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| Histogram |
Histogram:For the histogram used in digital image processing, see Color histogram.
In statistics, a histogram is a graphical display of tabulated frequencies. That is, a histogram is the graphical version of a table which shows what proportion of cases fall into each of several or many specified categories. The categories are usually specified as nonoverlapping intervals of some variable.
Examples
There are many different ways to display the same table, and two kinds of histograms are shown below. As an example we consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, [http://www.census.gov/prod/2004pubs/c2kbr-33.pdf], Table 5). The census found that there were 124 million people who work outside of their homes. People were asked how long it takes them to get to work, and their responses were divided into categories: less than 5 minutes, more than 5 minutes and less than 10, more than 10 minutes and less than 15, and so on. The tables shows the numbers of people per category in thousands, so that 4,180 means 4,180,000.
The data in the following tables are displayed graphically by the diagrams below. An interesting feature of both diagrams is the spike in the 30 to 35 minutes category. It seems likely that this is an artifact: half an hour is a common unit of informal time measurement, so people whose travel times were perhaps a little less than or a little greater than 30 minutes might be inclined to answer "30 minutes".
Data by absolute numbers
graphical display
| Interval | Width | Quantity | Quantity/width |
| 0 | 5 | 4,180 | 836 |
| 5 | 5 | 13,687 | 2,737 |
| 10 | 5 | 18,618 | 3,723 |
| 15 | 5 | 19,634 | 3,926 |
| 20 | 5 | 17,981 | 3,596 |
| 25 | 5 | 7,190 | 1,438 |
| 30 | 5 | 16,369 | 3,273 |
| 35 | 5 | 3,212 | 642 |
| 40 | 5 | 4,122 | 824 |
| 45 | 15 | 9,200 | 613 |
| 60 | 30 | 6,461 | 215 |
| 90 | 60 | 3,435 | 57 |
This histogram shows the number of cases per unit interval so that the height of each bar is equal to the proportion of total people in the survey who fall into that category. The area under the curve represents the total number of cases (124 million). This type of histogram is ideal for an overview of absolute numbers.
Data by proportion
unit interval
| Interval | Width | Quantity (Q) | Q/total/width |
| 0 | 5 | 4,180 | 0.0067 |
| 5 | 5 | 13,687 | 0.0220 |
| 10 | 5 | 18,618 | 0.0300 |
| 15 | 5 | 19,634 | 0.0316 |
| 20 | 5 | 17,981 | 0.0289 |
| 25 | 5 | 7,190 | 0.0115 |
| 30 | 5 | 16,369 | 0.0263 |
| 35 | 5 | 3,212 | 0.0051 |
| 40 | 5 | 4,122 | 0.0066 |
| 45 | 15 | 9,200 | 0.0049 |
| 60 | 30 | 6,461 | 0.0017 |
| 90 | 60 | 3,435 | 0.0004 |
This histogram differs from the first only in the vertical scale. The height of each bar is the decimal percentage of the total that each category represents, and the total height of all the bars is equal to 1, the decimal equivalent of 100%. This version is ideal for comparing proportions.
Mathematical Definition
In a more general mathematical sense, a histogram is simply a mapping that counts the number of observations that fall into various disjoint categories (known as bins), whereas the graph of a histogram, which is often taught at high-school, is merely one way to represent a histogram. Thus, if we let N be the total number of observations and n be the total number of bins, the histogram meets the following conditions:
where k is an index over the bins.
Cumulative Histogram
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 of a histogram is defined as:
External links
- [http://www.census.gov/population/www/socdemo/journey.html Journey To Work and Place Of Work] (location of census document cited in example)
- [http://www.vias.org/tmdatanaleng/cc_histogram.html Teach/Me Data Analysis]
- [http://www.luminous-landscape.com/tutorials/understanding-series/understanding-histograms.shtml Understanding histograms in digital photography]
Category:Charts
Category:Statistics
Category:Diagrams
Color histogramIn computer graphics, a color histogram is a representation of an image derived by counting the 'color' of each pixel. The idea was proposed by Michael Swain and Dana Ballard in 1991 and is primarily used in situations where speed of processing is a factor in the choice of algorithm, or where a specific object, rather than a more abstract class of objects is required to be identified; as noted by Swain and Ballard, "It may not be helpful to model coffee cups as being red and white, but yours may be."
