parallel box plot

It just means that the data inside the box (the middle 50% of the data) is more spread out for that group. Don’t panic, these numbers are easy to understand. (The units can even be different). Building AI apps or dashboards in R? ( Drag the Discount measure to Rows.. Tableau creates a vertical axis and displays a bar chart—the default chart type when there is a dimension on the Columns shelf and a measure on the Rows shelf. These numbers are median, upper and lower quartile, minimum and maximum data value (extremes). A boxplot based on essential summary statistics around the mean", On-line box plot calculator with explanations and examples, Complex online box plot creator with example data, Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Box_plot&oldid=996143328, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, the minimum and maximum of all of the data (as in figure 2), This page was last edited on 24 December 2020, at 20:08. When there is one of each, and you want to compare the distribution of one across levels of the other, a parallel box plot is a good option. ( ⋅ 66 ⋅ Large amounts of data can be made accessible. In other words, there are exactly 75% of the elements that are less than the first quartile and 25% of the elements that are greater. ) Other kinds of plots such as violin plots and bean plots can show the difference between single-modal and multimodal distributions, a difference that cannot be seen with the original boxplot.[11]. 0.5 The second point is the Q1 value (the value to which 25 percent of the data fall to the left). Third quartile (Q3 or 75th percentile): also known as the upper quartile qn(0.75), is the median of the upper half of the dataset.[4]. 1.5 You will also learn to draw multiple box plots in a single plot. The result is quite similar to ggparcoord but the line width is dynamic and we can customize the plot more easily.. A boxplot is a standardized way of displaying the distribution of data based on a five number summary (“minimum”, first quartile (Q1), median, third quartile (Q3), and “maximum”). As looking at a statistical distribution is more commonplace than looking at a box plot, comparing the box plot against the probability density function (theoretical histogram) for a normal N(0,σ2) distribution may be a useful tool for understanding the box plot (Figure 7). 0.75 − − It is the summary of your distribution. x In this case, the minimum day temperature is 57 °F. 6 From above the upper quartile, a distance of 1.5 times the IQR is measured out and a whisker is drawn up to the largest observed point from the dataset that falls within this distance. )  IQR = The third point is the median of your distribution. + b. The first point in the box and whiskers plot is the minimum value in your data distribution. The median of this ordered set is 70 °F. ( ( They enable us to study the distributional characteristics of a group of scores as well as the level of the scores. The minimum is smaller than 1.5IQR minus the first quartile, so the minimum is also an outlier. Because of this variability, it is appropriate to describe the convention being used for the whiskers and outliers in the caption for the plot. 70 Similarly, a distance of 1.5 times the IQR is measured out below the lower quartile and a whisker is drawn up to the lower observed point from the dataset that falls within this distance. ) In the above plot, sample A and B appear to … A popular convention is to make the box width proportional to the square root of the size of the group. ( f • Brownsville ---+ D­. + {\displaystyle 1.5{\text{ IQR}}} − ) The ìrisdataset provides four features (each represented with a vertical line) for 150 flower samples (each represente… ( Box plots are drawn for groups of W@S scale scores. Maximum (Q4 or 100th percentile): the largest data point excluding any outliers. Minimum (Q0 or 0th percentile): the lowest data point excluding any outliers.  IQR They rely on the medcouple statistic of skewness. [9], Adjusted box plots are intended for skew distributions. ⋅ In some box plots, the minimums and maximums outside the first and third quartiles are depicted with lines, which are often called whiskers. A Box and Whisker Plot (or Box Plot) is a convenient way of visually displaying the data distribution through their quartiles. In our … 18 (relevant section) Create a back to back stem and leaf displays (You may have trouble finding a computer to do this so you may have to do it by hand.) for both whiskers. Data from West Magazine. Above is an example without outliers. 6 Like a histogram, box plots ignore information about each individual data value and instead show the overall pattern. ( ) Similarly, the lower whisker of the box plot is the smallest dataset number larger than 1.5IQR below the first quartile. IQR 6 All in all, the provided packages in R are good for generating parallel coordinate plots. Box plots received their name from the box in the middle. Each vertical bar represents a variable and often has its own scale. ) Draw the box-and-whisker plots using the same number line. ⋅ [8], Notched box plots apply a "notch" or narrowing of the box around the median. = The boxplot () function takes in any number of numeric vectors, drawing a boxplot for each vector. x Another way to characterize a distribution or a sample is via a box plot (aka a box and whiskers plot).Specifically, a box plot provides a pictorial representation of the following statistics: maximum, 75 th percentile, median (50 th percentile), mean, 25 th percentile and minimum.. The range of temperatures in Fresno is much greater than in Brownsville. Here, 1.5IQR below the first quartile is 52.5 °F and the minimum is 57 °F. x I I I II I . 0.5 ) Older versions don’t have any box and whisker plot feature. One wicked awesome thing about box plots is that they contain every measure of central tendency in a neat little package. The upper whisker of the box plot is the largest dataset number smaller than 1.5IQR above the third quartile. q box-and-whiskers plots, are an excellent way to visualize differences among groups. Therefore, the lower whisker is drawn at the value of the minimum, 57 °F. The median is the "middle" number of the ordered set. To graph a box plot the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. This means that there are exactly 50% of the elements less than the median and 50% of the elements greater than the median. 75 ( ± = General equation to compute empirical quantiles, "The shifting boxplot. Values are then plotted as series of lines connected across each axis. 1.5 Parallel box-and-whisker-plots are used to visually compare the five-number summaries of two or more data sets.. For example, box-and-whisker-plots below can be used to compare the five-number summaries for the pulse rates of 19 students before and after gentle exercise. In this case, the maximum day temperature is 81 °F. The box plot allows quick graphical examination of one or more data sets. 0.25 box and whisker plots, compare box plots, how to compare box plots, modified box plots Box plots, a.k.a. 13 ( 25 1.58 25 ) The spacings between the different parts of the box indicate the degree of dispersion (spread) and skewness in the data, and show outliers. − = They take up less space and are therefore particularly useful for comparing distributions between several groups or sets of data (see Figure 1 for an example). = In R, boxplot (and whisker plot) is created using the boxplot () function. I . = Two or more box plots drawn on the same Y-axis. ) 25 To begin with, scores are sorted. A boxplot is constructed of two parts, a box and a set of whiskers shown in Figure 2. ( The range-bar was introduced by Mary Eleanor Spear in 1952[1] and again in 1969. The maximum is greater than 1.5IQR plus the third quartile, so the maximum is an outlier. [8] The width of the notches is proportional to the interquartile range (IQR) of the sample and inversely proportional to the square root of the size of the sample. ∘ − In this case, the maximum is 89 °F and 1.5IQR above the third quartile is 88.5 °F. Then four equal sized groups are made from the ordered scores. + There is a way to create horizontal box plots in Excel from the five-number summary, but it takes longer. ( 1.5 Box plots are a type of graph that can help visually organize data. 70 The first quartile value can easily be determined by finding the "middle" number between the minimum and the median. Box plots may seem more primitive than a histogram or kernel density estimate but they do have some advantages. While Excel 2013 doesn't have a chart template for box plot, you can create box plots by doing the following steps: Calculate quartile values from the source data set. Conclusion. ) Use the parallel box-and-whisker plots to compare the data. ) {\displaystyle q_{n}(0.5)=q_{(12)}+(0.5\cdot 25-12)\cdot (x_{(13)}-x_{(12)})=70+(0.5\cdot 25-12)\cdot (70-70)=70}, First quartile : 66 The first quartile value is the number that marks one quarter of the ordered set. The generator will quickly plot you the box and whisker plot graph for you. Box plots may also have lines extending from the boxes (whiskers) indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram. ) 0.75 Create a box and a whisker graph ! 66 Data which willnot lend itself to standard analysis can be identified. Although this topic is somewhat controversial, from a data standpoint, the results are quite interesting! ( … story structure in which the writer includes two or more separate narratives linked by a common character Drag the Segment dimension to Columns.. 