GOAL: I CAN USE TECHNOLOGY TO COMPUTE AND INTERPRET THE CORRELATION COEFFICIENT OF A LINEAR FIT. (S-ID.8) Data Analysis Correlation Coefficient

Correlation Describes the relationship between two variables in a scatterplot. Data can have a positive correlation, a negative correlation, or no correlation.

Correlation Coefficient Correlation coefficients measure the strength and direction of a relationship between two variables. Generally, the correlation coefficient is denoted by r

Correlation Coefficient How to Interpret a Correlation Coefficient The value of a correlation coefficient ranges between -1 and 1. The strongest linear relationship is indicated by a correlation coefficient of -1 or 1. The weakest linear relationship is indicated by a correlation coefficient equal to 0. A positive correlation means that if one variable gets bigger, the other variable tends to get bigger. (just like a positive slope) A negative correlation means that if one variable gets bigger, the other variable tends to get smaller. (just like a negative slope)

Correlation Coefficient Keep in mind that the Pearson product-moment correlation coefficient only measures linear relationships. Therefore, a correlation of 0 does not mean zero relationship between two variables; rather, it means zero linear relationship.

Scatter plots and Correlation Coefficients Maximum positive correlation (r = 1.0) Strong positive correlation (r = 0.80) Zero correlation (r = 0) Maximum negative correlation (r = -1.0) Moderate negative correlation (r = ) Strong correlation & outlier (r = 0.71)

Example The data below are the gestation periods, in months, of randomly selected animals and their corresponding life spans, in years. 1) Find the equation of the regression line for the given data. 2) What does the rate of change mean in the context of this problem? Gestation, x Life Span, y

Example The data below are the gestation periods, in months, of randomly selected animals and their corresponding life spans, in years. 3) What is the correlation coefficient? 4) What does the correlation coefficient tell you about the model? Gestation, x Life Span, y

Example The data below are the final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam. 1) Find the equation of the regression line for the given data. 2) What does the rate of change mean in the context of this problem? Hours, x Scores, y

Example The data below are the final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam. 3) What is the correlation coefficient? 4) What does the correlation coefficient tell you about the model? Hours, x Scores, y

Summary Several points are evident from the scatter plots. When the slope of the line in the plot is negative, the correlation is negative; and vice versa.slope The strongest correlations (r = 1.0 and r = -1.0 ) occur when data points fall exactly on a straight line. The correlation becomes weaker as the data points become more scattered. If the data points fall in a random pattern, the correlation is equal to zero. Correlation is affected by outliers. Compare the first scatter plot with the last scatter plot. The single outlier in the last plot greatly reduces the correlation (from 1.00 to 0.71).outliers