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4 examples toggle examples subsection 4 1 example without outliers 4 2 example with outliers 4 3 in the case of large datasets 4 3 1 general equation to compute empirical quantiles 5 visualization 6 see also 7 references 8 further reading 9 external links toggle the table of contents box plot 27 languages العربية català čeština dansk deutsch ελληνικά español euskara فارسی français हिन्दी bahasa indonesia italiano 日本語 한국어 nederlands norsk bokmål polski português română русский slovenščina svenska türkçe українська 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia data visualization box plot of data from the michelson experiment in descriptive statistics a box plot or boxplot is a method for demonstrating graphically the locality spread and skewness groups of numerical data through their quartiles 1 a box plot representing data in addition to the box on a box plot there can be lines which are called whiskers extending from the box indicating variability outside the upper and lower quartiles thus the plot is also called the box and whisker plot and the box and whisker diagram outliers that differ significantly from the rest of the dataset 2 may be plotted as individual points beyond the whiskers on the box plot box plots are non parametric they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution 3 though tukey s box plot assumes symmetry for the whiskers and normality for their length the spacings in each subsection of the box plot indicate the degree of dispersion spread and skewness of the data which are usually described using the five number summary in addition the box plot allows one to visually estimate various l estimators notably the interquartile range midhinge range mid range and trimean box plots can be drawn either horizontally or vertically history edit the range bar method was first introduced by mary eleanor spear in her book charting statistics in 1952 4 and again in her book practical charting techniques in 1969 5 the box and whisker plot was first introduced in 1970 by john tukey who later published on the subject in his book exploratory data analysis in 1977 6 elements edit box plot with whiskers from minimum to maximum the same box plot with whiskers drawn within the 1 5 iqr value a box plot is a standardized way of displaying the dataset based on the five number summary the minimum the maximum the sample median and the first and third quartiles minimum q 0 or 0th percentile the lowest data point in the data set excluding any outliers maximum q 4 or 100th percentile the highest data point in the data set excluding any outliers median q 2 or 50th percentile the middle value in the data set first quartile q 1 or 25th percentile also known as the lower quartile q n 0 25 it is the median of the lower half of the dataset third quartile q 3 or 75th percentile also known as the upper quartile q n 0 75 it is the median of the upper half of the dataset 7 in addition to the minimum and maximum values used to construct a box plot another important element that can also be employed to obtain a box plot is the interquartile range iqr as denoted below interquartile range iqr the distance between the upper and lower quartiles iqr q 3 q 1 q n 0 75 q n 0 25 displaystyle text iqr q_ 3 q_ 1 q_ n 0 75 q_ n 0 25 box edit the box is drawn from q 1 to q 3 with a horizontal line drawn inside it to denote the median some box plots include an additional character to represent the mean of the data 8 9 whiskers edit the whiskers must end at an observed data point but can be defined in various ways in the most straightforward method the boundary of the lower whisker is the minimum value of the data set and the boundary of the upper whisker is the maximum value of the data set because of this variability it is appropriate to describe the convention that is being used for the whiskers and outliers in the caption of the box plot another popular choice for the boundaries of the whiskers is based on the 1 5 iqr value from above the upper quartile q 3 a distance of 1 5 times the iqr is measured out and a whisker is drawn up to the largest observed data point from the dataset that falls within this distance similarly a distance of 1 5 times the iqr is measured out below the lower quartile q 1 and a whisker is drawn down to the lowest observed data point from the dataset that falls within this distance because the whiskers must end at an observed data point the whisker lengths can look unequal even though 1 5 iqr is the same for both sides all other observed data points outside the boundary of the whiskers are plotted as outliers 10 the outliers can be plotted on the box plot as a dot a small circle a star etc see example below there are other representations in which the whiskers can stand