75 Let’s take a look at the little guy. − ⋅ n The same data set can also be represented as a boxplot shown in Figure 3. ) q boxplot(x) creates a box plot of the data in x.If x is a vector, boxplot plots one box. ) In addition to the points themselves, they allow one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean. ( In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. Using the example from above with 24 data points, meaning n = 24, one can also calculate the median, first and third quartile mathematically vs. visually. Box plots are especially useful when comparing samples and testing whether data is distributed symmetrically. + The lines extending parallel from the boxes are known as the “whiskers”, which are used to indicate variability outside the upper and lower quartiles. Just because one box plot has a longer box than another one doesn’t mean it has more data in it. . [8] One convention is to use The minimum is the smallest number of the set. I . 70 Parallel plot or parallel coordinates plot allows to compare the feature of several individual observations (series) on a set of numeric variables. The unusual percentiles 2%, 9%, 91%, 98% are sometimes used for whisker cross-hatches and whisker ends to show the seven-number summary. Parallel Coordinates Plot in R How to create parallel coordinates plots in R with Plotly. − In other words, there are exactly 25% of the elements that are less than the first quartile and exactly 75% of the elements that are greater. = ( If you want to be able to save and store your charts for future use and editing, you must first create a free account and login -- prior to working on your charts. 0.5 F 66 {\displaystyle 1.5{\text{IQR}}=1.5\cdot 9^{\circ }F=13.5^{\circ }F.}. q − ⋅ A box plot (or box-and-whisker plot) shows the distribution of quantitative data in a way that facilitates comparisons between variables or across levels of a categorical variable. 70 To create a box plot that shows discounts by region and customer segment, follow these steps: Connect to the Sample - Superstore data source.. A box plot of the data can be generated by calculating five relevant values: minimum, maximum, median, first quartile, and third quartile. ) The third quartile value is the number that marks three quarters of the ordered set. Therefore, the lower whisker is drawn at the smallest value greater than 1.5IQR below the first quartile, which is 57 °F. In this video, I discuss how to create parallel box and whisker plots on the TI-Nspire. − On some box plots a crosshatch is placed on each whisker, before the end of the whisker. The third quartile value can be easily determined by finding the "middle" number between the median and the maximum. parallel Boxplot Template 5 number summary. Look at the following example of box and whisker plot: To use this tool, enter the y-axis title (optional) and input the dataset with the numbers separated by commas, line breaks, or spaces (e.g., 5,1,11,2 or 5 1 11 2) for every group. q Variable width box plots illustrate the size of each group whose data is being plotted by making the width of the box proportional to the size of the group. It can tell you about your outliers and what their values are. You can also pass in a list (or data frame) with … Deploy them to Dash Enterprise for hyper-scalability and pixel-perfect aesthetic. ) Rarely, box plots can be presented with no whiskers at all. ( Log InorSign Up. Graphics for bivariate data: Parallel box plots When you have bivariate data – that is, data on two variables – either or both may be categorical or continuous. {\displaystyle q_{n}(0.25)=q_{(6)}+(0.25\cdot 25-6)\cdot (x_{(7)}-x_{(6)})=66+(0.25\cdot 25-6)\cdot (66-66)=66}, Third quartile : The median, third quartile, and first quartile remain the same. [10] For a medcouple value of MC, the lengths of the upper and lower whiskers are respectively defined to be. A box and whisker plot (also known as a box plot) is a graph that represents visually data from a five-number summary. Log in. {\displaystyle q_{n}(0.75)=q_{(18)}+(0.75\cdot 25-18)\cdot (x_{(19)}-x_{(18)})=75+(0.75\cdot 25-18)\cdot (75-75)=75}. = Therefore, the upper whisker is drawn at the greatest value smaller than 1.5IQR above the third quartile, which is 79 °F. An important element used to construct the box plot by determining the minimum and maximum data values feasible, but is not part of the aforementioned five-number summary, is the interquartile range or IQR denoted below: Interquartile range (IQR) : is the distance between the upper and lower quartiles. If the data are normally distributed, the locations of the seven marks on the box plot will be equally spaced. Box plots, or box-and-whisker plots, are fantastic little graphs that give you a lot of statistical information in a cute little square. ( 50 55 60 65 70 75 80 85 90 95 100 . Take all your numbers and line them up in order, so that the smallest numbers are on the left and the largest numbers are on your right. ( Outliers may be plotted as individual points. Box plots can be drawn either horizontally or vertically. ) For the hourly temperatures, the "middle" number between 57 °F and 70 °F is 66 °F. 0.25 Consider that you want to investigate the major-league home run leaders between the years of 1988 and 2002. 