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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

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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 111115 27120 37.5100 48102 545160

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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 111 27 37.5 48 512 610 case # Shoe size 16.5 28 38 411 59 France Germany

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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 2Germany7 3 7.5 4France8 5 12 6France10

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Scatterplot CARSBOATS 4030 2025 3065 20 30 40 20 50 80 CARS BOATS

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Scatterplot CARSBOATS 4030 2025 3065 20 30 40 20 50 80 CARS BOATS

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cigarettes.xls

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

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Made-up Examples PERCENT TAKING SAT STATE AVE SCORE

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Made-up Examples SHOE SIZE IQ

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Made-up Examples BAKING TEMP JUDGE’S IMPRESSION 250 350 450

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Made-up Examples GDP PER CAPITA LIFE EXPECTANCY

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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 !!!!

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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.

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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?

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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.

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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

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

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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 +1 +1 means perfect positive correlation; dots on line -1 means perfect negative correlation; dots on line 0 means no relationship, OR no linear relationship

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

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Introduction to Quantitative Data Analysis (continued) Reading on Quantitative Data Analysis: Baxter and Babbie, 2004, Chapter 12.

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