Color histograms are a flexible construct that can be built from images in various color spaces, whether RGB, rg-chromaticity or any other color space of any dimension. A histogram of an image is produced first by discretising the colors in the image into a number of bins, and counting the number of image pixels in each bin:
| | red |
| 0-63 | 64-127 | 128-191 | 192-255 |
| blue |
0-63 | 43 | 78 | 18 | 0 |
| 64-127 | 45 | 67 | 33 | 2 |
| 128-191 | 127 | 58 | 25 | 8 |
| 192-255 | 140 | 47 | 47 | 13 |
This provides a far more compact overview of the data in an image than knowing the exact value of every pixel. The color histogram of an image is invariant with translation and rotation about the viewing axis, and varies only slowly with the angle of view. This makes the color histogram particularly suited to recognising an object of unknown position and rotation within a scene. Importantly, translation of an RGB image into the illumination invariant rg-chromaticity space allows the histogram to operate well in varying light levels.
The main drawback of histograms is that the representation is solely dependant of the color of the object being studied. There is no way to distinguish a red and white cup from a red and white plate. Put another way, histogram-based algorithms have no concept of a generic 'cup', and a model of our red and white cup is no use when given an otherwise identical blue and white cup.
There is a software that can compute and interactively display 3D color histograms and color distributions:
http://rsb.info.nih.gov/ij/plugins/color-inspector.html
Category:Computer graphics
StatisticsStatistics is a broad mathematical discipline which studies ways to collect, summarize and draw conclusions from data. It is applicable to a wide variety of academic disciplines from the physical and social sciences to the humanities, as well as to business, government, and industry.
Once data is collected, either through a formal sampling procedure or by recording responses to treatments in an experimental setting (cf experimental design), or by repeatedly observing a process over time (time series), graphical and numerical summaries may be obtained using descriptive statistics.
Patterns in the data are modeled to draw inferences about the larger population, using inferential statistics to account for randomness and uncertainty in the observations. These inferences may take the form of answers to essentially yes/no questions (hypothesis testing), estimates of numerical characteristics (estimation), prediction of future observations, descriptions of association (correlation), or modeling of relationships (regression).
The framework described above is sometimes referred to as applied statistics. In contrast, mathematical statistics (or simply statistical theory) is the subdiscipline of applied mathematics which uses probability theory and analysis to place statistical practice on a firm theoretical basis.
The word statistics is also the plural of statistic (singular), which refers to the result of applying a statistical algorithm to a set of data.
Origin
The word statistics ultimately derives from the modern Latin term statisticum collegium ("council of state") and the Italian word statista ("statesman" or "politician"). The German Statistik, first introduced by Gottfried Achenwall (1749), originally designated the analysis of data about the state. It acquired the meaning of the collection and classification of data generally in the early nineteenth century. It was introduced into English by Sir John Sinclair. Thus, the original principal purpose of statistics was data to be used by governmental and (often centralized) administrative bodies. The collection of data about states and localities continues, largely through national and international statistical services; in particular, censuses provide regular information about the population. Today, however, the use of statistics has broadened far beyond the service of a state or government, to include such areas as business, natural and social sciences, and medicine, among others.
Statistical methods
Experimental and observational studies
A common goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on a response or dependent variable. There are two major types of causal statistical studies, experimental studies and observational studies. In both types of studies, the effect of changes of an independent variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types is in how the study is actually conducted.
An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation may have modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Instead data is gathered and correlations between predictors and the response are investigated.
An example of an experimental study is the famous Hawthorne studies which attempted to test changes to the working environment at the Hawthorne plant of the Western Electric Company. The researchers were interested in whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured productivity in the plant then modified the illumination in an area of the plant to see if changes in illumination would affect productivity. Due to errors in experimental procedures, specifically the lack of a control group, the researchers while unable to do what they planned were able to provide the world with the Hawthorne effect.
