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| {{Hatnote|For the histograms used in digital image processing, see [[Image histogram]] and [[Color histogram]].}}
| | == fierce storm from the jungle 'shot' out == |
| {{Infobox quality tool
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| | image = Histogram of arrivals per minute.svg
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| | category =
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| | describer = [[Karl Pearson]]
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| | purpose = To roughly assess the [[probability distribution]] of a given variable by depicting the frequencies of observations occurring in certain ranges of values
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| }}
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| In [[statistics]], a '''histogram''' is a graphical representation of the distribution of data. It is an estimate of the [[probability distribution]] of a [[continuous variable]] and was first introduced by [[Karl Pearson]].<ref name="pearson">{{cite doi|10.1098/rsta.1895.0010}}</ref> A histogram is a representation of tabulated [[frequency (statistics)|frequencies]], shown as adjacent [[rectangle]]s, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval. The height of a rectangle is also equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval. The total area of the histogram is equal to the number of data. A histogram may also be [[normalization (statistics)|normalized]] displaying relative frequencies. It then shows the proportion of cases that fall into each of several [[Categorization|categories]], with the total area equaling 1. The categories are usually specified as consecutive, non-overlapping [[interval (mathematics)|interval]]s of a variable. The categories (intervals) must be adjacent, and often are chosen to be of the same size.<ref>Howitt, D. and Cramer, D. (2008) ''Statistics in Psychology''. Prentice Hall</ref> The rectangles of a histogram are drawn so that they touch each other to indicate that the original variable is continuous.<ref>Charles Stangor (2011) "Research Methods For The Behavioral Sciences". Wadsworth, Cengage Learning. ISBN 9780840031976.</ref>
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| Histograms are used to plot the density of data, and often for [[density estimation]]: estimating the [[probability density function]] of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the ''x''-axis are all 1, then a histogram is identical to a [[relative frequency]] plot.
| | Xiao Yan Yi Deng feet ground, the body is turned into a shadow, rapid inroads that faint black forest.<br><br>With the departure of Xiao Yan, here is slowly fell into silence, it lasts ten minutes after the last ten road shadow, fierce [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html 電波時計 カシオ] storm from the jungle 'shot' out, holding a sword, face 'color' dignified glance a look around, saw no movement after that just light relief, as one another, are all with a wry smile [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 ソーラー] and shook his head, 一枚 flares ready to go on the right hand clenched, and it was stuffed pregnant , for that can repel [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html 時計 メンズ カシオ] even the old sovereign Yunshan terrorist figures who these people are naturally afraid of God to be incomplete.<br><br>a leading cloud-lan apprentice slowly out, waving a sword, Jianguang flashing, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 時計 電波 ソーラー] on a tree stem, leaving an obscure sigil, done it afterwards, he just turned around and whispered: ' here has been the search is completed, if then down, then, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ ソーラー電波腕時計] is to be entered |
| | | 相关的主题文章： |
| An alternative to the histogram is [[kernel density estimation]], which uses a [[kernel (statistics)|kernel]] to smooth samples. This will construct a [[smooth function|smooth]] probability density function, which '''will in general more accurately reflect the underlying variable.
| | <ul> |
| '''
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| The histogram is one of the [[Seven Basic Tools of Quality|seven basic tools of quality control]].<ref>{{cite web | url = http://www.asq.org/learn-about-quality/seven-basic-quality-tools/overview/overview.html | author = Nancy R. Tague | title = Seven Basic Quality Tools | year = 2004 | work = The Quality Toolbox | publisher = [[American Society for Quality]] | location = [[Milwaukee|Milwaukee, Wisconsin]] | page = 15 | accessdate = 2010-02-05}}</ref>
| | <li>[http://www.nmmxc.com/bbs/forum.php?mod=viewthread&tid=869475 http://www.nmmxc.com/bbs/forum.php?mod=viewthread&tid=869475]</li> |
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| ==Etymology==
| | <li>[http://scottalanciolek.com/Main_Page http://scottalanciolek.com/Main_Page]</li> |
| [[File:Black cherry tree histogram.svg|thumb|left|An example histogram of the heights of 31 [[Prunus serotina|Black Cherry]] trees.]]
