The term bivariate analysis refers to the analysis of two variables. You can remember this because the prefix “bi” means “two.”

The purpose of bivariate analysis is to understand the relationship between two variables

There are three common ways to perform bivariate analysis:

1. Scatterplots

2. Correlation Coefficients

3. Simple Linear Regression

The following example shows how to perform each of these types of bivariate analysis using the following dataset that contains information about two variables: (1) Hours spent studying and (2) Exam score received by 20 different students:

#create data frame
df <- data.frame(hours=c(1, 1, 1, 2, 2, 2, 3, 3, 3, 3,
                         3, 4, 4, 5, 5, 6, 6, 6, 7, 8),
                 score=c(75, 66, 68, 74, 78, 72, 85, 82, 90, 82,
                         80, 88, 85, 90, 92, 94, 94, 88, 91, 96))

#view first six rows of data frame
head(df)

  hours score
1     1    75
2     1    66
3     1    68
4     2    74
5     2    78
6     2    72

1. Scatterplots

We can use the following syntax to create a scatterplot of hours studied vs. exam score in R:

#create scatterplot of hours studied vs. exam score
plot(df$hours, df$score, pch=16, col='steelblue',
     main='Hours Studied vs. Exam Score',
     xlab='Hours Studied', ylab='Exam Score')

The x-axis shows the hours studied and the y-axis shows the exam score received.

From the plot we can see that there is a positive relationship between the two variables: As hours studied increases, exam score tends to increase as well.

2. Correlation Coefficients

A Pearson Correlation Coefficient is a way to quantify the linear relationship between two variables.

We can use the cor() function in R to calculate the Pearson Correlation Coefficient between two variables:

#calculate correlation between hours studied and exam score received
cor(df$hours, df$score)

[1] 0.891306

The correlation coefficient turns out to be 0.891.

This value is close to 1, which indicates a strong positive correlation between hours studied and exam score received.

3. Simple Linear Regression

Simple linear regression is a statistical method we can use to find the equation of the line that best “fits” a dataset, which we can then use to understand the exact relationship between two variables.

We can use the lm() function in R to fit a for hours studied and exam score received:

#fit simple linear regression model
fit <- lm(score ~ hours, data=df)

#view summary of model
summary(fit)

Call:
lm(formula = score ~ hours, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-6.920 -3.927  1.309  1.903  9.385 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  69.0734     1.9651   35.15  < 2e-16 ***
hours         3.8471     0.4613    8.34 1.35e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.171 on 18 degrees of freedom
Multiple R-squared:  0.7944,	Adjusted R-squared:  0.783 
F-statistic: 69.56 on 1 and 18 DF,  p-value: 1.347e-07

The fitted regression equation turns out to be:

Exam Score = 69.0734 + 3.8471*(hours studied)

This tells us that each additional hour studied is associated with an average increase of 3.8471 in exam score.

We can also use the fitted regression equation to predict the score that a student will receive based on their total hours studied.

For example, a student who studies for 3 hours is predicted to receive a score of 81.6147:

• Exam Score = 69.0734 + 3.8471*(hours studied)
• Exam Score = 69.0734 + 3.8471*(3)
• Exam Score = 81.6147

Additional Resources

The following tutorials provide additional information about bivariate analysis: