Chapter 2: Examining Relationships

Section 2.2: Correlation

Linear relationships between two variables are important because a stright line is a simple pattern that is common. How close a scatterplot approaches a line depends on the scales of the axes.

The correlation r

Measures the strength and direction of the linear association between two quantitative variables.

The correlation coefficient can be calculated using



where x_i and y_i are the observations of one individual, s_x is the standard deviation for the x variable and s_y is the standard deviation for the y variable. Also, the factor standardizes all the x observations and the factor standardizes all the y observations. These two factors are multiplied for each individual then summed over all individuals. Lastly, the sum is divided by the factor n – 1.

Luckily, here are steps to find the correlation coefficient using the TI-83 calculator.

Facts about correlation

1. Correlation makes no distinction between explanatory and response variables.
2. Both variables MUST be quantitative.
3. Units of measurement do not affect r. Also, r has no units.
4. Positive r indicates positive association between the two variables. Negative r indicates negative association between the two variables.
5. r is always between –1 and +1. Values near 0 indicate a very weak linear relationship. Values near –1 or +1 indicate that the points in a scatterplot lie close to a straight line. Extreme values of ±1 occur only when the points lie exactly along a straight line.
6. Correlation measures only the strength of linear relationship between two variables. It does not describe curved (i.e., quadratic, exponential, logarithmic, etc.) relationships.
7. Correlation is not resistant to outliers. Use r with caution when there are possible outliers.

IMPORTANT: Correlation is not a complete description of two–variable data. You should also include the means and standard deviations of each variable.