mean = 76 St. Dev =

Calculator Lists…… Set Formula L1L2L3 = (L1-2.8)/ L4 = (L2-76)/ L5 = L3 x L4 xyz(of x)z (of y)z (of x) times z(of y)

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

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? AgeBlood Pressure

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…… L1L2L3L4L n=

Compute “r”……

Describe it……  Since r = Strong Positive Correlation

Example 3……  Find the correlation coefficient for the following data.  Do you remember what we found from the scatter plot? # of AbsencesFinal Grade

 X Values:  Y Values:

List Values you should have…… L1L2L3L4L n=

Compute “r”……

Describe it……  Since r = 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 ExerciseAmt of Milk

 X Values:  Y Values:

List Values you should have…… Hrs of ExerciseAmt of MilkL3L4L n=

Compute “r”……

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.

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.

Assignment……  Worksheet