for several other things such as one standard deviation above and below the mean of the data set the 9th percentile and the 91st percentile of the data set the 2nd percentile and the 98th percentile of the data set rarely box plot can be plotted without the whiskers this can be appropriate for sensitive information to avoid whiskers and outliers disclosing actual values observed 11 the unusual percentiles 2 9 91 98 are sometimes used for whisker cross hatches and whisker ends to depict the seven number summary if the data are normally distributed the locations of the seven marks on the box plot will be equally spaced on some box plots a cross hatch is placed before the end of each whisker variations edit four box plots with and without notches and variable width since the mathematician john w tukey first popularized this type of visual data display in 1969 several variations on the classical box plot have been developed and the two most commonly found variations are the variable width box plots and the notched box plots 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 a popular convention is to make the box width proportional to the square root of the size of the group 12 notched box plots apply a notch or narrowing of the box around the median notches are useful in offering a rough guide of the significance of the difference of medians if the notches of two boxes do not overlap this will provide evidence of a statistically significant difference between the medians the height of the notches is proportional to the interquartile range iqr of the sample and is inversely proportional to the square root of the size of the sample however there is an uncertainty about the most appropriate multiplier as this may vary depending on the similarity of the variances of the samples 12 the width of the notch is arbitrarily chosen to be visually pleasing and should be consistent amongst all box plots being displayed on the same page one convention for obtaining the boundaries of these notches is to use a distance of 1 58 iqr n displaystyle pm frac 1 58 text iqr sqrt n around the median 13 adjusted box plots are intended to describe skew distributions and they rely on the medcouple statistic of skewness 14 for a medcouple value of mc the lengths of the upper and lower whiskers on the box plot are respectively defined to be 1 5 iqr e 3 mc 1 5 iqr e 4 mc if mc 0 1 5 iqr e 4 mc 1 5 iqr e 3 mc if mc 0 displaystyle begin matrix 1 5 text iqr cdot e 3 text mc 1 5 text iqr cdot e 4 text mc text if text mc geq 0 1 5 text iqr cdot e 4 text mc 1 5 text iqr cdot e 3 text mc text if text mc leq 0 end matrix for a symmetrical data distribution the medcouple will be zero and this reduces the adjusted box plot to the tukey s box plot with equal whisker lengths of 1 5 iqr displaystyle 1 5 text iqr for both whiskers other kinds of box plots such as the violin plots and the bean plots can show the difference between single modal and multimodal distributions which cannot be observed from the original classical box plot 6 examples edit example without outliers edit a box plot with no outliers a series of hourly temperatures were measured throughout the day in degrees fahrenheit the recorded values are listed in order as follows f 57 57 57 58 63 66 66 67 67 68 69 70 70 70 70 72 73 75 75 76 76 78 79 81 a box plot of the data set can be generated by first calculating five relevant values of this data set minimum maximum median q 2 first quartile q 1 and third quartile q 3 the minimum is the smallest number of the data set in this case the minimum recorded day temperature is 57 f the maximum is the largest number of the data set in this case the maximum recorded day temperature is 81 f the median is the middle number of the ordered data set this means that exactly 50 of the elements are below the median and 50 of the elements are greater than the median the median of this ordered data set is 70 f the first quartile value q 1 or 25th percentile is the number that marks one quarter of the ordered data set 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 than it the first quartile value can be easily determined by finding the middle number between the minimum and the median for the hourly temperatures the middle number found between 57 f and 70 f is 66 f the third quartile value q 3 or 75th percentile is the number that marks three quarters of the ordered data set in other words there are exactly 75 of the elements that are less than the third quartile and 25 of the elements that are greater than it the third quartile value can be easily obtained by finding the middle number between the median and the maximum for the hourly temperatures the middle number between 70 f and 81 f is 75 f the interquartile range or iqr can be calculated by subtracting the first quartile