Why Do Box and Whisker Plots Show Uneven Whiskers?
Understanding the relationship between the interquartile range (IQR) and the visualization of data in box and whisker plots can be quite enlightening. While the IQR is defined as the difference between the third quartile (Q3) and the first quartile (Q1), box and whisker plots sometimes present uneven whiskers, which can lead to questions about the representation of data. In this article, we will explore why this happens and what it means for the analysis.
Introduction to Box and Whisker Plots
Box and whisker plots, also known as box plots, are a valuable tool in data visualization. They provide a concise summary of the distribution of a dataset. These plots display the quintiles of the data: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. The box itself shows the interquartile range (IQR), which is the difference between Q3 and Q1.
The Role of Whiskers in Box and Whisker Plots
Whiskers in a box plot are essential for understanding the spread of data outside the quartiles. The lower whisker represents the smallest value within the lower quartile (Q1) minus 1.5 times the IQR, while the upper whisker represents the largest value within the upper quartile (Q3) plus 1.5 times the IQR. Anything beyond these values is considered an outlier and is typically marked with a symbol like an asterisk.
Uneven Whiskers and Their Causes
The unequal length of whiskers in a box plot primarily arises due to the presence of outliers or a non-normal distribution of data. When the data is normally distributed, the whiskers would be of equal length. However, in real-world datasets, we often encounter skewed distributions or outliers that can influence the length of the whiskers.
Example with Skewed Data
Consider the following dataset:
1 1 1 1 1 3 3 3 5 6 7 8 9 10 11 11 11 11 12 20
Dividing the data into quartiles gives:
Q1: 1 1 1 1 1 Q2: 3 3 3 5 6 median: 6.5 Q3: 7 8 9 10 11 Q4: 11 11 11 12 20In this case, the lower whisker would stretch from 1 to 3, the lower box from 3 to 6.5, the upper box from 6.5 to 11, and the upper whisker from 11 to 20. Here, the data points are relatively evenly distributed, resulting in relatively equal whisker lengths.
Example with Outliers
Now, let's consider the same dataset but with the outlier '20' removed:
1 1 1 1 1 3 3 3 5 6 7 8 9 10 11 11 11 11 12
Re-calculating the quartiles:
Q1: 1 1 1 1 1 Q2: 3 3 3 5 6 median: 6 Q3: 7 8 9 10 11 Q4: 11 11 11 12The lower whisker would stretch from 1 to 3, the lower box from 3 to 6, the upper box from 6 to 11, and the upper whisker from 11 to 12. Here, the data is less skewed and shows different whisker lengths due to the absence of the outlier '20'. This change in the distribution affects the length of the upper whisker, making it shorter than the lower whisker.
Impact of Outliers
Outliers can significantly influence the length of the whiskers. In the first dataset, the outlier '20' extended the upper whisker, making it longer compared to the lower whisker. If the outlier is considered an outlier and is excluded, its influence is removed, leading to shorter whiskers.
Conclusion
Uneven whiskers in box and whisker plots are a reflection of the distribution of data. They provide valuable insights into the presence of outliers and the skewness of the dataset. Understanding this can help in making more informed decisions about the data and the underlying phenomena being studied. Whether the data is normally distributed or skewed, the whiskers' length can provide a visual summary of the spread and potential anomalies within the dataset.