Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship between and .
The correlation coefficient, , developed by Karl Pearson in the early 1900s, is a numerical measure of the strength of association between the independent variable and the dependent variable .
The correlation coefficient is calculated as
where = the number of data points.
If you suspect a linear relationship between and , then can measure how strong the linear relationship is.
Strong correlation does not suggest that causes or causes . We say "correlation does not imply causation". For example, every person who learned math in the 17th century is dead. However, learning math does not necessarily cause death!
a. A scatter plot showing data with a positive correlation. b. A scatter plot showing data with a negative correlation. c. A scatter plot showing data with zero correlation.
The formula for looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate . The correlation coefficient is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).
r2 is called the coefficient of determination. r2 is the square of the correlation coefficient , but is usually stated as a percent, rather than in decimal form. r2
has an interpretation in the context of the data:
Consider the third exam/final exam example introduced in the previous section
The line of best fit is:
The correlation coefficient is
The coefficient of determination is
Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be explained by the variation in the grades on the third exam, using the best fit regression line.
Therefore approximately 56% of the variation (1 - 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best fit regression line. (This is seen as the scattering of the points about the line).
**With contributions from Roberta Bloom.
A measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable. The formula is:
where is the number of data points. The coefficient cannot be more then 1 and less then -1. The closer the coefficient is to ±1, the stronger the evidence of a significant linear relationship between and .
Source: Barbara Illowsky, Ph.D.,Susan Dean, https://cnx.org/contents/XgdE-Z55@40.9:XEKQgmhr@12/Correlation-Coefficient-and-Coefficient-of-Determination
This work is licensed under a Creative Commons Attribution 3.0 License.