Probable prime - Revision history

en>Jnestorius: /* top */ rm 3rd para "A probable prime need not be prime...." 1st para already explains pseudoprime

2014-11-22T00:14:18Z

top: rm 3rd para "A probable prime need not be prime...." 1st para already explains pseudoprime

← Older revision		Revision as of 02:14, 22 November 2014
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en>FreeT: /* Variations */ added example

2014-02-07T19:09:20Z

Variations: added example

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en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-28T21:13:52Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 23:13, 28 January 2014
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	~~Eusebio Stanfill~~ is ~~what~~'s ~~displayed on my birth records although it~~ is ~~certainly~~ not the name on ~~particular birth certificate~~. ~~Vermont~~ is ~~just where my home was~~. ~~Software~~ making ~~has been my afternoon job for~~ a if. ~~To cook~~ is the ~~only hobby my wife doesn't agree~~ to. ~~You can believe my website here~~: http://~~prometeu~~.~~net~~<br><br>~~My site~~: ~~clash~~ of ~~clans hack android~~ ([http://~~prometeu~~.~~net hop over to this site~~])		[[Image:Michelsonmorley-boxplot.svg\|thumb\|300px\|Figure 1. Box plot of data from the [[Michelson–Morley experiment]]]]
			In [[descriptive statistics]], a '''box plot''' or '''boxplot''' is a convenient way of graphically depicting groups of numerical data through their [[quartile]]s. 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'''. [[Outlier]]s may be plotted as individual points.

			Box plots display differences between [[statistical population\|populations]] without making any assumptions of the underlying [[probability distribution\|statistical distribution]]: they are [[non-parametric]]. The spacings between the different parts of the box help indicate the degree of [[statistical dispersion\|dispersion]] (spread) and [[skewness]] in the data, and identify [[outlier]]s. In addition to the points themselves, they allow one to visually estimate various [[L-estimator]]s, notably the [[interquartile range]], [[midhinge]], [[range (statistics)\|range]], [[mid-range]], and [[trimean]]. Boxplots can be drawn either horizontally or vertically.

			==Types of boxplots==
			[[File:Box-Plot mit Min-Max Abstand.png\|thumb\|Figure 2. Boxplot with whiskers from minimum to maximum]]
			[[File:Box-Plot mit Interquartilsabstand.png\|thumb\|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<ref name="mcgill tukey larsen">{{Cite journal
			\|first1=Robert \|last1=McGill \|first2=John W. \|last2=Tukey \|author2-link=John W. Tukey \|first3=Wayne A. \|last3=Larsen
			\| title = Variations of Box Plots
			\| journal = [[The American Statistician]]
			\| volume = 32
			\| issue = 1
			\|date=February 1978
			\| pages = 12–16
			\| jstor = 2683468
			\| doi = 10.2307/2683468
			}}</ref> (as in Figure 2)
			* the lowest datum still within 1.5 [[Interquartile range\|IQR]] of the lower quartile, and the highest datum still within 1.5 [[Interquartile range\|IQR]] of the upper quartile<ref name="frigge hoaglin iglewicz">{{Cite journal
			\|first1=Michael \|last1=Frigge \|first2=David C. \|last2=Hoaglin \|first3=Boris \|last3=Iglewicz
			\| title = Some Implementations of the Boxplot
			\| journal = [[The American Statistician]]
			\| volume = 43
			\| issue = 1
			\|date=February 1989
			\| pages = 50–54
			\| jstor = 2685173
			\| doi = 10.2307/2685173
			}}</ref><ref name="Rboxplotstats" /> (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.<ref name="frigge hoaglin iglewicz" />

			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 [[normal distribution\|normally distributed]], the locations of the seven marks on the box plot will be equally spaced.

			==Variations==
			[[File:Fourboxplots.svg\|thumb\|Figure 4. Four box plots, with and without notches and variable width]]
			Since the American 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.<ref name="mcgill tukey larsen" />

			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.<ref name="mcgill tukey larsen" /> The width of the notches is proportional to the interquartile range 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).<ref name="mcgill tukey larsen" /> One convention is to use <math alt="+/-1.58*IQR/sqrt(n)">\pm 1.58 \times IQR \div \sqrt{n}</math>.<ref name="Rboxplotstats">{{Cite web \| title = R: Box Plot Statistics \| work = R manual \| url = http://stat.ethz.ch/R-manual/R-devel/library/grDevices/html/boxplot.stats.html \| accessdate = 26 June 2011}}</ref>

			== Visualization ==

			[[Image:Boxplot vs PDF.svg\|thumb\|Figure 5. Boxplot and a [[probability density function]] (pdf) of a Normal N(0,1σ<sup>2</sup>) Population]]

			The box plot is a quick way of examining one or more sets of data graphically. Box plots may seem more primitive than a [[histogram]] or [[kernel density estimation\|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 [[Histogram#Number of bins and width\|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 intuitive than looking at a box plot, comparing the box plot against the probability density function (theoretical histogram) for a normal N(0,1σ<sup>2</sup>) distribution may be a useful tool for understanding the box plot (Figure 5).

			==See also==
			* [[Bagplot\|Bivariate boxplot]]
			* [[Five-number summary]]
			* [[Exploratory data analysis]]
			* [[Functional boxplot]]
			* [[Violin plot]]
			* [[Fan chart (statistics)\|Fan chart]]

			==References==
			{{reflist\|30em}}

			==Further reading==
			* {{cite book \|author=John W. Tukey \| authorlink=John Tukey \|year=1977 \|title=Exploratory Data Analysis \|publisher=[[Addison-Wesley]]}}
			* {{cite journal \|jstor=2685133 \|pages=257–262 \|last1=Benjamini \|first1=Y. \|title=Opening the Box of a Boxplot \|volume=42 \|issue=4 \|journal=The American Statistician \|year=1988 \|doi=10.2307/2685133}}
			* {{cite journal \|jstor=2686061 \|pages=382–387 \|last1=Rousseeuw \|first1=P. J.\|author1-link=Peter Rousseeuw\|last2=Ruts \|first2=I. \|last3=Tukey \|first3=J. W. \|author3-link=John Tukey\|title=The Bagplot: A Bivariate Boxplot \|volume=53 \|issue=4 \|journal=The American Statistician \|year=1999 \|doi=10.2307/2686061}}

			==External links==
			{{commons category\|Box plots}}
			* [http://www.lcgceurope.com/lcgceurope/data/articlestandard/lcgceurope/132005/152912/article.pdf Visual Presentation of Data by Means of Box Plots]
			* [http://www.physics.csbsju.edu/stats/box2.html On-line box plot calculator with explanations and examples] (Has beeswarm example)
			* [http://www.r-statistics.com/2011/03/beeswarm-boxplot-and-plotting-it-with-r/ Beeswarm Boxplot] - superimposing a frequency-jittered stripchart on top of a boxplot

			{{Statistics\|descriptive}}

			[[Category:Statistical charts and diagrams]]
			[[Category:Statistical outliers]]

en>Bender235: /* External links */ navbox

2012-02-07T21:46:48Z

External links: navbox

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