Two Quantitative Variables Scatterplots examples how to draw them Association what to look for in a scatterplot Correlation strength of a linear relationship how to calculate good news and bad news

Paired vs. Unpaired Variables Paired variables come from the same data table. Each record has one value of X and one value of Y, and they go together a pair. case # Shoe size IQ

Paired vs. Unpaired Variables Unpaired variables come from different tables …or from different lines of one table. IN CHAPTER TWO WE’RE DEALING WITH PAIRED VARIABLES. ca se # Shoe size case # Shoe size France Germany

Paired vs. Unpaired Variables Unpaired variables come from different tables …or from different lines of one table. IN CHAPTER TWO WE’RE DEALING WITH PAIRED VARIABLES. case # CountryShoe size 1France11 2Germany France France10

Scatterplot CARSBOATS CARS BOATS

Scatterplot CARSBOATS CARS BOATS

cigarettes.xls

Kinds of Association… Positive vs. Negative Strong vs. Weak Linear vs. Non-linear

Made-up Examples PERCENT TAKING SAT STATE AVE SCORE

Made-up Examples SHOE SIZE IQ

Made-up Examples BAKING TEMP JUDGE’S IMPRESSION

Made-up Examples GDP PER CAPITA LIFE EXPECTANCY

What to look for in a scatterplot… Do the cases break up into separate clusters? Are there outliers? Is there an ASSOCIATION between the variables? OR are they INDEPENDENT? ALWAYS DRAW THE PICTURE !!!!

Scatterplots: Which variable goes where? RESPONSE VARIABLE goes on Y axis (“Y”)(“dependent variable”) EXPLANATORY VARIABLE goes on X axis (“X”)(“independent variable”) If neither is really a response variable, it doesn’t matter which variable goes where.

Scatterplots: Drawing Considerations Don’t show the axes without a good reason Don’t show gridlines without a good reason Scales should cover the ranges of the variables-- —outliers? —no need to include 0 —what if same units?

CORRELATION (or, the CORRELATION COEFFICIENT) measures the strength of a linear relationship. If the relationship is non-linear, it measures the strength of the linear part of the relationship. But then it doesn’t tell the whole story. Correlation can be positive or negative.

Computing correlation… 1.Replace each variable with its standardized version. 2.Multiply each pair ( x i ’ times y i ’ ) 3.Take an “average” of the products

Computing correlation r, or R, or greek  (rho) n-1, not n sum of all the products

Good things about correlation It’s symmetric ( correlation of x and y means same as correlation of y and x ) It doesn’t depend on scale or units — adding or multiplying either variable by a constant doesn’t change r — of course not; r depend only on the standardized versions r is always in the range from -1 to means perfect positive correlation; dots on line -1 means perfect negative correlation; dots on line 0 means no relationship, OR no linear relationship

Bad things about correlation Sensitive to outliers Misses non-linear relationships Doesn’t imply causality