2.4: Box Plots, Quartiles, and Percentiles
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Introduce box plots
- Define quartiles
- Define percentiles
- Calculate percentiles
- Calculate values for a five-number summary
Section \(2.4\) Excel File (contains all of the data sets for this section)
Using Box Plots to Visualize Data
Frequency distributions and their graphs (bar graphs and histograms) provide insight into data by grouping observations into classes and then determining each class's frequency or relative frequency. The classes depend on the values our data takes on, and there is some freedom regarding the number of classes we might choose to separate our data into.
Another method of graphing ordinal, interval, or ratio level data, called a box plot (or a box-and-whisker plot), groups data into four classes based on order, each containing approximately \(25\%\) of the observations. Consider the following figure containing three box plots relating students' final grades in statistics across different universities.
Figure \(\PageIndex{1}\): Box plots of final statistics grades for three universities
We know that there are four classes for each box plot. Note the five vertical lines; these values correspond to the boundaries of the classes. The left-most line corresponds to the data set's smallest value, the minimum. The first class extends from the minimum to the left side of the "box" and includes \(25\%\) of the observations; we call the upper bound for this first class, the first quartile \(Q_1\) (one-quarter of the observations are less than or equal to it). The second class again needs to have \(25\%\) of the observations with a lower bound of \(Q_1\) and an upper bound \(Q_2\). We call the upper bound of the second class the second quartile \(Q_2\), which is most commonly referred to as the median, meaning 50% of the data fall below this value. The third class again needs to have \(25\%\) of the observations with a lower bound of \(Q_2\) and an upper bound of the third quartile \(Q_3\); \(75\%\) of the observations are less than or equal to \(Q_3\). The final class consists of the remaining \(25\%\) of observations with lower bound \(Q_3\) and upper bound of the largest value, the maximum, of the data set. The first and fourth classes form the whiskers of the box plot, while the second and third form the box. These five numbers (minimum, \(Q_1\), \(Q_2\), \(Q_3\), and maximum) form what we call the five-number summary of the data.
Consider Figure \(2.4.1\) above and classify each box plot as positively skewed, negatively skewed, or symmetric. Explain.
- Answer
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We preface our answer by noting that box plots, like histograms and bar graphs from grouped frequency distributions, are formed by grouping observations. Since the graph does not provide information about each data value, we are primarily making a claim about a characteristic of the graphical representation.
Recall that a graph is positively skewed if the right tail extends further than the left tail and negatively skewed if the left tail extends further than the right tail. A graph is symmetric if we can fold it in half so that the left and right sides roughly match. The tails and the whiskers fall in similar parts of the graphs discussed. If one whisker is longer, we can say that the box plot is skewed in the direction of the longer whisker. We classify the box plot of University C as negatively skewed and the box plot of University B as positively skewed. The whiskers seem to be of equal length for University A, but that is not enough to assert symmetry. We also want the two halves of the box to be the same. This is the case for the box plot of University A. We classify the box plot of University A as symmetric.
Given a data set, we can quickly identify the minimum and maximum values. Determining the quartiles, however, presents more of a challenge. With an ordered data set, we understand where a quarter, half, and three-quarters of the data would fall. Consider the following data sets with their five-number summaries.
Figure \(\PageIndex{2}\): Three data sets with intuitive placement of the box plot boundaries
We need help determining our five-number summary for these data sets.
The first two data sets each contain \(8\) observations, while the third contains \(9\) observations. We can easily group the first two data sets in groups of \(2\) to get \(25\%\) of the observations in each class. With \(9\) observations, however, we cannot get exactly \(25\%\) in each class. However, note that there are equal numbers of observations above and below \(Q_2\) for each data set.
If we try to attach values to quartiles, we face another challenge. Let us begin with \(Q_1\). In the first data set, we see that \(Q_1\) naturally falls between \(4\) and \(6,\) but what value should we assign? Our box plots would look significantly different if we used \(6\) as opposed to \(4\). There is no easy solution to this challenge, and statisticians have developed a variety of approaches. We will provide a simple approach later in this section; please remember that it is not the only approach. However, most of the various approaches produce measures that are reasonably close to each other.
