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 in Python using the following pandas DataFrame that contains information about two variables: (1) Hours spent studying and (2) Exam score received by 20 different students:

import pandas as pd

#create DataFrame
df = pd.DataFrame({'hours': [1, 1, 1, 2, 2, 2, 3, 3, 3, 3,
                             3, 4, 4, 5, 5, 6, 6, 6, 7, 8],
                   'score': [75, 66, 68, 74, 78, 72, 85, 82, 90, 82,
                             80, 88, 85, 90, 92, 94, 94, 88, 91, 96]})

#view first five rows of DataFrame
df.head()

	hours	score
0	1	75
1	1	66
2	1	68
3	2	74
4	2	78

1. Scatterplots

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

import matplotlib.pyplot as plt

#create scatterplot of hours vs. score
plt.scatter(df.hours, df.score)
plt.title('Hours Studied vs. Exam Score')
plt.xlabel('Hours Studied')
plt.ylabel('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 corr() function in pandas to create a correlation matrix:

#create correlation matrix
df.corr()

	hours	score
hours	1.000000	0.891306
score	0.891306	1.000000

The correlation coefficient turns out to be 0.891. This 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 quantify the relationship between two variables.

We can use the OLS() function from the statsmodels package to quickly fit a for hours studied and exam score received:

import statsmodels.apias sm

#define response variable
y = df['score']

#define explanatory variable
x = df[['hours']]

#add constant to predictor variables
x = sm.add_constant(x)

#fit linear regression model
model = sm.OLS(y, x).fit()

#view model summary
print(model.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                  score   R-squared:                       0.794
Model:                            OLS   Adj. R-squared:                  0.783
Method:                 Least Squares   F-statistic:                     69.56
Date:                Mon, 22 Nov 2021   Prob (F-statistic):           1.35e-07
Time:                        16:15:52   Log-Likelihood:                -55.886
No. Observations:                  20   AIC:                             115.8
Df Residuals:                      18   BIC:                             117.8
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         69.0734      1.965     35.149      0.000      64.945      73.202
hours          3.8471      0.461      8.340      0.000       2.878       4.816
==============================================================================
Omnibus:                        0.171   Durbin-Watson:                   1.404
Prob(Omnibus):                  0.918   Jarque-Bera (JB):                0.177
Skew:                           0.165   Prob(JB):                        0.915
Kurtosis:                       2.679   Cond. No.                         9.37
==============================================================================

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: