Descriptive Statistics: CorrelationDescriptive Statistics: Correlation

Describes the relationship between two or more variables.

Describes the strength of the relationship in terms of a number from -1.0 to +1.0.Describes the direction of the relationship as positive or negative.

Types of CorrelationsTypes of Correlations

Variable X increases 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 CorrelationsTypes of Correlations

Variable X decreases 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 CorrelationsTypes of Correlations

Variable X increases Variable Y decreases

Negative Correlation

Value ranging from -1.00 to .00

Example: the older you are, the less flexible your body is

Types of CorrelationsTypes of Correlations

Variable X decreases Variable Y increases

Negative Correlation

Value ranging from -1.00 to .00

Example: the less time you study, the more errors you will make

Correlation StrengthCorrelation Strength

.00 - .20 Weak or none .20 - .40 Weak .40 - .60 Moderate .60 - .80 Strong .80 - 1.00 Very strong

Positive or Negative?Positive or Negative?

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

Caution!Caution!

Correlation does not indicate causation. 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: Amount of ice sold and number of bee stings. SAT scores and GPA in college.

A Picture of CorrelationA Picture of Correlation

A scatter plot visually represents a correlation The X axis is on the horizontal

The variable on the x-axis is called the explanatory, or predictor variable.

The Y axis is on the vertical. The variable on the y axis is called the response

variable.

Correlation: IQ and GPACorrelation: IQ and GPA

IQ GPA 110 2.5 140 4.0 80 1.0 100 2.0 130 3.5 90 1.5 120 3.0 70 .5

43210

140

130

120

110

100

90

80

70

GPA

IQ

Correlation: IQ and ErrorsCorrelation: IQ and Errors IQ

Errors 80 14 120 6 100 10 90 12 130 4 110 8 140 2 70 16

151050

140

130

120

110

100

90

80

70

Errors

IQ

Correlation: IQ and WeightCorrelation: IQ and Weight IQ Weight 120 170 100 160 70 120 140 130 90 200 130 110 80 150 110 140

200190180170160150140130120110

140

130

120

110

100

90

80

70

Weight

IQ

Correlation PropertiesCorrelation Properties

The sign of a correlation coefficient gives the direction of the association.

Correlation will range between -1 and +1. A correlation of + 1 means that all the data points fall perfectly on a single, straight line.

Correlation has no units – it is based on z-scores.

More Properties of More Properties of CorrelationCorrelation

Correlation is unaffected by shifting or rescaling of the data set.

Correlation measures the strength of LINEAR association.

Factors Influencing Factors Influencing CorrelationCorrelation

Higher correlations are expected in a heterogeneous population than in a homogeneous one. Example: In elementary and high school, there

is a positive correlation between height and success in basketball.

Example: In the pros, there is no such correlation.

Conditions to CheckConditions to Check

Quantitative Variable Condition: correlation only applies to quantitative variables. Make sure you know what your units are, and what they measure.

Straight Enough Condition: correlation only applies to linear relationships.

Outlier Condition: some outliers can alter correlation greatly. Be sure to check correlation with the point, and again without the point.

Formula for CorrelationFormula for Correlation

Creating Scatterplots on the Creating Scatterplots on the CASCAS

You will first need a data set. From the home screen of the CAS, go to NEW DOCUMENT (enter), ADD LIST & SPREADSHEETS (enter).

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 Creating Scatterplots on the CASCAS

Now, go to the top of Column B, and name this column “MPG” for our data Highway Gas Mileage.

Go to cell B1 and begin entering data: 32, 30, 30, 32, 22, 20, 21

Next, press “CTRL” “I” “ADD DATA & STATISTICS”

Move your cursor down to the x axis.

Creating Scatterplots on the Creating Scatterplots on the CASCAS

You should see the outline of a small box, and the text “click to add a variable”.

Click inside the box and add “HP” Do the same for the y axis, “click to add a

variable”, add “MPG”. You now have a scatterplot.

Practice:Practice:

Describe the association of the scatterplot on your CAS.

More Goodies on the CASMore Goodies on the CAS

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

Finding the Correlation Finding the Correlation Coefficient on the CASCoefficient 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 Finding the Correlation Coefficient on the CASCoefficient on the CAS

Tab all the way down to the last box that reads “1st 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[]”.

Press enter, and you will see many things fill column C

Finding the Correlation Finding the Correlation Coefficient on the CASCoefficient on the CAS

Now, you try one by yourselfNow, you try one by yourself

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