We face another challenge when we try to understand \(Q_1\) in the second data set. We would naturally assign a value of \(4\) to \(Q_1\) because the only number between \(4\) and \(4\) is \(4\). We wanted \(25\%\) of the observations to be less than or equal to \(Q_1,\) but we have \(3\) values that are less than or equal to \(4\) in our data set, that is \(37.5\%\) of our observations.
We highlight these challenges to frame our understanding appropriately. We use box plots, quartiles, and percentiles (which we will get to shortly) to get a general, intuitive feel about our data using methods that may differ from field to field, statistician to statistician, and program to program. When consuming statistics or conducting analysis, know which method is in use.
Quartiles are descriptive statistics that express at what values there will be about \(25\%,\) \(50\%,\) or \(75\%\) of the observations at or below that value. There is nothing extraordinarily unique about \(25\%,\) \(50\%,\) or \(75\%.\) We could choose \(10\%\) or \(99\%.\) When we expand our ideas to include different percentages of observations, we call them percentiles. \(Q_1\) is the \(25^{th}\) percentile. \(Q_2\) is the \(50^{th}\) percentile. \(Q_3\) is the \(75^{th}\) percentile.
Percentiles have utility beyond building summary visualizations; they help us understand how individual observations compare to the entire data set. They measure relative position within an ordered data set. For example, a test score by itself is usually difficult to interpret. For instance, if one of us had a score on a measure of shyness of \(35\) out of a possible \(50,\) we would have little idea how shy that person was compared to others. It would be helpful to know the percentage of people with equal or lower shyness scores. If \(65\%\) of the scores were at or below this person's score, then the score would be at the \(65^{th}\) percentile.
- If Helen's score was at the \(95^{th}\) percentile, what percentage of scores are at or below Helen's?
- Answer
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The percentile means that \(95\%\) of the scores are at or below Helen's score.
- If the scores ranged from \(1\) to \(100\) on an exam and Helen earned a score of \(95,\) does this necessarily mean that her score is at the \(95^{th}\)percentile?
- Answer
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No, the percentile gives a relative position of the scores. The number of scores at or below her score determines the percentile measure. If everyone did well and only \(70%\) of the scores fell at or below Helen's, she would be at the \(70^{th}\) percentile even though she got \(95\) out of a \(100\) points.
Calculating Percentiles
We already indicated that there are several different ways to calculate quartiles. This is because quartiles are percentiles, and there are several ways to calculate percentiles, which may lead to different values in different situations. The method that we present is one of the simplest calculations.
The \(P^{th}\) percentile is a value such that \(P\%\) of the observations fall at or below that value. We need the data to be counted and ordered from smallest to largest. Let \(n\) be the number of observations in our data set. Next, we calculate the number of observations that make up \(P\%\) of the observations. We call this number the rank \(R\) of the percentile.\[R=\frac{P}{100}\cdot n\nonumber\]Now there are two possibilities for the \(R\) value; either it will be a natural number \(\{1,\) \(2,\) \(3,\) \(4,\) \(5,\) \(...\}\) or not.
- If \(R\) is a natural number, we find the midpoint between the \(R^{th}\) and \((R+1)^{st}\) values.
- If \(R\) is not a natural number, round \(R\) up to the following natural number and take the value in that position.
The real number system has the following designations.
Natural Numbers: \(1,\) \(2,\) \(3,\) \(\dots\)
Whole Numbers: \(0,\) \(1,\) \(2,\) \(3,\) \(\dots\)
Integers: \(\dots,\) \(-3,\) \(-2,\) \(-1,\) \(0,\) \(1,\) \(2,\) \(3,\) \(\dots\)
Rational Numbers: Any number that can be written as a fraction
Irrational Numbers: Any number that cannot be written as a fraction. Examples include \(e,\) \(\pi,\) \(\sqrt{2}\)
Consider the \(20\) quiz scores shown in the table below and compute the five-number summary and \(82^{nd}\) percentile. After completing the calculation by hand, use the Section \(2.4\) Excel file to calculate each percentile using the functions \(\text{PERCENTILE.INC}\) and \(\text{PERCENTILE.EXC}.\) Compare the values.
