The Box Plots Alternative for Visualizing Quantitative Data

Although bar charts are popular among researchers and are ubiquitous in quantitative software packages, they do not always provide the best visualization for a dataset. This column discusses a simple, alternative graphical method that is often underappreciated: the box plot, also known as the box-and-whisker plot. The basic elements of the box plot are presented, along with how to correctly interpret box plots, variations that are available to provide more information, and free online software that researchers can use to create box plots for publication. Typical bar chart for the balance scores of a group of patients before therapy, after therapy, and at 1-year follow-up. Mean scores for each group are shown with the top line of the bar only. Error bars show 1 standard error (SE) of the mean. N = 80, 40, and 10 as a result of patients lost to follow-up. Princeton statistician John Tukey designed the box plot as an easy-to-draw data visualization as part of exploratory data analysis 5. The box plot has persisted into the computer age as an information-rich graphic that conveys key features of a numeric dataset at a glance. Unlike the bar chart, it uses statistical summaries (median and interquartile range) that are robust in the presence of skewness and outliers and require no assumptions about the population. It shows the full range of the sample data, provides information about the tails, and indicates the shape of the data. It can be used for samples as small as n = 5 and allows for quick side-by-side comparisons between groups. Annotated box plot of 1000 points from a normal distribution with a mean of 100 and a standard deviation of 20. The data in Figure 1 yield more information when expressed as box plots (Figure 3). Here the viewer can compare the medians among the groups, and other information is also apparent. Patients before therapy had a median balance score of about 39, which rose to about 53 after therapy and dropped to about 48 a year later. The positive skew of the patients' scores before therapy can be seen from the long right (upper) whisker, revealing that whereas most patients initially tended to score fairly low, a few (including 2 high outliers) scored much higher. Immediately after therapy, the patients' scores were much higher, showing almost no overlap with scores before therapy. The data had a negative skew, seen in the longer left (lower) whisker and the one moderately low outlier. The maximum possible score of the Berg Balance Scale is 56, would explain the ceiling effect in this group of data. One year after therapy, patients' scores dropped somewhat, but the data were fairly symmetric and the range containing the middle 50% of the patients was quite wide, which indicates the scores were sparsely spread out. Simple side-by-side box plots for the data in Figure 1. Center lines show the medians, box limits indicate the 25th and 75th percentiles as determined by R software, whiskers extend 1.5 times the interquartile range from the 25th and 75th percentiles, and outliers are represented by dots. n = 80, 40, and 10 sample points. To aid in visualization, the original data can be seen overlaid the box plots in Figure 4. Data values superimposed on the box plots to aid in visualization. Center lines show the medians, box limits indicate the 25th and 75th percentiles as determined by R software, whiskers extend 1.5 times the interquartile range from the 25th and 75th percentiles, outliers are represented by dots, and data points are plotted as open circles. n = 80, 40, and 10 sample points. Samples with a large size or from a normal distribution: Whiskers are defined to cover 100% of the data (thus no outliers can be identified). Center lines show the medians, box limits indicate the 25th and 75th percentiles as determined by R software, and whiskers extend to minimum and maximum values. n = 80, 40, and 10 sample points. Whiskers are defined to cover 90% of the data. Center lines show the medians, box limits indicate the 25th and 75th percentiles as determined by R software, whiskers extend to 5th and 95th percentiles, and outliers are represented by dots. n = 80, 40, 10 sample points. The width of boxes is proportional to the square root of the sample size for each group, and thus wider boxes reflect increased precision of estimates. Center lines show the medians, box limits indicate the 25th and 75th percentiles as determined by R software, whiskers extend 1.5 times the interquartile range from the 25th and 75th percentiles, and outliers are represented by dots. n = 80, 40, and 10 sample points. Notches are ±1.58*IQR/sqrt(n) and represent the 95% confidence interval for each median. Non-overlapping notches give roughly 95% confidence that 2 medians differ, such that in 19 of 20 cases, the population medians (estimated based on the samples) are in fact different. Center lines show the medians, box limits indicate the 25th and 75th percentiles as determined by R software, whiskers extend 1.5 times the interquartile range from the 25th and 75th percentiles, outliers are represented by dots, and the width of the boxes is proportional to the square root of the sample size. n = 80, 40, and 10 sample points. Sample means (indicated by crosses) and 95% confidence intervals of the means (indicated by shaded bars) superimposed on the box plots show the same information traditionally seen on bar charts. Note that confidence intervals rely are not recommended for skewed data or small sample sizes, both of which are present here. Center lines show the medians, box limits indicate the 25th and 75th percentiles as determined by R software, whiskers extend 1.5 times the interquartile range from the 25th and 75th percentiles, outliers are represented by dots, and the width of the boxes is proportional to the square root of the sample size. n = 80, 40, and 10 sample points. Box plots are available on most software packages, including SPSS, SAS, Stata, and R. A free online tool, BoxPlotR, is also available at http://boxplot.tyerslab.com/. This program allows users to upload data, create box plots (including the variants discussed here), and download plots in various graphical formats for inclusion in a manuscript 9.