Visualizing Distribution of Data

Why box-plot?

Box-plot is one of the effective ways to visually represent the distribution of data and it gives you an overall idea about how the data looks. Also, it is one of the best tools to identify the outliers to check if an association you find in your analysis can be explained by the presence of potential unusual observations.

The graph shown below is an example of box-plot where the box shows the interquartile range (IQR), the horizontal line inside the box presents the median (50th percentile), and the whiskers (lines extending from the box) represent maximum value that is not an outlier (1.5 IQR above the third quartile) and minimum value that is not an outlier (1.5 IQR below the first quartile). All individual points outside the whiskers are considered as outliers.

First quartile (Q1 / 25^th Percentile) also knows as lower quartile is the center of the lower half of the data.
Median (Q2 / 50^th Percentile) is the middle value of the data.
Third quartile (Q3 / 75^th Percentile) also known as upper quartile is center of the upper half of the data.
Interquartile range (IQR) shows the middle half of the data. (distance between first and third quartile)

Box-plots are usually a useful way to compare the distribution of data among two or more groups, or to compare the distribution of data among one or more than one group across time.

Box-plot using the base R function

Let’s use the below data as an example. The first column shows the gender of each observation, the values in the second column are pre-test scores and the values in the third column are post-test scores.

df <- data.frame(
  gender    = c("Male", "Male", "Male", "Male", "Male", "Male", "Male", "Male", "Male", "Female", "Female", "Female", "Female", "Female", "Female", "Female", "Female", "Female", "Female", "Female"),
  pre_test =    c(30,   26, 24, 19, 27, 21, 31, 16, 16, 26, 28, 29, 36, 25, 27, 24, 32, 30, 33, 26),
  post_test =   c(35,   31, 29, 24, 32, 26, 36, 21, 21, 31, 33, 34, 41, 30, 32, 29, 37, 35, 38, 31)
)

df

##    gender pre_test post_test
## 1    Male       30        35
## 2    Male       26        31
## 3    Male       24        29
## 4    Male       19        24
## 5    Male       27        32
## 6    Male       21        26
## 7    Male       31        36
## 8    Male       16        21
## 9    Male       16        21
## 10 Female       26        31
## 11 Female       28        33
## 12 Female       29        34
## 13 Female       36        41
## 14 Female       25        30
## 15 Female       27        32
## 16 Female       24        29
## 17 Female       32        37
## 18 Female       30        35
## 19 Female       33        38
## 20 Female       26        31

The base R function to build a boxplot is boxplot().

boxplot(df$pre_test, main="Pre-test")

Let’s combine both plots for better comparison.

boxplot(df[,c("pre_test", "post_test")], main = "Pre-test vs Post test")

Box-plot using the ggplot2 package

I prefer the ggplot2 plots because the output is nicer and importantly it can be easily modified compared to base R graphs, moreover, it is more popular among R users.

The geom_box() function from the ggplot2 package can be used to create a box-plot.

# install.packages("ggplot2")
library(ggplot2)

ggplot(df) +
  geom_boxplot(aes(y = pre_test, x = "1: Pre-test")) +
  geom_boxplot(aes(y = post_test, x = "2: Post-test")) +
  theme_bw() +
  labs(title = "Pre-test vs. Post-test scores", x = NULL, y = NULL)

The plot on the left side shows the distribution of pre-test scores and the one on the right side shows the distribution of post-test scores. The above plot shows that average post-test score (31.5) is higher compare to the pre-test (26.5).

It is also possible to use box-plot to visualize the distribution of data by categories of another variable as well, see below example for instance.

ggplot(df) +
  geom_boxplot(aes(y = pre_test, x = "1: Pre-test", fill = gender), alpha = 0.25) +
  geom_boxplot(aes(y = post_test, x = "2: Post-test", fill = gender), alpha = 0.25) +
  theme_bw() +
  labs(title = "Pre-test vs. Post-test scores, by gender", x = NULL, y = NULL)

The above plot shows that on average females have higher score (boxes in red color) compare to those of males (boxes in blue color) in both pre-test and post-test.