Table \(\PageIndex{1}\): \(20\) quiz scores with corresponding rank
| Number | \(4\) | \(4\) | \(4\) | \(5\) | \(6\) | \(6\) | \(6\) | \(6\) | \(6\) | \(6\) | \(7\) | \(7\) | \(8\) | \(8\) | \(8\) | \(9\) | \(9\) | \(10\) | \(10\) | \(10\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Rank | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) | \(16\) | \(17\) | \(18\) | \(19\) | \(20\) |
- Answer
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We first note that the data is already ordered from smallest to largest with \(20\) observations. A secondary row has been created to index the observations. Note that the row heading is Rank; consider how this ties back to rank \(R\) in our calculation.
Let us begin with the five-number summary.
The minimum value is \(4.\)
\(Q_1,\) the \(25^{th}\) percentile. \(R=\frac{25}{100}\cdot 20\) \(=\frac{1}{4}\cdot 20\) \(= \frac{20}{4}\) \(=5.\) Note \(5\) is a natural number. We then look at the \(5^{th}\) and \(6^{th}\) observation values, which are both \(6\) and find the midpoint between them. Thus \(Q_1\) \(=6.\) Using Excel, we get \(6;\) they are all the same.
\(Q_2,\) the \(50^{th}\) percentile. \(R=\frac{50}{100}\cdot 20\) \(=\frac{1}{2}\cdot 20\) \(=\frac{20}{2}\) \(=10.\) Note \(10\) is a natural number. We then look at the \(10^{th}\) and \(11^{th}\) observation values, which are \(6\) and \(7,\) respectively and find the midpoint between them. Thus \(Q_2\) \(=\frac{1}{2}(6+7)\) \(=\frac{13}{2}\) \(=6.5.\) Excel computes \(6.5\) and \(6.5;\) they are all the same.
\(Q_3,\) the \(75^{th}\) percentile. \(R=\frac{75}{100}\cdot 20\) \(=\frac{3}{4}\cdot 20\) \(= \frac{60}{4}\) \(=15.\) Note \(15\) is a natural number. We then look at the \(15^{th}\) and \(16^{th}\) observation values, which are \(8\) and \(9,\) respectively and find the midpoint between them. Thus \(Q_3\) \(=\frac{1}{2}(8+9)\) \(=\frac{17}{2}\) \(=8.5.\) Excel gives \(8.25\) and \(8.75;\) they are all different.
The maximum value is \(10.\)
Let us look at the \(82^{nd}\) percentile. \(R=\frac{82}{100}\cdot 20\) \(=\frac{41}{50}\cdot 20\) \(=\frac{820}{50}\) \(= \frac{82}{5}\) \(=16.4.\) Notice \(16.4\) is not a natural number. We must look at the \(17^{th}\) observation value showing the \(82^{nd}\) percentile is \(9.\) Excel calculates \(9.22\) and \(9;\) they are different and the same respectively.
We have seen that different methods of calculation can produce slightly different values. With large data sets, we generally resort to technology to produce our measures and might not have control over the precise methodology employed therein. As such, we remember that percentiles provide rough measures for the distribution of our data sets and nuance our understanding that roughly this percent of observations fall below roughly this value. When large data sets or limited time make hand computation prohibitive, we recommend using functions such as \(\text{PERCENTILE.INC}\) and \(\text{QUARTILE.INC}.\)
Consider the preceding example. If we looked at the \(83^{rd}\) percentile \((R\) \(=16.6)\) which would also be the \(17^{th}\) value, which is \(9.\) So, the \(83^{rd}\) percentile is the same as the \(82^{nd}\) percentile. This happens since there are only \(20\) data points; we cannot subdivide \(20\) indefinitely. The two percentiles would typically differ with large data sets.
Box Plots: Constructing and Interpreting
As part of the "Stroop Interference Case Study," students in introductory statistics were presented with a page containing \(30\) colored rectangles. Their task was to name the colors as quickly as possible. Their times (in seconds) were recorded. Compare the scores for the \(15\) men and \(30\) women who participated in the experiment by making separate box plots for each gender.
Table \(\PageIndex{2}\): Women's times (left) and men's times (right)
| \(14\) | \(17\) | \(18\) | \(19\) | \(20\) | \(21\) | \(16\) | \(19\) | \(23\) | |
| \(15\) | \(17\) | \(18\) | \(19\) | \(20\) | \(22\) | \(17\) | \(20\) | \(24\) | |
| \(16\) | \(17\) | \(18\) | \(19\) | \(20\) | \(23\) | \(18\) | \(22\) | \(26\) | |
| \(16\) | \(17\) | \(18\) | \(20\) | \(20\) | \(24\) | \(19\) | \(22\) | \(26\) | |
| \(17\) | \(18\) | \(18\) | \(20\) | \(21\) | \(24\) | \(19\) | \(23\) | \(28\) |
- Answer
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To construct box plots, we need the five-number summaries.