An example of an observational study is a study which explores the correlation between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then perform statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers and then look at the number of cases of lung cancer in each group.
The basic steps for an experiment are to:
# plan the research including determining information sources, research subject selection, and ethical considerations for the proposed research and method,
# design the experiment concentrating on the system model and the interaction of independent and dependent variables,
# summarize a collection of observations to feature their commonality by suppressing details (descriptive statistics),
# reach consensus about what the observations tell us about the world we observe (statistical inference),
# document and present the results of the study.
Levels of measurement
There are four types of measurements or measurement scales used in statistics. The four types or levels of measurement (ordinal, nominal, interval, and ratio) have different degrees of usefulness in statistical research. Ratio measurements, where both a zero value and distances between different measurements are defined, provide the greatest flexibility in statistical methods that can be used for analysing the data. Interval measurements, with meaningful distances between measurements but no meaningful zero value (such as IQ measurements or temperature measurements in degrees Celsius). Ordinal measurements have imprecise differences between consecutive values but a meaningful order to those values. Nominal measurements have no meaningful rank order among values.
Statistical techniques
Some well known statistical tests and procedures for research observations are:
- Student's t-test
- chi-square
- analysis of variance (ANOVA)
- Mann-Whitney U
- regression analysis
- correlation
- Fischer's Least Significant Difference test
- Pearson product-moment correlation coefficient
- Spearman's rank correlation coefficient
Probability
The probability of an event is often defined as a number between one and zero. In reality however there is virtually nothing that has a probability of 1 or 0. You could say that the sun will certainly rise in the morning, but what if an extremely unlikely event destroys the sun? What if there is a nuclear war and the sky is covered in ash and smoke?
We often round the probability of such things up or down because they are so likely or unlikely to occur, that it's easier to recognize them as a probability of one or zero.
However, this can often lead to misunderstandings and dangerous behaviour, because people are unable to distinguish between, e.g., a probability of 10−4 and a probability of 10−9, despite the very practical difference between them. If you expect to cross the road about 105 or 106 times in your life, then reducing your risk of being run over per road crossing to 10−9 will make it unlikely that you will be run over while crossing the road for your whole life, while a risk per road crossing of 10−4 will make it very likely that you will have an accident, despite the intuitive feeling that 0.01% is a very small risk.
Use of prior probabilities of 0 (or 1) causes problems in Bayesian statistics, since the posterior distribution is then forced to be 0 (or 1) as well. In other words, the data is not taken into account at all! As Lindley puts it, if a coherent Bayesian attaches a prior probability of zero to the hypothesis that the Moon is made of green cheese, then even whole armies of astronauts coming back bearing green cheese cannot convince him. Lindley advocates never using prior probabilities of 0 or 1. He calls it Cromwell's rule, from a letter Oliver Cromwell wrote to the synod of the Church of Scotland on August 5th, 1650 in which he said "I beseech you, in the bowels of Christ, consider it possible that you are mistaken."
Important contributors to statistics
- Carl Gauss
- Blaise Pascal
- Sir Francis Galton
- William Sealey Gosset (known as "Student")
- Karl Pearson
- Sir Ronald Fisher
- Gertrude Cox
- Charles Spearman
- Pafnuty Chebyshev
- Aleksandr Lyapunov
- Isaac Newton
- Abraham De Moivre
- Adolph Quetelet
- Florence Nightingale
- John Tukey
- George Dantzig
See also list of statisticians.
Specialized disciplines
Some sciences use applied statistics so extensively that they have specialized terminology. These disciplines include:
- Biostatistics
- Business statistics
- Data mining (applying statistics and pattern recognition to discover knowledge from data)
- Economic statistics (Econometrics)
- Engineering statistics
- Statistical physics
- Demography
- Psychological statistics
- Social statistics (for all the social sciences)
- Statistical literacy
- Process analysis and chemometrics (for analysis of data from analytical chemistry and chemical engineering)
- Reliability engineering
- Statistics in various sports, particularly baseball and cricket
Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions. In these roles it is a key tool, and perhaps the only reliable tool.