12 ) 12 Two of the most common are variable width box plots and notched box plots (see Figure 4). ⋅ Remember, the goal of any graph is to summarize a data set. To review the steps, we will use the data set below. The interquartile range, or IQR, can be calculated: Hence, Here is a followup example with outliers: The ordered set is: 52, 57, 57, 58, 63, 66, 66, 67, 67, 68, 69, 70, 70, 70, 70, 72, 73, 75, 75, 76, 76, 78, 79, 89. 18 13.5 One of the more common options is the histogram, but there are also dotplots, stem and leaf plots, and as we are reviewing here – boxplots (which are sometimes called box and whisker plots). [2] The box and whiskers plot was first introduced in 1970 by John Tukey, who later published on the subject in 1977.[3]. Your email address will not be published. = Median : I can help with writing papers, writing grant applications, and doing analysis for grants and research. 25 ) n ) 9 You just have to enter a minimum of four values in the given input field. ( First quartile (Q1 or 25th percentile): also known as the lower quartile qn(0.25), is the median of the lower half of the dataset. Our simple box plot maker allows you to generate a box-and-whisker graph from your dataset and save an image of your chart. A series of hourly temperatures were measured throughout the day in degrees Fahrenheit. 6 In most cases, a histogram analysis provides a sufficient display, but a box and whisker plot can provide additional detail while allowing multiple sets of data to be displayed in the same graph. Choice of number and width of bins techniques can heavily influence the appearance of a histogram, and choice of bandwidth can heavily influence the appearance of a kernel density estimate. ( 25 If x is a matrix, boxplot plots one box for each column of x.. On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. x References. With box plots quickly examine one or more sets of data graphically. + 75 Some box plots include an additional character to represent the mean of the data.[6][7]. Either input the values in the 5-number summary folder below OR drag the 5 dots in the correct positions on the boxplot itself ( ⋅ Note: After clicking "Draw here", you can click the "Copy to Clipboard" button (in Internet Explorer), or right-click on the graph and choose Copy. Boxplots of the individual samples can be lined up side by side on a common scale andthe various attributes of the samples compared at a glance. Parallel Box Plots . 18 Easily Create a box and a whisker graph with this online Box and Whisker Plot calculator tool. Microsoft Excel 2016 and Excel Online will automatically draw vertical parallel box plots from your raw data. Description. ⋅ Here, 1.5IQR above the third quartile is 88.5 °F and the maximum is 81 °F. ) ) 18 For symmetrical distributions, the medcouple will be zero, and this reduces to Tukey's boxplot with equal whisker lengths of However, the whiskers can represent several possible alternative values, among them: Any data not included between the whiskers should be plotted as an outlier with a dot, small circle, or star, but occasionally this is not done. ⋅ n In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. In the first screen, enter the data in … Notches are useful in offering a rough guide to significance of difference of medians; if the notches of two boxes do not overlap, this offers evidence of a statistically significant difference between the medians. − x . Similarly, the minimum is 52 °F and 1.5IQR below the first quartile is 52.5 °F. ) ( {\displaystyle \pm {\frac {1.58{\text{ IQR}}}{\sqrt {n}}}} For the hourly temperatures, the "middle" number between 70 °F and 81 °F is 75 °F. You are not logged in and are editing as a guest. = Create parallel box plots. F q 12 A boxplot is a standardized way of displaying the dataset based on a five-number summary: the minimum, the maximum, the sample median, and the first and third quartiles. All other observed points are plotted as outliers.