Table \(\PageIndex{3}\): Five number summaries for the data presented in Table \(\PageIndex{1}\)
Females Males Box Plot Component Minimum \(14\) \(16\) End of Left Whisker \(Q_1\) \(17\)
(\(R=\frac{25}{100}\cdot 30=7.5\))
\(19\)(\(R=\frac{25}{100}\cdot 15=3.75\))
Left Side of Box \(Q_2\)=median
\(18.5\)(\(R=\frac{50}{100}\cdot 30=15\))
\(22\)(\(R=\frac{50}{100}\cdot 15=7.5\))
Line in Box \(Q_3\)
\(20\)(\(R=\frac{75}{100}\cdot 30=22.5\))
\(24\)
(\(R=\frac{75}{100}\cdot 15=11.25\))
Right Side of Box Maximum \(24\) \(28\) End of Right Whisker 
Figure \(\PageIndex{3}\): Box plots for male and female times for naming the colors of various rectangles
The men tended to take longer than the women. About \(25\%\) of male times were longer than the maximum female time. At least \(75\%\) of the male times were longer than the median female time.
Suppose data came from a task that aims to move a computer mouse to a target on the screen as fast as possible. On \(20\) of the trials, the target was a small rectangle; on the other \(20,\) the target was a large rectangle. The time to reach the target was recorded on each trial. The box plots of the two distributions are shown below. What can we conclude by looking at the two box plots?
Figure \(\PageIndex{4}\): Box plots for the response times by small and large target
- Answer
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We can see that although there is some overlap in times, it generally took longer to move the mouse to the small target than to the large one. The minimum time for the small target is longer than the median time of the large target. At least \(50\%\) of times for the small target are longer than all of the large target times.
Construct two data sets, treated as observations from a discrete quantitative variable, consisting of \(12\) values each so that the box plots are identical, but the bar graph of one data set is perfectly symmetric while the other is not.
- Answer
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If the box plots are going to be identical, the five-number summaries must be the same. When arranged from least to greatest, \(Q_1\) is the average of the \(3^{rd}\) and \(4^{th}\) values, \(Q_2\) is the average of the \(6^{th}\) and \(7^{th}\) values, and \(Q_3\) is the average of the \(9^{th}\) and \(10^{th}\) values. We can construct a perfectly symmetric data set by pairing observations by proximity to the center. Since the \(6^{th}\) and \(7^{th}\) values are in the middle, they would be paired together, the \(5^{th}\) with the \(8^{th}\), and the \(4^{th}\) and \(9^{th}\), and so forth. We want the values in these positions to be equally distant from the median value.
We can start by picking the first data set that is not perfectly symmetric. We will produce such a data set if we use a pattern of one observation value followed by a different value repeated twice. We picked even numbers starting at \(0\) to ensure that our quartiles were nice values. Not all procedures would produce a data set suitable for this example because we need our second data set to be symmetric. We must check this because we want the minimum and maximum to be equally far from the median and the first and third quartiles.
\[\{0,2,2,4,6,6,8,10,10,12,14,14\}\nonumber\]
We have fixed our five number summary: \(\min=0\),\(Q_1=3\), \(Q_2=7\), \(Q_3=11\), and \(\max=14\). Our procedure produced values allowing us to produce a symmetric data set with the same five-number summary. The average of two of the same numbers is that number, so, for ease, we can start our symmetric data set with the following numbers. \[ \{0,~~,3,3,~~,7,7,~~,11,11,~~,14\}\nonumber \] All we need to do is fill in the remaining spots, ensuring symmetry and preserving order.
\[\{0,2,2,4,6,6,8,10,10,12,14,14\}\nonumber\\ \{0,1,3,3,4,7,7,10,11,11,13,14\} \]
Figure \(\PageIndex{5}\): Two identical box plots
Figure \(\PageIndex{6}\): Non-identical bar graphs of the two data sets that produced identical box plots.