value q 1 from the third quartile value q 3 iqr q 3 q 1 75 f 66 f 9 f displaystyle text iqr q_ 3 q_ 1 75 circ f 66 circ f 9 circ f hence 1 5 iqr 1 5 9 f 13 5 f displaystyle 1 5 text iqr 1 5 cdot 9 circ f 13 5 circ f 1 5 iqr above the third quartile is q 3 1 5 iqr 75 f 13 5 f 88 5 f displaystyle q_ 3 1 5 text iqr 75 circ f 13 5 circ f 88 5 circ f 1 5 iqr below the first quartile is q 1 1 5 iqr 66 f 13 5 f 52 5 f displaystyle q_ 1 1 5 text iqr 66 circ f 13 5 circ f 52 5 circ f the upper whisker boundary of the box plot is the largest data value that is within 1 5 iqr above the third quartile here 1 5 iqr above the third quartile is 88 5 f and the maximum is 81 f therefore the upper whisker is drawn at the value of the maximum which is 81 f similarly the lower whisker boundary of the box plot is the smallest data value that is within 1 5 iqr below the first quartile here 1 5 iqr below the first quartile is 52 5 f and the minimum is 57 f therefore the lower whisker is drawn at the value of the minimum which is 57 f example with outliers edit a box plot with outliers above is an example without outliers here is a follow up example for generating box plot with outliers the ordered set for the recorded temperatures is f 52 57 57 58 63 66 66 67 67 68 69 70 70 70 70 72 73 75 75 76 76 78 79 89 in this example only the first and the last number are changed the median third quartile and first quartile remain the same in this case the maximum value in this data set is 89 f and 1 5 iqr above the third quartile is 88 5 f the maximum is greater than 1 5 iqr plus the third quartile so the maximum is an outlier therefore the upper whisker is drawn at the greatest value smaller than 1 5 iqr above the third quartile which is 79 f similarly the minimum value in this data set is 52 f and 1 5 iqr below the first quartile is 52 5 f the minimum is smaller than 1 5 iqr minus the first quartile so the minimum is also an outlier therefore the lower whisker is drawn at the smallest value greater than 1 5 iqr below the first quartile which is 57 f in the case of large datasets edit an additional example for obtaining box plot from a data set containing a large number of data points is general equation to compute empirical quantiles edit q n p x k α x k 1 x k displaystyle q_ n p x_ k alpha x_ k 1 x_ k with k p n 1 and α p n 1 k displaystyle text with k p n 1 text and alpha p n 1 k here x k displaystyle x_ k stands for the general ordering of the data points i e if i k displaystyle i k then x i x k displaystyle x_ i x_ k using the above example that has 24 data points n 24 one can calculate the median first and third quartile either mathematically or visually median q n 0 5 x 12 0 5 25 12 x 13 x 12 70 0 5 25 12 70 70 70 f displaystyle begin aligned q_ n 0 5 x_ 12 0 5 cdot 25 12 cdot x_ 13 x_ 12 5pt 70 0 5 cdot 25 12 cdot 70 70 70 circ text f end aligned first quartile q n 0 25 x 6 0 25 25 6 x 7 x 6 66 0 25 25 6 66 66 66 f displaystyle begin aligned q_ n 0 25 x_ 6 0 25 cdot 25 6 cdot x_ 7 x_ 6 5pt 66 0 25 cdot 25 6 cdot 66 66 66 circ text f end aligned third quartile q n 0 75 x 18 0 75 25 18 x 19 x 18 75 0 75 25 18 75 75 75 f displaystyle begin aligned q_ n 0 75 x_ 18 0 75 cdot 25 18 cdot x_ 19 x_ 18 5pt 75 0 75 cdot 25 18 cdot 75 75 75 circ text f end aligned box plot and a probability density function pdf of a normal n 0 1σ 2 population box plots displaying the skewness of the data set mathematics portal visualization edit although box plots may seem more primitive than histograms or kernel density estimates they do have a number of advantages first the box plot enables statisticians to do a quick graphical examination on one or more data sets box plots also take up less space and are therefore particularly useful for comparing distributions between several groups or sets of data in parallel lastly the overall structure of histograms and kernel density estimate can be strongly influenced by the choice of number and width of bins techniques and the choice of bandwidth respectively although looking at a statistical distribution is more common than looking at a box plot it can be useful to compare the box plot against the probability density function theoretical histogram for a normal n 0 σ 2 distribution and observe their characteristics directly see also edit bagplot contour boxplot data and information visualization exploratory data analysis fan chart five number summary functional boxplot seasonality seven number summary sina plot violin plot references edit c dutoit s h 2012 graphical exploratory data analysis springer isbn 978 1 4612 9371 2 oclc 1019645745 cite book cs1 maint multiple names authors list link grubbs frank e february 1969 procedures for detecting outlying observations in samples technometrics 11...
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