### Topic Page: Box plot

**Box-and-Whisker Plot**

*Encyclopedia of Research Design*

A box-and-whisker plot, or box plot, is a tool used to visually display the range, distribution symmetry, and central tendency of a distribution in order to illustrate the variability and the concentration of values within a distribution. The box plot is a graphical representation of the five-number summary, or a quick way of summarizing the center and dispersion of data for a variable. The five-number summary includes the minimum value, 1st (lower) quartile (Q1), median, 3rd (upper) quartile (Q3), and the maximum value. Outliers are also indicated on a box plot. Box plots are especially useful in research methodology and data analysis as one of the many ways to visually represent data. From this visual representation, researchers glean several pieces of information that may aid in drawing conclusions, exploring unexpected patterns in the data, or prompting the researcher to develop future research questions and hypotheses. This entry provides an overview of the history of the box plot, key components and construction of the box plot, and a discussion of the appropriate uses of a box plot.

A box plot is one example of a graphical technique used within exploratory data analysis (EDA). EDA is a statistical method used to explore and understand data from several angles in social science research. EDA grew out of work by John Tukey and his associates in the 1960s and was developed to broadly understand the data, graphically represent data, generate hypotheses and build models to guide research, add robust measures to an analysis, and aid the researcher in finding the most appropriate method for analysis. EDA is especially helpful when the researcher is interested in identifying any unexpected or misleading patterns in the data. Although there are many forms of EDA, researchers must employ the most appropriate form given the specific procedure's purpose and use.

One of the first steps in any statistical analysis is to describe the central tendency and the variability of the values for each variable included in the analysis. The researcher seeks to understand the center of the distribution of values for a given variable (central tendency) and how the rest of the values fall in relation to the center (variability). Box plots are used to visually display variable distributions through the display of robust statistics, or statistics that are more resistant to the presence of outliers in the data set. Although there are somewhat different ways to construct box plots depending on the way in which the researcher wants to display outliers, a box plot always provides a visual display of the five-number summary. The median is defined as the value that falls in the middle after the values for the selected variable are ordered from lowest to highest value, and it is represented as a line in the middle of the rectangle within a box plot. As it is the central value, 50% of the data lie above the median and 50% lie below the median. When the distribution contains an odd number of values, the median represents an actual value in the distribution. When the distribution contains an even number of values, the median represents an average of the two middle values.

To create the rectangle (or box) associated with a box plot, one must determine the 1st and 3rd quartiles, which represent values (along with the median) that divide all the values into four sections, each including approximately 25% of the values. The 1st (lower) quartile (Q1) represents a value that divides the lower 50% of the values (those below the median) into two equal sections, and the 3rd (upper) quartile (Q3) represents a value that divides the upper 50% of the values (those above the median) into two equal sections. As with calculating the median, quartiles may represent the average of two values when the number of values below and above the median is even. The rectangle of a box plot is drawn such that it extends from the 1st quartile through the 3rd quartile and thereby represents the interquartile range (IQR; the distance between the 1st and 3rd quartiles). The rectangle includes the median.

In order to draw the “whiskers” (i.e., lines extending from the box), one must identify fences, or values that represent minimum and maximum values that would not be considered outliers. Typically, fences are calculated to be Q − 1.5 IQR (lower fence) and Q_{3} + 1.5 IQR (upper fence). Whiskers are lines drawn by connecting the most extreme values that fall within the fence to the lines representing Q1 and Q3. Any value that is greater than the upper fence or lower than the lower fence is considered an outlier and is displayed as a special symbol beyond the whiskers. Outliers that extend beyond the fences are typically considered mild outliers on the box plot. An extreme outlier (i.e., one that is located beyond 3 times the length of the IQR from the 1st quartile (if a low outlier) or 3rd quartile (if a high outlier) may be indicated by a different symbol. Figure 1 provides an illustration of a box plot.

Box plots can be created in either a vertical or a horizontal direction. (In this entry, a vertical box plot is generally assumed for consistency.) They can often be very helpful when one is attempting to compare the distributions of two or more data sets or variables on the same scale, in which case they can be constructed side by side to facilitate comparison.

The following six steps are used to create a vertical box plot:

Order the values within the data set from smallest to largest and calculate the median, lower quartile (Q

_{1}), upper quartile (Q_{3}), and minimum and maximum values.Calculate the IQR.

Determine the lower and upper fences.

Using a number line or graph, draw a box to mark the location of the 1st and 3rd quartiles. Draw a line across the box to mark the median.

Make a short horizontal line below and above the box to locate the minimum and maximum values that fall within the lower and upper fences. Draw a line connecting each short horizontal line to the box. These are the box plot whiskers.

