Bivariate Data Scatter Plots and Correlation Coefficient Section

Bivariate Data – Scatter Plots and Correlation Coefficient…… Section 3. 1 and 3. 2

2 Quantitative Variables…… o We represent 2 variables that are quantitative by using a scatter plot. o 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…… o X: a. Input Variable b. Independent Var c. Controlled Var o Y: a. Output Variable b. Dependent Var c. Results from the Controlled variable

Example…… o When dealing with height and weight, which variable would you use as the input variable and why? o Answer: o Height would be used as the input variable because weight is often predicted based on a person’s height.

Constructing a scatter plot…… o Do a scatter plot of the following data: Indepe n d e Depend nt ent Variab le Variable Age Blood Pre ssu re 43 128 48 120

What do we look for? . . . o A. Is it a positive correlation, negative correlation, or no correlation? o B. Is it a strong or weak correlation? o C. What is the shape of the graph?

Answer……With TI Age Blood Pr e s s ur e 43 128 48 120 56 135 61 143 67 141

Notice…… o Notice the following: o A. Strong Positive – as x increases, y also increases. B. Linear - it is a graph of a line.

Example 2……By Hand Independent Dependent Variable # of Absences Final Grade 6 82 2 86 15 43 9 74 12 58 5 90 8 78

Example 2……With TI Independent Dependent Variable # of Absences Final Grade 6 82 2 86 15 43 9 74 12 58 5 90 8 78

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

Example 3……By Hand Independent Dependent 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 TI Independent Dependent 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…… o Notice: o 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…… o Put x’s in L 1 and y’s in L 2 o Click on “ 2 nd y=“ o Set scatter plot to look like the screen to the right. o Press zoom 9 or set your own window and then press graph.

Linear Correlation Section 3. 2

Correlation…… o Definition – a statistical method used to determine whether a relationship exists between variables. o 3 Types of Correlation: A. Positive B. Negative C. No Correlation

o o o 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 …… o Primary Purpose: to measure the strength of the relationship between the variables. o *This is a test question!!!!

Coefficient of Linear Correlation o The numerical measure of the strength and the direction between 2 variables. o This number is called the correlation coefficient. o The symbol used to represent the correlation coefficient is “r. ”

The range of “r” values…… o The range of the correlation coefficient is -1 to +1. o The closer to 0 you get, the weaker the correlation.

Range…… Strong Negative o No Linear Relationship Strong Positive __________________ -1 0 +1

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

Example 1…… o o 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…… o 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. o o Put x’s in L 1 Put y’s in L 2 Run stat calc one var stats L 1 – Write down mean & st. dev. Run stat calc one var stats L 2 – Write down mean & st. dev.

o X values: o Y values:

Write down on your paper……You’ll use them later. o X Values: Mean = 2. 8 St. Dev = 1. 643167673 o Y Values: Mean = 76 St. Dev = 11. 40175425

Calculator Lists…… Set Formula L 1 L 2 L 3 = (L 1 -2. 8)/1. 643167673 L 4 = (L 2 -76)/11. 40175425 L 5 = L 3 x L 4 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”…… o o From the lists…. . n = 5

What does that mean? o Since r = 0. 61, the correlation is a moderate correlation. o Do we want to make predictions from this? o It depends on how precise the answer needs to be.

Example 2…… o o 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…… o o 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.

o X Values: o Y Values:

List values you should have…… n=6 L 1 L 2 L 3 L 4 L 5 43 128 -1. 368 -0. 7458 1. 0205 48 120 -0. 8965 -1. 448 1. 2978 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”……

Describe it…… o Since r = 0. 897 Strong Positive Correlation

Example 3…… o o 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

o X Values: o Y Values:

List Values you should have…… n=7 L 1 L 2 L 3 L 4 L 5 6 82 -0. 4898 0. 53626 -0. 2626 2 86 -1. 404 0. 7746 -1. 088 15 43 1. 5673 -1. 788 -2. 802 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”……

Describe it…… o Since r = -0. 944 Strong Negative Correlation

Example 4…… o o 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

o X Values: o Y Values:

List Values you should have…… n=9 Hrs of Exercise Amt of Milk L 3 L 4 L 5 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 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”……

Describe It…… o Since r =. 067 No Correlation…. . No correlation exists

What is o It is the coefficient of determination. o It is the percentage of the total variation in y which can be explained by the relationship between x and y. o 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…… o If it means that 89% of the variation in y can be explained by the relationship between x and y. o It is a good fit.

Assignment…… o Worksheet