Software
Modern statistics is supported by computers to perform some of the very large and complex calculations required.
Whole branches of statistics have been made possible by computing, for example neural networks.
The computer revolution has implications for the future of statistics, with a new emphasis on 'experimental' and 'empirical' statistics.
Statistical packages in common use include:
See also
- Analysis of variance (ANOVA)
- Extreme value theory
- Instrumental variables estimation
- List of academic statistical associations
- List of national and international statistical services
- List of publications in statistics
- List of statistical topics
- List of statisticians
- Machine learning
- Misuse of statistics
- Multivariate statistics
- Permutation test
- Regression analysis
- Statistical package
- Statistical phenomena
External links
- [http://www.hkshum.net/stats/ Clear explanation of the three Statistical Distributions studied throughout secondary school] great for younger students.
General sites and organizations
- [http://lib.stat.cmu.edu/ Statlib: Data, Software and News from the Statistics Community (Carnegie Mellon)]
- [http://www.cbs.nl/isi/ International Statistical Institute]
- [http://www.mathcs.carleton.edu/probweb/probweb.html The Probability Web]
Link collections
- [http://www.cbs.nl/isi/FreeTools.htm Free Statistical Tools on the WEB (at ISI)]
- [http://www.york.ac.uk/depts/maths/histstat Materials for the History of Statistics (Univ. of York)]
- [http://www.xycoon.com/ Statistics resources and calculators (Xycoon)]
- [http://members.aol.com/johnp71/javastat.html StatPages.net (statistical calculations, free software, etc.)]
- [http://www.nih.gov/sigs/bioethics/casestudies.html Bioethics Resources on the Web from the U.S. National Institute of Health (links to tutorials, case studies, and on-line courses)]
Online courses and textbooks
- [http://www.statsoft.com/textbook/stathome.html Electronic Statistics Textbook (StatSoft,Inc.)]
- [http://www.vias.org/tmdatanaleng/ Teach/Me Data Analysis (a Springer-Verlag book)]
- [http://www.richland.cc.il.us/james/lecture/m170/ Statistics: Lecture Notes (from a professor at Richland Community College)]
- [http://statistics.cyberk.com/splash/ CyberStats: Electronic Statistics Textbook (CyberGnostics, Inc)]
- [http://www.stat.ucla.edu/%7Edinov/courses_students.html A variety of class notes and educational materials on probability and statistics]
Statistical software
- [http://www.r-project.org/ R Project for Statistical Computing (free software)]
- [http://www.socr.ucla.edu/ Statistics Online Computational Resource (UCLA)]
- [http://root.cern.ch/ Root Analysis Framework (CERN)]
- [http://www.newmdsx.com/ Multidimensional Scaling Software]
- [http://www.rosuda.org/Software/ Software for interactive graphical analyses]
- [http://www.rank1st.com/website_monitoring/index.html Website Analytics and Monitoring]
- [http://www.csdassn.org/software_reports.cfm Software Reports] by Statistical Software Newsletter
- [http://chirouble.univ-lyon2.fr/~ricco/tanagra/ Tanagra (free software)], including machine learning and data mining techniques
Other resources
- [http://www.sixsigmafirst.com/anova.htm ANOVA]
- [http://www.math.uah.edu/stat/index.html Virtual Laboratories in Probability and Statistics (Univ. of Alabama)] (requires MathML and Java 2 Runtime Environment)
- [http://www.ericdigests.org/2000-2/resources.htm Resources for Teaching and Learning about Probability and Statistics (ERIC Digests)]
- [http://www.ericdigests.org/1993/marriage.htm Resampling: A Marriage of Computers and Statistics (ERIC Digests)]
- [http://www.execpc.com/~helberg/statframes.html Statistical Resources on the Web]
- [http://www.conceptstew.co.uk/PAGES/s4t_glossary_A.html Statistics glossary]
- [http://www.statistics.com/content/glossary/index.php3 Statistics Glossary at statistics.com]
- [http://jobs.strategy-blogs.com/Statisticians.html Statistician Job Outlook - Analysis of wages and working environment for the occupation]
- [http://www.amstat.org/sections/sis/ Statistics in Sports (Section of the ASA)]
- [http://meta.wikimedia.org/wiki/Statistics Statistics - Meta], statistics of Wikimedia projects
Additional references
Lindley, D. Making Decisions. John Wiley. Second Edition 1985. ISBN 0471908088
Category:Mathematical science occupations
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Category:Applied mathematics
Category:Academic disciplines
ms:Statistik
ja:統計学
simple:Statistics
th:สถิติศาสตร์
fiu-vro:Statistiga
Graphical display
Information graphics or infographics are visual representations of information, data or knowledge. These graphics are used anywhere where information needs to be explained quickly or simply, such as in signs, maps, journalism, technical manuals, and education. They are also used extensively as tools by computer scientists, mathematicians, and statisticians to ease the process of developing and communicating conceptual information.
