Descriptive Statistics: Correlation u Describes the relationship between two or more variables. u Describes the strength of the relationship in terms of a number from -1.0 to u Describes the direction of the relationship as positive or negative.

Types of Correlations u Variable X increases u Variable Y increases Positive Correlation Value ranging from.00 to 1.00 Example: the more you eat, the more weight you will gain

Types of Correlations u Variable X decreases u Variable Y decreases Positive Correlation Value ranging from.00 to 1.00 Example: the less you study, the lower your test score will be

Types of Correlations u Variable X increases u Variable Y decreases Negative Correlation Value ranging from to.00 Example: the older you are, the less flexible your body is

Types of Correlations u Variable X decreases u Variable Y increases Negative Correlation Value ranging from to.00 Example: the less time you study, the more errors you will make

Correlation Strength u Weak or none u Weak u Moderate u Strong u Very strong

Positive or Negative? u IQ and reading achievement u Anxiety and test scores u Amount of calories consumed and weight gain. u Amount of exercise and weight gain u Reading achievement and math achievement u Foot size and math ability

Caution! u Correlation does not indicate causation. u Correlation only establishes that a relationship exists; it reflects the amount of variability that is shared between two variables and what they have in common. Examples: u Amount of ice sold and number of bee stings. u SAT scores and GPA in college.

A Picture of Correlation u A scatter plot visually represents a correlation u The X axis is on the horizontal u The variable on the x-axis is called the explanatory, or predictor variable. u The Y axis is on the vertical. u The variable on the y axis is called the response variable.

Correlation: IQ and GPA u IQGPA u u u u u u u u 70.5

Correlation: IQ and Errors u IQ Errors u 8014 u 1206 u u 9012 u 1304 u 1108 u 1402 u 7016

Correlation: IQ and Weight u IQWeight u u u u u u u u

Correlation Properties u The sign of a correlation coefficient gives the direction of the association. u Correlation will range between -1 and +1. A correlation of + 1 means that all the data points fall perfectly on a single, straight line. u Correlation has no units – it is based on z- scores.

More Properties of Correlation u Correlation is unaffected by shifting or rescaling of the data set. u Correlation measures the strength of LINEAR association.

Factors Influencing Correlation u Higher correlations are expected in a heterogeneous population than in a homogeneous one. u Example: In elementary and high school, there is a positive correlation between height and success in basketball. u Example: In the pros, there is no such correlation.

Conditions to Check u Quantitative Variable Condition: correlation only applies to quantitative variables. Make sure you know what your units are, and what they measure. u Straight Enough Condition: correlation only applies to linear relationships. u Outlier Condition: some outliers can alter correlation greatly. Be sure to check correlation with the point, and again without the point.

Formula for Correlation u

Creating Scatterplots on the CAS u You will first need a data set. From the home screen of the CAS, go to NEW DOCUMENT (enter), ADD LIST & SPREADSHEETS (enter). u Go to the top of Column A, and name your data. For now, use “HP” for our data Horsepower. Go to cell A1, and start entering the data: 200, 230, 200, 148, 291, 300, 295

Creating Scatterplots on the CAS u Now, go to the top of Column B, and name this column “MPG” for our data Highway Gas Mileage. u Go to cell B1 and begin entering data: 32, 30, 30, 32, 22, 20, 21 u Next, press “CTRL” “I” u “ADD DATA & STATISTICS” u Move your cursor down to the x axis.

Creating Scatterplots on the CAS u You should see the outline of a small box, and the text “click to add a variable”. u Click inside the box and add “HP” u Do the same for the y axis, “click to add a variable”, add “MPG”. u You now have a scatterplot.

Practice: u Describe the association of the scatterplot on your CAS.

More Goodies on the CAS u You can also find the correlation coefficient on your CAS. Follow the steps on the next slide to do so.

Finding the Correlation Coefficient on the CAS Go back to your spreadsheet (shortcut: press CTRL and the left arrow key of the center NAV PAD of the CAS) Press MENU, STATISTICS, STAT CALCS, LINEAR REGRESSION ( a + bx) …… or Menu, 4, 1, 4 A box will pop open. Only do the following: XLIST: choose the column you entered the x variable, (make sure that “a[]” is in this box) YLIST: choose the column you entered the y variable, (for this example, “b[]” )

Finding the Correlation Coefficient on the CAS u Tab all the way down to the last box that reads “1 st Result Column”. This is where you tell the CAS the column you would like the results in. Choose anything but “a[]”, and “b[]”. I usually put the results in “c[]”. u Press enter, and you will see many things fill column C

Finding the Correlation Coefficient on the CAS u

Now, you try one by yourself u Find the correlation coefficient for the following data on several fast food burgers: Fat (g): 19, 31, 34, 35, 39, 39, 43 Sodium (mg): 920, 1500, 1310, 860, 1180, 940, 1260