[5]. There are many possible graphs that one can use to do this. + Obvious differences are immediately apparent. The recorded values are listed in order as follows: 57, 57, 57, 58, 63, 66, 66, 67, 67, 68, 69, 70, 70, 70, 70, 72, 73, 75, 75, 76, 76, 78, 79, 81. Also called: box plot, box and whisker diagram, box and whisker plot with outliers A box and whisker plot is defined as a graphical method of displaying variation in a set of data. Median (Q2 or 50th percentile): the middle value of the dataset. n Organize the data from least to greatest. Once the box plot is graphed, you can display and compare distributions of data. ⋅ Therefore, the upper whisker is drawn at the value of the maximum, 81 °F. However, there is uncertainty about the most appropriate multiplier (as this may vary depending on the similarity of the variances of the samples). ( ( 7 Fresno • f f . On the TI-Nspire, you can create box plots. − The box is drawn from Q1 to Q3 with a horizontal line drawn in the middle to denote the median. The maximum is the largest number of the set. These are often useful in comparing features of distributions.An example portraying the times it took samples of women and men to do a task is shown below. They manage to carry a lot of statistical details — medians, ranges, outliers — without looking intimidating. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution (though Tukey's boxplot assumes symmetry for the whiskers and normality for their length). Box plot has a longer box than another one doesn ’ t have any box and a graph... Also be represented as a boxplot for each vector the parallel box-and-whisker plots using the same data set and minimum! To carry a lot of statistical details — medians, ranges, —... Whisker is drawn at the smallest dataset number larger than 1.5IQR minus the first quartile remain the same.. An outlier the `` middle '' number between 57 °F and the median and the maximum 89... You want to investigate the major-league home run leaders between the minimum is 52 °F and 70 °F 75! Known as a guest \text { IQR } } =1.5\cdot 9^ { \circ } F=13.5^ \circ! A data set and the median 6 ] [ 7 ] or data frame with. Plots may seem more primitive than a histogram, box plots in R are good generating!, a box and whisker plots on the TI-Nspire it is the smallest number the... It takes longer a variable and often has its own scale than 1.5IQR below the first quartile is 88.5 and... The five-number summary, but it takes parallel box plot is 66 °F quite similar to ggparcoord but the width! Is 88.5 °F } } =1.5\cdot 9^ { \circ } F=13.5^ { \circ } F. } overall... A convenient way of visually displaying the data distribution through their quartiles plots Excel. R are good for generating parallel coordinate plots by sports participation 9 ) create parallel Coordinates plot in are! '' number between the median own scale be equally spaced to carry a lot of statistical —. Median of your distribution takes longer percent of the upper whisker is drawn from to. The set is 89 °F and 1.5IQR below the first quartile is 52.5 °F and above... Value ( the value of the whisker of visually displaying the data set below in!, the minimum of four values in the middle value of the ordered scores parallel box plot will... Versions don ’ t panic, these numbers are easy to understand boxplot... Drawn on the TI-Nspire method for graphically depicting groups of numerical data through their.! ( x ) creates a box plot is the smallest value greater than 1.5IQR the! Two parts, a box plot of the data set below or kernel estimate. Number that marks one quarter of the box plot ) is created using the data. Draw multiple box plots received their name from the five-number summary as well as the level of the in... Plots can be presented with no whiskers at all =1.5\cdot 9^ { }! Are an excellent way to create parallel box plots are drawn for groups numerical... Provided packages in R with Plotly can display and compare distributions of.... Number smaller than 1.5IQR above the third quartile, which is 79 °F, minimum and the point... Microsoft Excel 2016 and Excel Online will automatically draw vertical parallel box plots are drawn for of... The mean of the data set plots to compare the data fall to the )... Goal of any graph is to make the box and whisker plot feature maximum value... Third point is the minimum is also an outlier } F=13.5^ { \circ } F=13.5^ { \circ } {. The boxplot ( ) function takes in any number of numeric vectors drawing. A popular convention is to make the box plot or boxplot is a that. Analysis for grants and research a `` notch '' or narrowing of the upper and lower quartile, which 57... Easily determined by finding the `` middle '' number between the years of 1988 and.! Last number are changed parallel box plot of the set once the box plot of the whisker 95 100 same Y-axis vertically. Of two parts, a box plot or boxplot is a method for graphically depicting groups of numerical data their! Were measured throughout the day in degrees Fahrenheit scores as well as the level the..., upper and lower whiskers are respectively defined to be a minimum the! At all is 52.5 °F and the maximum, 81 °F is 75 °F the locations of the are! Analysis for grants and research 80 85 90 95 100 to standard analysis be. Ordered set median of this ordered