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| The etymology of the word ''histogram'' is uncertain. Sometimes it is said to be derived from the [[Greek language|Greek]] ''histos'' 'anything set upright' (as the masts of a ship, the bar of a loom, or the vertical bars of a histogram); and ''gramma'' 'drawing, record, writing'. It is also said that [[Karl Pearson]], who introduced the term in 1891, derived the name from "historical diagram".<ref>{{cite journal
| | <li>[http://www.goldshopmall.com/plus/feedback.php?aid=26 http://www.goldshopmall.com/plus/feedback.php?aid=26]</li> |
| | author =M. Eileen Magnello
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| |date=December 2006
| | </ul> |
| | title =Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician
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| | journal =The New Zealand Journal for the History and Philosophy of Science and Technology
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| | volume = 1 volume
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| | oclc =682200824
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| | url =http://www.rutherfordjournal.org/article010107.html
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| }}
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| </ref>
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| {{-}}
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| ==Examples==
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| The [[United States Census Bureau|U.S. Census Bureau]] found that there were 124 million people who work outside of their homes.<ref>[http://www.census.gov/prod/2004pubs/c2kbr-33.pdf US 2000 census].</ref> Using their data on the time occupied by travel to work, Table 2 below shows the absolute number of people who responded with travel times "at least 30 but less than 35 minutes" is higher than the numbers for the categories above and below it. This is likely due to people rounding their reported journey time.{{Citation needed|date=August 2010}} The problem of reporting values as somewhat arbitrarily [[round number|rounded numbers]] is a common phenomenon when collecting data from people.{{Citation needed|date=June 2011}}
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| [[File:Travel time histogram total n Stata.png|thumb|350px|Histogram of travel time (to work), US 2000 census. Area under the curve equals the total number of cases. This diagram uses Q/width from the table.]]
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| :{| class="wikitable" style="text-align:center"
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| |+Data by absolute numbers
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| ! Interval !! Width !! Quantity !! Quantity/width
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| | 0 || 5 || 4180 || 836
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| | 5 || 5 || 13687 || 2737
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| | 10 || 5 || 18618 || 3723
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| |-
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| | 15 || 5 || 19634 || 3926
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| |-
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| | 20 || 5 || 17981 || 3596
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| | 25 || 5 || 7190 || 1438
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| | 30 || 5 || 16369 || 3273
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| | 35 || 5 || 3212 || 642
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| | 40 || 5 || 4122 || 824
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| | 45 || 15 || 9200 || 613
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| |-
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| | 60 || 30 || 6461 || 215
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| |-
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| | 90 || 60 || 3435 || 57
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| |}
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| This histogram shows the number of cases per [[unit interval]] as the height of each block, so that the area of each block is equal to the number of people in the survey who fall into its category. The area under the curve represents the total number of cases (124 million). This type of histogram shows absolute numbers, with Q in thousands.
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| <br clear="all" />
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| [[File:Travel time histogram total 1 Stata.png|thumb|350px|Histogram of travel time (to work), US 2000 census. Area under the curve equals 1. This diagram uses Q/total/width from the table.]]
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| :{| class="wikitable" style="text-align:center"
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| |+Data by proportion
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| ! Interval !! Width !! Quantity (Q) !! Q/total/width
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| | 0 || 5 || 4180 || 0.0067
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| | 5 || 5 || 13687 || 0.0221
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| | 10 || 5 || 18618 || 0.0300
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| | 15 || 5 || 19634 || 0.0316
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| | 20 || 5 || 17981 || 0.0290
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| | 25 || 5 || 7190 || 0.0116
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| | 30 || 5 || 16369 || 0.0264
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| | 35 || 5 || 3212 || 0.0052
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| | 40 || 5 || 4122 || 0.0066
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| | 45 || 15 || 9200 || 0.0049
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| | 60 || 30 || 6461 || 0.0017
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| | 90 || 60 || 3435 || 0.0005
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| |}
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| This histogram differs from the first only in the [[vertical direction|vertical]] scale. The area of each block is the fraction of the total that each category represents, and the total area of all the bars is equal to 1 (the fraction meaning "all"). The curve displayed is a simple [[density estimation|density estimate]]. This version shows proportions, and is also known as a unit area histogram.
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| <br clear="all" /> | |
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| In other words, a histogram represents a frequency distribution by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding frequencies: the height of each is the average frequency density for the interval. The intervals are placed together in order to show that the data represented by the histogram, while exclusive, is also contiguous. (E.g., in a histogram it is possible to have two connecting intervals of 10.5–20.5 and 20.5–33.5, but not two connecting intervals of 10.5–20.5 and 22.5–32.5. Empty intervals are represented as empty and not skipped.)<ref>Dean, S., & Illowsky, B. (2009, February 19). Descriptive Statistics: Histogram. Retrieved from the Connexions Web site: http://cnx.org/content/m16298/1.11/</ref>
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| ==Mathematical definition==
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| [[File:Cumulative vs normal histogram.svg|thumb|350px|An ordinary and a cumulative histogram of the same data. The data shown is a random sample of 10,000 points from a normal distribution with a mean of 0 and a standard deviation of 1.]]