History of information graphics
Early experiments
In prehistory, early humans created the first information graphics: maps. Map-making began several millennia before writing, and map at Çatalhöyük dates from around 7500 BCE.
In 1626 Christopher Scheiner published the Rosa Ursina sive Sol which used a variety of graphics to reveal his astronomical research on the sun. He used a series of images to explain the rotation of the sun over time (by tracking sunspots).
In 1786, William Playfair published the first data graphs in his book The Commercial and Political Atlas. The book is filled with statistical graphs that represent the economy of 18th century England using bar charts and histograms. Playfair innovated again in 1801 with the first area chart in Statistical Breviary.
James Joseph Sylvester introduced the term "graph" in 1878 and published a set of diagrams showing the relationship between chemical bonds and mathematical properties. These were also the first mathematic graphs.
mathematic graphs
1861 saw the release of a seminal information graphic on the subject of Napoleon's disastrous march on Moscow. The creator, Charles Joseph Minard, captured four different changing variables that contributed to the failure, in a single two-dimensional image:
- the army's distance and direction as they travelled
- the altitude the troops passed through
- the size of the army as troops died from hunger and wounds
- the freezing temperatures they experienced
Development of a visual language
In 1936 Otto Neurath introduced a system of pictographs intended to function as an international visual or picture language. Isotype included a set of stylized human figures which were the basis for the ubiquitous modern stick figures.
The 1972 Munich Olympics were the venue for Otl Aicher to introduce a new set of pictograms that proved to be extremely popular, and influenced the ubiquitous modern stick figures used in public signs.
Modern uses
- 1982 - USA Today's weather page
Elements of information graphics
The basic material of an information graphic is the data, information, or knowledge that the graphic presents. In the case of data, the creator may make use of automated tools such as graphing software to represent the data in the form of lines, boxes, arrows, and various symbols and pictograms. The information graphic might also features a key which defines the visual elements in plain English. A scale and labels are also common.
Modern practitioners of information graphics / design
A statistician and sculptor, Edward Tufte has written a series of highly regarded books on the subject of information graphics. Tufte also delivers lectures and workshops on a regular basis. He describes the process of incorporating many dimensions of information into a two-dimensional image as 'escaping flatland.'
Nigel Holmes is an established commercial creator of what he calls "explanation graphics". His works deal not only with the visual display of information but also of knowledge - how to do things. He created graphics for Time magazine for 16 years, and is the author of several books on the subject.
Close and strongly related to the field of information graphics, is information design. Actually, making infographics is a certain discipline within the information design world. Modern day American information designers, like Nigel Holmes, Edward Tufte, Peter Sullivan and Sam Ward, Donald Norman; are accompanied by a very active Dutch information designer: Paul Mijksenaar. His Amsterdam and New York based design studio is specialized in the development of visual oriented information systems. They create so called wayfinding and waysigning systems for all kinds of large public transport systems and infrastructures. Examples of their work are, the signing systems for airports in the Netherlands (Schiphol, Amsterdam), but also for airports in Italy and the United States like: JFK Airport and Dallas Forth Worth.
