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Chapter 4 - Scatterplots and Correlation Dealing with several variables within a group vs. the same variable for different groups. Response Variable: measures the outcome of a study. Explanatory Variable: attempts to explain the observed outcomes. ex: Body Temp vs. Alcohol (Mice) ex: Predicting SAT Math if you know SAT Verbal

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WARNING! WARNING! EXPLANATORY VARIABLES DO NOT NECESSARILY CAUSE CHANGES IN RESPONSE VARIABLES!!!

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Scatter Plot: The most effective way to display the relationship between two quantitative variables measured on the same individuals. (2.1 cont ’ d) Horizontal Axis (x) = explanatory variable (if there is one) Vertical Axis (y) = response variable (if there is one) If there is no exp/resp distinction, it can be plotted either way…

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Interpreting Scatter Plots Look for overall pattern Direction / Form / Strength Direction = “ Positive ” or “ Negative “ Association: Positive Association: Above average values of one variable tend to accompany above average values of the other variable. Negative Association: Above average values of one variable tend to accompany below average values of the other variable. Form - can be linear / curved / clustered Strength Stronger = less scatter - closer to a straight line… Weaker = more scatter, not as linear…

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Direction = Positive Form = Linear Strength = Fairly Strong Direction = Positive Form = Scattered Strength = Weak

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Direction = NegativeForm = ScatteredStrength = Weak

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Direction = NegativeForm = Curved / ClusteredStrength = Weak

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Calculator Steps for Scatter plot 1) Enter data into list 1 & 2 ex 2.5 pg 99: 2) 2ndY= 3) Enter 4) Select Type Set Xlist / Frequency

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Calculator Steps for Scatter plot (cont ’ d) 5) Turn On 6) Set Window to match Data Window 7) Graph

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Adding categorical variables to scatter plots Use different colors or symbols to indicate a categorical variable or duplicate values…

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1.0 2.0 3.0 4.0 255075100125150175200225250 Cell Minutes per Week vs. GPA + + + + + + + + + + + + + + + + + + + + + + Seniors Juniors Soph + + Duplicate

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Correlation Correlation = a numerical measure of how strong a linear relationship is. Visually, correlation is hard to judge. Our eyes can be fooled by white space around a scatterplot and the plotting scales. **Same Data – Different Scales** ex:

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Formula: Correlation variable … of the sum … of the products ex: Correlation between height and weight – height is x / weight is y…. … of the standardized heights … and the standardized weights …for each measurement n - 1 … is an average

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Calculator Procedure ex: Fossil Data Step 1) Insert Data into lists ** Set DiagnosticOn** (one time step) Step 2) Run Stat Calc LinReg

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Correlation Facts Correlation (r) always falls between -1 and 1. The closer to 0 r is, the weaker the relationship. Positive r = positive association / negative r = negative association. Because r uses standardized values, r has no units. Correlation measures the strength of only LINEAR relationships. It cannot be used to describe curved relationships no matter how strong they are.

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WARNING! WARNING! CORRELATION IS STRONGLY AFFECTED BY OUTLIERS!! WARNING! WARNING! CORRELATION IS NOT A COMPLETE DESCRIPTION OF 2-VARIABLE DATA!!

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The image above shows scatterplots of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. The four y variables have the same mean (7.5), standard deviation (4.12) and correlation (0.81)

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