Bivariate Descriptive Statistics

Bivariate analysis examines the relationship between two variables. The approach you take depends on whether your variables are numerical (continuous) or categorical (discrete groups).

This chapter covers the two main combinations you’ll encounter:

  1. Numerical explanatory variable → Numerical outcome variable
  2. Categorical explanatory variable → Numerical outcome variable
# Load required packages
library(dplyr)
library(ggplot2)

# We'll use the built-in mtcars dataset for examples
data(mtcars)

This is a dataset about different cars (from the 1970s) and some related variables (design, etc)

head(mtcars)
                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Case 1: Numerical → Numerical

When both your explanatory variable (independent variable) and outcome variable (dependent variable) are numerical, you’re examining how one continuous measure relates to another.

Example Variables

  • Explanatory variable: wt (car weight in 1000 lbs)
  • Outcome variable: mpg (miles per gallon)

Descriptive Statistics for Both Variables

First, examine the distribution of each variable individually using summary():

# Summary statistics for the explanatory variable (weight)
summary(mtcars$wt)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.513   2.581   3.325   3.217   3.610   5.424 
# Summary statistics for the outcome variable (mpg)
summary(mtcars$mpg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  10.40   15.43   19.20   20.09   22.80   33.90

The summary() function provides:

  • Min: Minimum value
  • 1st Qu.: First quartile (25th percentile)
  • Median: Middle value (50th percentile)
  • Mean: Average value
  • 3rd Qu.: Third quartile (75th percentile)
  • Max: Maximum value

Correlation Analysis

The correlation coefficient (Pearson’s r) measures the strength and direction of the linear relationship between two numerical variables:

# Calculate correlation
cor(mtcars$wt, mtcars$mpg)
[1] -0.8676594

Interpreting correlation:

  • Values range from -1 to +1
  • Positive values: Variables increase together
  • Negative values: As one increases, the other decreases
  • Magnitude:
    • 0.0 - 0.1: Small correlation
    • 0.1 - 0.3: Moderate correlation
    • 0.3 - 0.6: Moderately Strong correlation
    • 0.6 - 0.9: Strong correlation
    • 0.9 - 1.0: Very Strong correlation

Visualization: Scatterplot

A scatterplot is the best way to visualise the relationship between two numerical variables:

plot(mtcars$wt, mtcars$mpg, main = "Relationship between Car Weight and Fuel Efficiency", xlab = "Weight (1000 lbs)",ylab = "Miles per Gallon (mpg)")

Case 2: Categorical → Numerical

When your explanatory variable is categorical (e.g., gender, region, treatment group) and your outcome variable is numerical, you cannot do a simple correlation, so one alternative is to compare means across groups.

Example Variables

  • Explanatory variable: am (transmission type: 0 = automatic, 1 = manual)
  • Outcome variable: mpg (miles per gallon)

First, let’s convert the transmission variable to a factor with meaningful labels:

mtcars <- mtcars %>%
  mutate(transmission = factor(am, 
                               levels = c(0, 1),
                               labels = c("Automatic", "Manual")))

Descriptive Statistics for Outcome Variable

Examine the overall distribution of the outcome variable:

# Summary statistics for mpg
summary(mtcars$mpg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  10.40   15.43   19.20   20.09   22.80   33.90

Frequency Distribution of Categorical Variable

For the categorical variable, examine the distribution of cases across categories:

Using Base R Tables

# Frequency table
table(mtcars$transmission)

Automatic    Manual 
       19        13 
# Proportion table
prop.table(table(mtcars$transmission))

Automatic    Manual 
  0.59375   0.40625

Group Averages by Category

The key analysis is comparing the mean of the outcome variable across categories:

# Calculate group means and other statistics
mtcars %>%
  group_by(transmission) %>%
  summarise(
    n = n(),
    mean_mpg = mean(mpg),
    median_mpg = median(mpg),
    sd_mpg = sd(mpg),
    min_mpg = min(mpg),
    max_mpg = max(mpg)
  )
# A tibble: 2 × 7
  transmission     n mean_mpg median_mpg sd_mpg min_mpg max_mpg
  <fct>        <int>    <dbl>      <dbl>  <dbl>   <dbl>   <dbl>
1 Automatic       19     17.1       17.3   3.83    10.4    24.4
2 Manual          13     24.4       22.8   6.17    15      33.9

Key statistics:

  • n: Sample size in each group
  • mean_mpg: Average fuel efficiency
  • median_mpg: Middle value (useful if data is skewed)
  • sd_mpg: Standard deviation (measure of spread)
  • min_mpg / max_mpg: Range of values

Visualization Options

Summary: Choosing Your Approach

Explanatory Variable Outcome Variable Key Statistics Best Visualization
Numerical Numerical summary(), cor() Scatterplot with regression line
Categorical Numerical count(), group_by() + summarise() Box plot, violin plot, or bar plot

Key Functions Reference

For numerical variables: - summary(): Min, quartiles, mean, max - cor(): Correlation coefficient

For categorical variables: - table() + prop.table(): Base R frequency tables - count() + mutate(n/sum(n)): dplyr frequency and proportions - group_by() + summarise(): Group-level statistics