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

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

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…

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…

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

Direction = NegativeForm = ScatteredStrength = Weak

Direction = NegativeForm = Curved / ClusteredStrength = Weak

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

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

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

Cell Minutes per Week vs. GPA Seniors Juniors Soph + + Duplicate

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:

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

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

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.

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

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)