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| In a more general mathematical sense, a histogram is a function ''m''<sub>''i''</sub> that counts the number of observations that fall into each of the disjoint categories (known as ''bins''), whereas the graph of a histogram is merely one way to represent a histogram. Thus, if we let ''n'' be the total number of observations and ''k'' be the total number of bins, the histogram ''m''<sub>''i''</sub> meets the following conditions:
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| : <math>n = \sum_{i=1}^k{m_i}.</math>
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| ===Cumulative histogram===
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| A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram ''M''<sub>''i''</sub> of a histogram ''m''<sub>''j''</sub> is defined as:
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| : <math>M_i = \sum_{j=1}^i{m_j}.</math>
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| ===Number of bins and width===
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| There is no "best" number of bins, and different bin sizes can reveal different features of the data. Grouping data is at least as old as [[John Graunt|Graunt]]'s work in the 17th century, but no systematic guidelines were given<ref name=scott92/> until [[Herbert Sturges|Sturges]]'s work in 1926.<ref name=sturges/>
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| Using wider bins where the density is low reduces noise due to sampling randomness; using narrower bins where the density is high (so the signal drowns the noise) gives greater precision to the density estimation. Thus varying the bin-width within a histogram can be beneficial. Nonetheless, equal-width bins are widely used.
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| Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate, so experimentation is usually needed to determine an appropriate width. There are, however, various useful guidelines and rules of thumb.<ref>''e.g.'' § 5.6 "Density Estimation", W. N. Venables and B. D. Ripley, ''Modern Applied Statistics with S'' (2002), Springer, 4th edition. ISBN 0-387-95457-0.</ref>
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| The number of bins ''k'' can be assigned directly or can be calculated from a suggested bin width ''h'' as:
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| :<math>k = \left \lceil \frac{\max x - \min x}{h} \right \rceil.</math>
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| The braces indicate the [[Floor function#The ceiling function|ceiling function]].
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| ; Square-root choice:
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| :<math>k = \sqrt{n}, \, </math>
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| which takes the square root of the number of data points in the sample (used by Excel histograms and many others).<ref>[http://cameron.econ.ucdavis.edu/excel/ex11histogram.html EXCEL 2007: Histogram]</ref>
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| ;Sturges' formula
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| Sturges' formula<ref name=sturges>{{cite journal |last=Sturges |first=H. A. |year=1926 |title=The choice of a class interval |journal=Journal of the American Statistical Association | pages=65–66 | jstor=2965501 }}</ref> is derived from a binomial distribution and implicitly assumes an approximately normal distribution.
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| :<math>k = \lceil \log_2 n + 1 \rceil, \, </math>
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| It implicitly bases the bin sizes on the range of the data and can perform poorly if ''n'' < 30.{{Citation needed|date=March 2012}} It may also perform poorly if the data are not normally distributed.
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| ;Rice Rule
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| :<math>k = \lceil 2 n^{1/3}\rceil,</math>
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| The Rice Rule <ref>Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University (chapter 2 "Graphing Distributions", section "Histograms")</ref> is presented as a simple alternative to Sturges's rule.
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| ;Doane's formula
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| Doane's formula<ref name=Doane1976>Doane DP (1976) Aesthetic frequency classification. American Statistician, 30: 181–183</ref> is a modification of Sturges' formula which attempts to improve its performance with non-normal data.
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| :<math> k = 1 + \log_2( n ) + \log_2 \left( 1 + \frac { |g_1| }{\sigma_{g_1}} \right) </math>
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| where <math>g_1</math> is the estimated 3rd-moment-[[skewness]] of the distribution and
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| :<math> \sigma_{g_1} = \sqrt { \frac { 6(n-2) }{ (n+1)(n+3) } } </math>
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| ;Scott's normal reference rule:
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| :<math>h = \frac{3.5 \hat \sigma}{n^{1/3}},</math> | |
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| where <math>\hat \sigma</math> is the sample [[standard deviation]]. Scott's normal reference rule<ref name=scott79>{{cite journal |last=Scott |first=David W. |year=1979 |title=On optimal and data-based histograms |journal=Biometrika |volume=66 |issue=3|pages=605–610 |doi=10.1093/biomet/66.3.605}}</ref> is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.<ref name=scott92>{{cite book|last=Scott|first=David W.|title=Multivariate Density Estimation: Theory, Practice, and Visualization|publisher=John Wiley|location=New York|year=1992|ref=harv}}</ref>
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| ;Freedman–Diaconis' choice
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| The [[Freedman–Diaconis rule]] is:<ref>{{cite journal |last=Freedman |first=David |coauthors=Diaconis, P. |year=1981 |title=On the histogram as a density estimator: ''L''<sub>2</sub> theory |journal=Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete |volume=57 |issue=4 |pages=453–476 |doi=10.1007/BF01025868}}</ref><ref name=scott92/>
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| :<math>h = 2 \frac{\operatorname{IQR}(x)}{n^{1/3}},</math>
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| which is based on the [[interquartile range]], denoted by IQR. It replaces 3.5σ of Scott's rule with 2 IQR, which is less sensitive than the standard deviation to outliers in data.