Mark each outlier with an asterisk or an “o.”

R. Lyman Ott and Michael Longnecker described five inferences that one can make from a box plot. First, the researcher can easily identify the median of the data by locating the line drawn in the middle of the box. Second, the researcher can easily identify the variability of the data by looking at the length of the box. Longer boxes illustrate greater variability whereas shorter box lengths illustrate a tighter distribution of the data around the median. Third, the researcher can easily examine the symmetry of the middle 50% of the data distribution by looking at where the median line falls in the box. If the median is in the middle of the box, then the data are evenly distributed on either side of the median, and the distribution can be considered symmetrical. Fourth, the researcher can easily identify outliers in the data by the asterisks outside the whiskers. Fifth, the researcher can easily identify the skewness of the distribution. On a distribution curve, data skewed to the right show more of the data to the left with a long “tail” trailing to the right. The opposite is shown when the data are skewed to the left. To identify skewness on a box plot, the researcher looks at the length of each half of the box plot. If the lower or left half of the box plot appears longer than the upper or right half, then the data are skewed in the lower direction or skewed to the left. If the upper half of the box plot appears longer than the lower half, then the data are skewed in the upper direction or skewed to the right. If a researcher suspects the data are skewed, it is recommended that the researcher investigate further by means of a histogram.

Over the past few decades, the availability of several statistical software packages has made EDA easier for social science researchers. However, these statistical packages may not calculate parts of a box plot in the same way, and hence some caution is warranted in their use. One study conducted by Michael Frigge, David C. Hoaglin, and Boris Iglewicz found that statistical packages calculate aspects of the box plot in different ways. In one example, the authors used three different statistical packages to create a box plot with the same distribution. Though the median looked approximately the same across the three box plots, the differences appeared in the length of the whiskers. The reason for the differences was the way the statistical packages used the interquartile range to calculate the whiskers. In general, to calculate the whiskers, one multiplies the interquartile range by a constant and then adds the result to Q_{3} and subtracts it from Q_{1}. Each package used a different constant, ranging from 1.0 to 3.0. Though packages typically allow the user to adjust the constant, a package typically sets a default, which may not be the same as another package's default. This issue, identified by Frigge and colleagues, is important to consider because it guides the identification of outliers in the data. In addition, such variations in calculation lead to the lack of a standardized process and possibly to consumer confusion. Therefore, the authors provided three suggestions to guide the researcher in using statistical packages to create box plots. First, they suggested using a constant of 1.5 when the number of observations is between 5 and 20. Second, they suggested using a constant of 2.0 for outlier detection and rejection. Finally, they suggested using a constant of 3.0 for extreme cases. In the absence of standardization across statistical packages, researchers should understand how a package calculates whiskers and follow the suggested constant values.

As with all forms of data analysis, there are many advantages and disadvantages, appropriate uses, and certain precautions researchers should consider when using a box plot to display distributions. Box plots provide a good visualization of the range and potential skewness of the data. A box plot may provide the first step in exploring unexpected patterns in the distribution because box plots provide a good indication of how the data are distributed around the median. Box plots also clearly mark the location of mild and extreme outliers in the distribution. Other forms of graphical representation that graph individual values, such as dot plots, may not make this clear distinction. When used appropriately, box plots are useful in comparing more than one sample distribution side by side. In other forms of data analysis, a researcher may choose to compare data sets using a t test to compare means or an F test to compare variances. However, these methods are more vulnerable to skewness in the presence of extreme values. These methods must also meet normality and equal variance assumptions. Alternatively, box plots can compare the differences between variable distributions without the need to meet certain statistical assumptions.

However, unlike other forms of EDA, box plots show less detail than a researcher may need. For one, box plots may display only the five-number summary. They do not provide frequency measures or the quantitative measure of variance and standard deviation. Second, box plots are not used in a way that allows the researcher to compare the data with a normal distribution, which stem plots and histograms do allow. Finally, box plots would not be appropriate to use with a small sample size because of the difficulty in detecting outliers and finding patterns in the distribution.

Besides taking into account the advantages and disadvantages of using a box plot, one should consider a few precautions. In a 1990 study conducted by John T. Behrens and colleagues, participants frequently made judgment errors in determining the length of the box or whiskers of a box plot. In part of the study, participants were asked to judge the length of the box by using the whisker as a judgment standard. When the whisker length was longer than the box length, the participants tended to overestimate the length of the box. When the whisker length was shorter than the box length, the participants tended to underestimate the length of the box. The same result was found when the participants judged the length of the whisker by using the box length as a judgment standard. The study also found that compared with vertical box plots, box plots positioned horizontally were associated with fewer judgment errors.

Exploratory Data Analysis, Histogram, Outlier

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