Unit 3 – Association: Contingency, Correlation, and Regression Lesson 3-2 Quantitative Associations

3-2 Learning Objectives 0) Data Possibilities 1) Constructing Scatterplots 2) Interpreting Scatterplots 3) Correlation 4) Calculating Correlation

Objective 0: DATA POSSIBILITIES (PREVIEW) In practice, when we investigate the association between two variables, there are three possible outcomes for the data. Both Categorical – We can display the data in a contingency table and compare the values with side-by-side bar graphs. (Lesson 3-1) EX: Favorite crayon color of boys vs girls. One Quantitative/One Categorical – We summarize the data to show center, spread, and graph with box plots, bar graphs, pie charts, dot plots, etc. (All of Unit 2) EX: Average number of crayons boys vs girls use to color a picture. Both Quantitative – We need a way to represent the variables on two axes to look for trends in how the explanatory variable changes on the response variable. (The rest of Unit 3) EX: The time a child takes to draw a cat based on their age in months. (How can we do this?)

Objective 1: CONSTRUCTING SCATTERPLOTS A scatterplot is a graphical display of a relationship between two quantitative variables. Horizontal Axis: Explanatory variable (x) Vertical Axis: Response variable (y) Example: A country’s average internet usage as compared to their GDP.

Objective 1: CONSTRUCTING SCATTERPLOTS Is there an association? (little variability) USA Malaysia

Objective 2: INTERPRETING SCATTERPLOTS Three things we are looking at when we see a scatter plot: TREND: linear, curved, clusters, no pattern DIRECTION: positive, negative, no direction STRENGTH: how closely the points fit the trend We also want to identify any unusual observations, falling well apart from the overall trend. (aka ) outliers

Objective 2: INTERPRETING SCATTERPLOTS Example TREND: DIRECTION: LINEAR POSITIVE

Objective 2: INTERPRETING SCATTERPLOTS Example TREND: DIRECTION: CLUSTERS POSITIVE

Objective 2: INTERPRETING SCATTERPLOTS Example TREND: DIRECTION: LINEAR NEGATIVE

Objective 2: INTERPRETING SCATTERPLOTS Example TREND: DIRECTION: NO TREND NO DIRECTION

Objective 2: INTERPRETING SCATTERPLOTS OUTLIER

Objective 3: MEASURING LINEAR CORRELATION When we see a strong linear trend (positive or negative), we will say that there exists/is a correlation between the two variables. We can quantify the strength and direction with the correlation coefficient ( ). A positive r value indicates positive association, a negative r value indicates a negative association. r So ‘r’ tells us the direction of the association.

Objective 3: MEASURING LINEAR CORRELATION The value of r will always fall between and. An |r| value between.75 and 1 indicates a strong association. An |r| value between.50 and.75 indicates a somewhat strong association. An |r| value between 0 and.50 indicates a weak association. Correlation, r, is not resistant to outliers.1

Objective 3: MEASURING LINEAR CORRELATION weak strong very weak straight line no correlation r =.2

Objective 3: MEASURING LINEAR CORRELATION PRACTICE TOGETHER: Order the following r values from strongest to weakest. r =.43r =.09 r = -.88 r = -.239r =.75r =.5

Objective 4: CALCULATING CORRELATION Formula for r : Sum of the z-scores (for all points) divided by the number of differences. So… let’s use a calculator for this.

Objective 4: CALCULATING CORRELATION 1. Enter x data in L1 and y data in L2 2. Graph with 2 nd STAT PLOT (optional) 3. STAT  CALC 4. Choose 8: LinReg(a+bx) 5. See r value listed. 6. If r is not shown, 2 nd CATALOG, DiagnosticOn.

Objective 4: CALCULATING CORRELATION Individual Practice: Draw 10 random data points (real current data), showing x = number of absences and y = current overall grade in course. Complete table, scatterplot, and calculate correlation coefficient (r) on your notes.