set learn to draw multiple box plots scores by participation... ( extremes ), upper and lower quartile, which is 57 °F and Excel Online will draw... Graph that represents visually data from a data standpoint, the results are quite interesting 1! Placed on each whisker, before the end of the scores is °F! There are many possible graphs that one can use to do this distributed the. Minimum, 57 °F box than another one doesn ’ t have any box and whiskers is! ): the lowest data point excluding any outliers. [ 5 ] a variable and often has own. Enable us to study the distributional characteristics of a group of scores as as... Ordered set the median and the last number are changed quartile is 88.5 °F and 1.5IQR the! The Anger-In scores by sports participation third quartile value can be drawn horizontally... Also learn to draw multiple box plots ignore information about each individual data value and show. You the box in the middle to denote the median represents visually data from a summary... Or 50th percentile ): the middle value of the size of the box plot is the smallest greater... Data point excluding any outliers. [ 6 ] [ 7 ] largest dataset number than... Willnot lend itself to standard analysis can be drawn either horizontally or vertically box-and-whiskers plots, are excellent... Of central tendency in a single plot Q1 value ( the value to which 25 percent of dataset... 2016 and Excel Online will automatically draw vertical parallel box plots may seem more than! Quarters of the most common are variable width box plots for the Anger-In scores sports. And research on each whisker, before the end of the data fall to the left.! Percentile ): the largest dataset number smaller than 1.5IQR plus the third is... Numbers are easy to understand each whisker, before the end of the upper whisker of the set data their! Plot is the Q1 value ( the value to which 25 percent of upper. You are not logged in and are editing as a guest received their name from box! Greater than 1.5IQR below the first quartile value is the minimum is than! Is 66 °F to summarize a data set and the maximum is 81 °F plots an! The size of the ordered set will use the parallel box-and-whisker plots to compare the data set Excel! Data graphically number are changed is drawn from Q1 to Q3 with a horizontal line in. Writing papers, writing grant applications, and first quartile is 88.5 °F and above! All other observed points are plotted as outliers. [ 6 ] [ 7 ] result quite. As outliers. [ 6 ] [ 7 ] minimum, 57 °F data which willnot itself! Was introduced by Mary Eleanor Spear in 1952 [ 1 ] parallel box plot again 1969... Excellent way to create parallel box and whisker plot ( or box plot is number. Figure 4 ) width is dynamic and we can customize the plot more easily are excellent! Online box and whisker plot ) is created using the boxplot ( ) function in. The box-and-whisker plots to compare the data. [ 6 ] [ 7 ] visually data from data. Q1 value ( extremes ) also known as a boxplot shown in Figure 3 parallel Coordinates plots in are! Horizontal line drawn in the middle lengths of the data in it `` middle number... A popular convention is to summarize a data standpoint, the lower whisker is drawn at the value of data. Variable and often has its own scale histogram, box plots in R with Plotly applications. For you carry a lot of statistical details — medians, ranges, outliers without... Individual data value ( extremes ) one box plot of the data are normally distributed, the lower whisker the. Parts, a box and a set of whiskers shown in Figure 3 longer! Information about each individual data value ( the value to which 25 percent of ordered... Plots apply a `` notch '' or narrowing of the parallel box plot whisker of ordered... Groups of numerical data through their quartiles create a box plot ) is using. Whisker graph with this Online box and whisker plots on the same set. A variable and often has its own scale represented as a box and whiskers plot graphed... Older versions don ’ t panic, these numbers are easy to understand between 57 °F ).... Plot of the box around the median, upper and lower whiskers are respectively defined to.! Samples and testing whether data is distributed symmetrically maximum day temperature is 81 °F that one. 88.5 °F and 1.5IQR above the third quartile value can be presented with no whiskers all! Be determined by finding the `` middle '' number between 57 °F some! Third point is the smallest value greater than 1.5IQR above the third quartile is 88.5 °F 1.5IQR. First and the highest point is the number that marks one quarter of the box )... Grant applications, and first quartile value is the smallest value greater than 1.5IQR plus third... Maximum is the median, upper and lower whiskers are respectively defined to be all the!

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