Box plot - wikipedia Box plot Figure 1. Box plot of data from the Michelson -- Morley experiment In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending vertically from the boxes ( whiskers ) indicating variability outside the upper and lower quartiles, hence the terms box - and - whisker plot and box - and - whisker diagram. Outliers may be plotted as individual points. Box plots are non-parametric : they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution. The spacings between the different parts of the box indicate the degree of dispersion ( spread ) and skewness in the data, and show outliers. 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. Box plots can be drawn either horizontally or vertically. Box plots received their name from the box in the middle. Contents 1 Types of box plots 2 Variations 3 Visualization 4 See also 5 References 6 Further reading 7 External links Types of box plots Figure 2. Boxplot with whiskers from minimum to maximum Figure 3. Same Boxplot with whiskers with maximum 1.5 IQR Box and whisker plots are uniform in their use of the box : the bottom and top of the box are always the first and third quartiles, and the band inside the box is always the second quartile ( the median ). But the ends of the whiskers can represent several possible alternative values, among them : the minimum and maximum of all of the data ( as in figure 2 ) the lowest datum still within 1.5 IQR of the lower quartile, and the highest datum still within 1.5 IQR of the upper quartile ( often called the Tukey boxplot ) ( as in figure 3 ) one standard deviation above and below the mean of the data the 9th percentile and the 91st percentile the 2nd percentile and the 98th percentile. 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. Some box plots include an additional character to represent the mean of the data. On some box plots a crosshatch is placed on each whisker, before the end of the whisker. Rarely, box plots can be presented with no whiskers at all. Because of this variability, it is appropriate to describe the convention being used for the whiskers and outliers in the caption for the plot. The unusual percentiles 2 %, 9 %, 91 %, 98 % are sometimes used for whisker cross-hatches and whisker ends to show the seven - number summary. If the data is normally distributed, the locations of the seven marks on the box plot will be equally spaced. Variations Figure 4. Four box plots, with and without notches and variable width Since the mathematician John W. Tukey introduced this type of visual data display in 1969, several variations on the traditional box plot have been described. Two of the most common are variable width box plots and notched box plots ( see Figure 4 ). 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. Notched box plots apply a `` notch '' or narrowing of the box around the median. 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. 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. However, there is uncertainty about the most appropriate multiplier ( as this may vary depending on the similarity of the variances of the samples ). One convention is to use ± 1.58 I Q R n ( displaystyle pm ( \frac ( 1.58 IQR ) ( sqrt ( n ) ) ) ). Adjusted box plots are intended for skew distributions. They rely on the medcouple statistic of skewness. For a medcouple value of MC, the lengths of the upper and lower whiskers are respectively defined to be 1.5 I Q R ⋅ e 3 M C, 1.5 I Q R ⋅ e − 4 M C if M C ≥ 0 1.5 I Q R ⋅ e 4 M C, 1.5 I Q R ⋅ e − 3 M C if M C ≤ 0 ( displaystyle ( \begin ( matrix ) 1.5 IQR cdot e ^ ( 3MC ), &1. 5IQR cdot e ^ ( - 4MC ) ( \text ( if ) ) MC geq 0 1.5 IQR cdot e ^ ( 4MC ), &1. 5IQR cdot e ^ ( - 3MC ) ( \text ( if ) ) MC leq 0 end ( matrix ) ) ). Observe that for symmetrical distributions, the medcouple will be zero, and this reduces to Tukey 's boxplot with equal whisker lengths of 1.5 I Q R ( displaystyle 1.5 IQR ) for both whiskers. Visualization Figure 5. Boxplot and a probability density function ( pdf ) of a Normal N ( 0, 1σ ) Population The box plot allows quick graphical examination of one or more data sets. Box plots may seem more primitive than a histogram or kernel density estimate but they do have some advantages. 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 ). 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. 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, 1σ ) distribution may be a useful tool for understanding the box plot ( Figure 5 ). See also Bivariate boxplot Candlestick chart Exploratory data analysis Fan chart Five - number summary Functional boxplot Seven - number summary Violin plot References ^ Jump up to : McGill, Robert ; Tukey, John W. ; Larsen, Wayne A. ( February 1978 ). `` Variations of Box Plots ''. The American Statistician. 32 ( 1 ) : 12 -- 16. doi : 10.2307 / 2683468. JSTOR 2683468. ^ Jump up to : Frigge, Michael ; Hoaglin, David C. ; Iglewicz, Boris ( February 1989 ). `` Some Implementations of the Boxplot ''. The American Statistician. 43 ( 1 ) : 50 -- 54. doi : 10.2307 / 2685173. JSTOR 2685173. ^ Jump up to : `` R : Box Plot Statistics ''. R manual. Retrieved 26 June 2011. Jump up ^ M. Hubert ; E. Vandervieren ( 2008 ). `` An adjusted boxplot for skewed distributions ''. Computational Statistics and Data Analysis. 52 ( 12 ) : 5186 -- 5201. doi : 10.1016 / j. csda. 2007.11. 008. Further reading John W. Tukey ( 1977 ). Exploratory Data Analysis. Addison - Wesley. Benjamini, Y. ( 1988 ). `` Opening the Box of a Boxplot ''. The American Statistician. 