# Relationship between Variables

Scatterplot

• Used to determine relationship between two continuous variables.
• One variable plotted on x-axis, another on y-axis.
• Positive Correlation: Higher x-values correspond to higher y-values.
• Negative Correlation: Higher x-values correspond to lower y-values.
• Examples:
• Body weight and BMI
• Height and Pressure etc.

Scatterplot Matrix

Linear vs Nonlinear Relationship

Variables changing proportionately in response to each other show linear relationship. Linear relationship is an abstract concept it depends what can be called linear and what can’t in a given context. A linear relationship may exist locally with a non linear relationship globally.

Summary Tables
Method for understanding the relationship between two variables when at least one the variables is discrete.
Example: Summary information about ages of active psychologists by demographics.

Ages (1) Total Active Psychologists Active Psychologists by Gender Active Psychologists by Race/Ethnicity
(2) Female (3) Male (4) Asian (5) Black/ African American (6) Hispanic (7) White
Mean 50.5 47.9 55.1 46.5 47.9 46.4 51.1
Median 51 48 57 43 46 44 53
Std. Dev. 12.5 12.4 11.4 13.3 10.3 11.2 12.6

Discrete Variable(s): Demography: (1), (2), (3), (4), (5), (6), (7)
Continuous Variable: Age

• Cross-Tabulation Tables/ Crosstabs/ Contingency Tables
• Method for summarizing two categorical variables
• In practice, continuous variables may be at times summarized as categorical variables.
• Example: Age could be divided into categories as young, adult and senior citizen, etc. Income could be divided into categories as poor, middle class, upper middle class, wealthy, etc.
• Correlation Coefficient
• A quantification of the linear relationship between two variables
• Ranges from -1 to +1
• Used for variables on an interval or ratio scale

$$r_{xy}$$ = $$\frac{\sum_{i=1}^{i=n}\left({x_i\,-\,\bar{x}}\right)\left({y_i\,-\,\bar{y}}\right)}{\left(n\,-\,1\right)s_{x}s_{y}}$$ = $$\frac{\sum{\left( {x_i\,-\,\bar{x}} \right)}\left( {y_i\,-\,\bar{y}} \right)}{\sqrt{\sum\left({x_i\,-\,\bar{x}}\right)^2\sum\left({y_i\,-\,\bar{y}}\right)^2}}$$

NOTE: Correlation coefficient does not capture nonlinear relationships. Many nonlinear relationships might exist which are not captured ($$r$$ = 0) by correlation coefficient.