Another good example of modern day practitioners of information graphics is the French bureau d'études. A bureau which visualizes a lot of complex matters like the governmental structures of power in the United States. Or the way the media in the States are linked and related together in an data driven war. They create maps about autonomous knowledges and powers, art and economies, governing by networks, world governance by private bankers, or maps about the contemporary bio-control systems.
The bureau visualizes these complex organization structures by mapping them, which should be a very effective way of making complex information more easily accessible. But what they want to show, is mainly how complex certain (media) structures are and what the actual relations are, without simplifying them. And without making it easy to overview and to understand. So basically the main objective within information design is not pursued and what you get as a 'reader' is an information overload. But it serves another goal and the public is very select.
Types of information graphic
- chart
- flowchart
- histogram
- graph
- map
- diagram
- contour map
- isotherm
- mind map
- signage systems
Interpreting information graphics
Many information graphics are specialised forms of depiction that represent their content in sophisticated and often abstract ways. In order to interpret the meaning of these graphics appropriately, the viewer requires a suitable level of graphicacy. In many cases, the required graphicacy involves comprehension skills that are learned rather than innate. At a fundamental level, the skills of decoding individual graphic signs and symbols must be acquired before sense can be made of an information graphic as a whole. Howevever, knowledge of the conventions for distributing and arranging these individual components is also necessary for the building of understanding.
References
- "Blood, Dirt, and Nomograms: A Particular History of Graphs", Thomas L. Hankins, Isis, University of Chicago Press (1999, 90: 50-80). [http://www.journals.uchicago.edu/Isis/journal/demo/v000n000/000000/000000.text.html]
- "Designing Infographics" (1998), Eric K. Meyer
See also
- information visualization
External links
- [http://www.edwardtufte.com/tufte/ Edward Tufte]
- [http://informationdesign.org/archives/cat_information_graphics.php informationdesign.org]
- [http://www.nixlog.com/infographics/ Examples of information graphics]
- [http://www.mijksenaar.com/ Paul Mijksenaar]
- [http://www.math.yorku.ca/SCS/Gallery/milestone/ Milestones in the History of Thematic Cartography, Statistical Graphics and Data Visualization]
- [http://www.nigelholmes.com/ Nigel Holmes, the Explanation Graphics and the Power of Pictures]
- [http://www.understandingusa.com/ Richard Saul Wurman's excellent visual overview of information about the United States]
- [http://bureaudetudes.free.fr/ mappings by bureau d'étueds]
- [http://www.theworldasflatland.net/ The World As Flatland]
- [http://www.funnelinc.com/ Funnel Inc. - infodesign for marketing, branding and wayfinding]
Category:Illustration
-
VerticalAn object is in a vertical position when it is aligned in an "up-down" direction, perpendicular to the horizon.
- In science the vertical (or plumb line) is the direction of the force of gravity.
- A pair of angles are said to be vertical if they share the same vertex and are bounded by the same pair of lines but are opposite to each other.
- In music the vertical aspect is simultaneity, either intervals or harmony, as opposed to succession or the linear aspect
- In marketing and business in general, a vertical informally means a vertical market.
ja:上下
Category:ChartsA chart is a type of information graphic that represents tabular numeric data.
Category:Infographics
Category:Statistics
Category:Visualization
Category:DiagramsMain article: Diagram.
Category:Infographics 953年
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大事记
- 德国科隆莱茵河上的古罗马桥被拆除，一直要到900多年后的1855年才会有一座新的桥在科隆跨越莱茵河。
- 德国哥庭根建城。
- 中国沧州铁狮子铸成。
出生
- 薛颜，北宋大臣（逝世于1025年）
逝世
- 述律皇后，耶律阿保机的皇后（出生于879年）
Category:10世纪
ko:953년
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