42 ( 4 ) : 257 -- 262. doi : 10.2307 / 2685133. JSTOR 2685133. Rousseeuw, P.J. ; Ruts, I. ; Tukey, J.W. ( 1999 ). `` The Bagplot : A Bivariate Boxplot ''. The American Statistician. 53 ( 4 ) : 382 -- 387. doi : 10.2307 / 2686061. JSTOR 2686061. External links Wikimedia Commons has media related to Box plots. Visual Presentation of Data by Means of Box Plots On - line box plot calculator with explanations and examples ( Has beeswarm example ) Beeswarm Boxplot - superimposing a frequency - jittered stripchart on top of a box plot Complex online box plot creator with example data - see also BoxPlotR : a web tool for generation of box plots Spitzer et al. Nature Methods 11, 121 -- 122 ( 2014 ) Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Central limit theorem Moments Skewness Kurtosis L - moments Count data Index of dispersion Summary tables Grouped data Frequency distribution Contingency table Dependence Pearson product - moment correlation Rank correlation Spearman 's rho Kendall 's tau Partial correlation Scatter plot Graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q -- Q plot Run chart Scatter plot Stem - and - leaf display Radar chart Data collection Study design Population Statistic Effect size Statistical power Sample size determination Missing data Survey methodology Sampling stratified cluster Standard error Opinion poll Questionnaire Controlled experiments Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment Uncontrolled studies Observational study Natural experiment Quasi-experiment Statistical inference Statistical theory Population Statistic Probability distribution Sampling distribution Order statistic Empirical distribution Density estimation Statistical model L space Parameter location scale shape Parametric family Likelihood ( monotone ) Location -- scale family Exponential family Completeness Sufficiency Statistical functional Bootstrap U V Optimal decision loss function Efficiency Statistical distance divergence Asymptotics Robustness Frequentist inference Point estimation Estimating equations Maximum likelihood Method of moments M - estimator Minimum distance Unbiased estimators Mean - unbiased minimum - variance Rao -- Blackwellization Lehmann -- Scheffé theorem Median unbiased Plug - in Interval estimation Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife Testing hypotheses 1 - & 2 - tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons Parametric tests Likelihood - ratio Wald Score Specific tests Z - test ( normal ) Student 's t - test F - test Goodness of fit Chi - squared G - test Kolmogorov -- Smirnov Anderson -- Darling Lilliefors Jarque -- Bera Normality ( Shapiro -- Wilk ) Likelihood - ratio test Model selection Cross validation AIC BIC Rank statistics Sign Sample median Signed rank ( Wilcoxon ) Hodges -- Lehmann estimator Rank sum ( Mann -- Whitney ) Nonparametric anova 1 - way ( Kruskal -- Wallis ) 2 - way ( Friedman ) Ordered alternative ( Jonckheere -- Terpstra ) Bayesian inference Bayesian probability prior posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator Correlation Regression analysis Correlation Pearson product - moment Partial correlation Confounding variable Coefficient of determination Regression analysis Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines ( MARS ) Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity Generalized linear model Exponential families Logistic ( Bernoulli ) / Binomial / Poisson regressions Partition of variance Analysis of variance ( ANOVA, anova ) Analysis of covariance Multivariate ANOVA Degrees of freedom Categorical / Multivariate / Time - series / Survival analysis Categorical Cohen 's kappa Contingency table Graphical model Log - linear model McNemar 's test Multivariate Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model Factor analysis Multivariate distributions Elliptical distributions Normal Time - series General Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality Specific tests Dickey -- Fuller Johansen Q - statistic ( Ljung -- Box ) Durbin -- Watson Breusch -- Godfrey Time domain Autocorrelation ( ACF ) partial ( PACF ) Cross-correlation ( XCF ) ARMA model ARIMA model ( Box -- Jenkins ) Autoregressive conditional heteroskedasticity ( ARCH ) Vector autoregression ( VAR ) Frequency domain Spectral density estimation Fourier analysis Wavelet Survival Survival function Kaplan -- Meier estimator ( product limit ) Proportional hazards models Accelerated failure time ( AFT ) model First hitting time Hazard function Nelson -- Aalen estimator Test Log - rank test Applications Biostatistics Bioinformatics Clinical trials / studies Epidemiology Medical statistics Engineering statistics Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification Social statistics Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics Spatial statistics Cartography Environmental statistics Geographic information system Geostatistics Kriging Portal Commons WikiProject Retrieved from `` https://en.wikipedia.org/w/index.php?title=Box_plot&oldid=819355496 '' Categories : Statistical charts and diagrams Statistical outliers Talk Contents Interaction About Wikipedia Català Čeština Dansk Deutsch Español Euskara فارسی Français 한국어 Italiano Nederlands 日本 語 Polski Português Română Русский Slovenščina Svenska Türkçe Українська 中文 Edit links This page was last edited on 8 January 2018, at 22 : 42. 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Question: when is a box and whisker plot used
Answer: for whisker cross-hatches and whisker ends to show the seven-number summary