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Boxplots (also known as box and whister plots)

By , published 2019


What is a boxplot?

Boxplots are a type of chart often used in explanatory data analysis to visually show the distribution of numerical data and skewness through displaying the data quartiles (or percentiles) and averages.

features of a boxplot

Definitions

Minimum Score

The lowest score, excluding outliers (shown at the end of the left whisker).

Lower Quartile

Twenty-five percent of scores fall below the lower quartile value (also known as the first quartile).

Median

The median marks the mid-point of the data and is shown by the line that divides the box into two parts (sometimes known as the second quartile). Half the scores are greater than or equal to this value and half are less.

Upper Quartile

Seventy-five percent of the scores fall below the upper quartile value (also known as the third quartile). Thus, 25% of data are above this value.

Maximum Score

The highest score, excluding outliers (shown at the end of the right whisker).

Whiskers

The upper and lower whiskers represent scores outside the middle 50% (i.e. the lower 25% of scores and the upper 25% of scores).

The Interquartile Range (or IQR)

This is the boxplot showing the middle 50% of scores.

Why are boxplots useful?

Boxplots divide the data into sections that each contain approximately 25% of the data in that set.

boxplot quartiles

Boxplots are useful as they provide a visual summary of the data enabling researchers to quickly identify mean values, the dispersion of the data set, and signs of skewness.

Note the image above represents data which is a perfect normal distribution and most boxplots will not conform to this symmetry (where each quartile is the same length).

Boxplots are useful as they show the average score of a data set.

The median is the average value from a set of data and is shown by the line that divides the box into two parts. Half the scores are greater than or equal to this value and half are less.

Boxplots are useful as they show the skewness of a data set

The boxplot shape will show if a statistical data set is normally distributed or skewed.

boxplots showing skewness of a data set

When the median is in the middle of the box, and the whiskers are about the same on both sides of the box, then the distribution is symmetric.

When the median is closer to the bottom of the box, and if the whisker is shorter on the lower end of the box, then the distribution is positively skewed (skewed right).

When the median is closer to the top of the box, and if the whisker is shorter on the upper end of the box, then the distribution is negatively skewed (skewed left).

Boxplots are useful as they show the dispersion of a data set.

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed.

The smallest value and largest value are found at the end of the ‘whiskers’ and are useful for providing a visual indicator regarding the spread of scores (e.g. the range).

boxplot showing the interquartile range

The interquartile range (IQR) is the boxplot showing the middle 50% of scores and can be calculated by subtracting the lower quartile from the upper quartile (e.g. Q3−Q1).

Boxplots are useful as they show outliers within a data set.

An outlier is an observation that is numerically distant from the rest of the data.

When reviewing a boxplot, an outlier is defined as a data point that is located outside the whiskers of the boxplot.

boxplot outliers

Source: https://towardsdatascience.com/understanding-boxplots-5e2df7bcbd51

For example, outside 1.5 times the interquartile range above the upper quartile and below the lower quartile (Q1 - 1.5 * IQR or Q3 + 1.5 * IQR).

How to compare box plots

Box plots are a useful way to visualize differences among different samples or groups. They manage to provide a lot of statistical information, including — medians, ranges, and outliers.

Note, although boxplots have been presented horizontally in this article, it is more common to view them vertically in research papers

Step 1: Compare the medians of boxplots

Compare the respective medians of each boxplot. If the median line of a boxplot lies outside of the box of a comparison boxplot, then there is likely to be a difference between the two groups.

compare boxplot medians

Source: https://blog.bioturing.com/2018/05/22/how-to-compare-box-plots/


Step 2: Compare the interquartile ranges and whiskers of boxplots

Compare the interquartile ranges (that is, the box lengths), to examine how the data is dispersed between each sample. The longer the box the more dispersed the data. The smaller the less dispersed the data.

compare boxplot range and IQR

Next, look at the overall spread as shown by the extreme values at the end of two whiskers. This shows the range of scores (another type of dispersion). Larger ranges indicate wider distribution, that is, more scattered data.

Step 3: Look for potential outliers (see above image)

When reviewing a boxplot, an outlier is defined as a data point that is located outside the whiskers of the boxplot.


Step 4: Look for signs of skewness

If the data do not appear to be symmetric, does each sample show the same kind of asymmetry?

boxplots showing skewness of a data set compared with distribution curves


How to reference this article:

McLeod, S. A. (2019, July 19). Boxplots (also known as box and whister plots). Retrieved from https://www.simplypsychology.org/boxplots.html

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