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| ; Choice based on minimization of an estimated ''L''<sup>2</sup><ref>{{cite journal | last = Shimazaki | first =H. | authorlink = | coauthors =Shinomoto, S. | title =A method for selecting the bin size of a time histogram | journal = Neural Computation | volume =19 | issue = 6 | pages =1503–1527 |publisher = | year =2007 | url = http://www.mitpressjournals.org/doi/abs/10.1162/neco.2007.19.6.1503 | pmid = 17444758 | doi = 10.1162/neco.2007.19.6.1503}}</ref> [[risk function]]:
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| :<math> \underset{h}{\operatorname{arg\,min}} \frac{ 2 \bar{m} - v } {h^2} </math>
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| where <math>\textstyle \bar{m}</math> and <math>\textstyle v</math> are mean and biased variance of a histogram with bin-width <math>\textstyle h</math>, <math>\textstyle \bar{m}=\frac{1}{k} \sum_{i=1}^{k} m_i</math> and <math>\textstyle v= \frac{1}{k} \sum_{i=1}^{k} (m_i - \bar{m})^2 </math>.
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| ; Remark
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| A good reason why the number of bins should be proportional to <math>n^{1/3}</math> is the following: suppose that
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| the data are obtained as <math>n</math> independent realizations of a bounded probability distribution with smooth density.
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| Then the histogram remains equally »rugged« as <math>n</math> tends to infinity. If <math>s</math> is the »width« of the
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| distribution (e. g., the standard deviation or the inter-quartile range), then the number of units in a bin (the frequency) is
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| of order <math>n h/s</math> and the ''relative'' standard error is of order <math>\sqrt{s/(n h)}</math>.
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| Comparing to the next bin, the relative change of the frequency is of order <math>h/s</math> provided that the
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| derivative of the density is non-zero. These two are of the same order if <math>h</math> is of order <math>s/n^{1/3}</math>,
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| so that <math>k</math> is of order <math>n^{1/3}</math>.
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| This simple cubic root choice can also be applied to bins with non-constant width.
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| ==See also==
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| {{Commons category|Histograms}}
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| {{Portal|Statistics}}
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| * [[Data binning]]
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| * [[Density estimation]]
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| ** [[Kernel density estimation]], a smoother but more complex method of density estimation
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| * [[Freedman–Diaconis rule]]
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| * [[Image histogram]]
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| * [[Pareto chart]]
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| * [[Seven Basic Tools of Quality]]
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| * [[V-optimal histograms]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * Lancaster, H.O. ''An Introduction to Medical Statistics.'' John Wiley and Sons. 1974. ISBN 0-471-51250-8
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| ==External links==
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| {{Wiktionary}}
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| * [http://www.census.gov/population/www/socdemo/journey.html Journey To Work and Place Of Work] ''(location of census document cited in example)''
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| * [http://www.mathworks.com/matlabcentral/fileexchange/30480-histconnect Smooth histogram for signals and images from a few samples]
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| * [http://quarknet.fnal.gov/toolkits/ati/histograms.html Histograms: Construction, Analysis and Understanding with external links and an application to particle Physics.]
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| * [http://2000.jukuin.keio.ac.jp/shimazaki/res/histogram.html A Method for Selecting the Bin Size of a Histogram]
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| * [http://www.shodor.org/interactivate/activities/histogram/ Interactive histogram generator]
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| * [http://www.mathworks.com/matlabcentral/fileexchange/27388-plot-and-compare-nice-histograms-by-default Matlab function to plot nice histograms]
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| * [http://excelandfinance.com/histogram-in-excel/ Dynamic Histogram in MS Excel]
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| * Histogram [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_ModelerActivities_MixtureModel_1 construction] and [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs manipulation] using Java applets, and [http://www.socr.ucla.edu/htmls/SOCR_Charts.html charts] on [[SOCR]]
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| {{Statistics|descriptive}}
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| [[Category:Statistical charts and diagrams]]
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| [[Category:Quality control tools]]
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| [[Category:Estimation of densities]]
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| [[Category:Non-parametric statistics]]
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| [[Category:Statistics articles needing expert attention]]
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