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Stats: Modeling the WorldChapter 7

Scatterplots, Association, and Correlation

Making a Picture of Bivariate Data

Relationships between variables are often what we truly want to know about our data.

Visually, you can show the associations between quantative variables using a scatterplot.

Looking at ScatterplotsAfter plotting two variables on a scatterplot, we describe the relationship by examining the form, direction, and strength of the association. We look for an overall pattern

Form: linear, curved, clusters, no pattern

Association/Direction: positive, negative, no direction

Scatterplot of the size of a diamond ring in carats and the price in dollars.

Looking at Scatterplots

Strength: how closely the points fit the “form”

Outliers: deviations from the pattern

Scatterplot of the size of a diamond ring in carats and the price in dollars.

Which of the scatterplots show:

a) Little or no association?

b) A negative association?

c) A linear association?

d) A moderately strong association?

e) A very strong association?

POD #30 10/18/2011

Roles for VariablesExplanatory (or predictor) variable goes on the x-axisResponse (or predicted) variable goes on the y-axis

**If the relationship between the variables is unclear, it does not matter which one we identify as the explanatory/response variable. Always THINK about the dataset and what you are measuring!!!!

Creating a ScatterplotBy hand:

- Graph on a normal x-y plane- Make sure to label and scale axes (including units if known!)- You do not have to show the origin!

By TI:- Enter data- 2nd:Stat Plot – 1st type of graph

Quantifying StrengthWhen determining the strength of a scatterplot, we would like a numerical value that indicates the strength of the relationship. This numerical value is called the correlation coefficient.

Correlation Coefficient (aka “r”)

The correlation coefficient (r) gives us a numerical measurement of the strength of the linear relationship between the explanatory and response variables.

Strength and Direction Direction:

Positive “r” indicates a positive association Negative “r” indicates a negative association

Strength: Values close to 0 indicate weak relations As r gets closer to 1, the relationship is stronger Values of exactly 1 indicate a perfect line

“r” ranges from −1 to +1

“r” quantifies the strength and direction of a linear relationship between two quantitative variables.

Strength: How closely the points follow a straight line.

Direction is positive when individuals with higher x values tend to have higher values of y.

POD #31 10/24/2011

When to use Correlation Quantitative Variables – r cannot be applied to

categorical data! Make sure you understand your variables

Linear data – r can always be calculated, but correlation only measures strength of linear relationships, so watch for curvature!

Outliers – Since r is calculated using z-scores (and hence the mean and st. dev), it is non-resistant to outliers!

Properties of Correlation Sign of r gives the direction of association Correlation is always between -1 and +1 Flipping x and y does NOT affect r R has NO units!! It has been standardized Changing units on x or y does not affect r R measures a LINEAR relationship only! R is non-resistant to outliers

Finding Correlation Using the TI

Stat: Calc: 4:LinReg

If your r does not show, you will need to turn DiagnosticsOn. Go to 2nd:0 (Catalog), scroll down to DiagnosticsOn and hit Enter

twice.