An example histogram of the heights of 31 Black Cherry trees. The Black Cherry ( Prunus serotina, also occasionally Wild Black Cherry Rum Cherry or Mountain Black Cherry is a species of cherry, native to eastern

In statistics, a histogram is a graphical display of tabulated frequencies. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Information graphics or infographics are visual representations of Information, Data or Knowledge. In Statistics the frequency of an event i is the number ni of times the event occurred in the Experiment or the study It shows what proportion of cases fall into each of several categories. Categorization is the process in which ideas and objects are recognized, differentiated and understood. A histogram differs from a bar chart in that it is the area of the bar that denotes the value, not the height, a crucial distinction when the categories are not of uniform width (Lancaster, 1974). See CategoryBar chart templates for Wikipedia templates to make bar charts The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent.

The word histogram is derived from Greek: histos 'anything set upright' (as the masts of a ship, the bar of a loom, or the vertical bars of a histogram); gramma 'drawing, record, writing'. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly The histogram is one of the seven basic tools of quality control, which also include the Pareto chart, check sheet, control chart, cause-and-effect diagram, flowchart, and scatter diagram. A Pareto chart is a special type of bar chart where the values being plotted are arranged in descending order The check sheet is a simple document that is used for collecting data in real-time and at the location where the data is generated In Statistical process control, the control chart, also known as the ' Shewhart chart' or ' process-behaviour chart' is a tool used to determine whether a The Ishikawa diagram (or fishbone diagram or also cause-and-effect diagram) are Diagrams that shows the Causes of a certain Event A flowchart is a Schematic representation of an Algorithm or a stepwise process, showing the steps as boxes of various kinds and their order by connecting A scatter graph or scatter plot is a type of Display using Cartesian coordinates to display values for two Variables for a set of data A generalization of the histogram is kernel smoothing techniques. A kernel is a weighting function used in Non-parametric estimation techniques This will construct a very smooth probability density function from the supplied data. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability

Examples

As an example we consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, [1], Table 5). The United States Census Bureau (officially Bureau of the Census as defined in Title) is the government agency that is responsible for the United States Census The census found that there were 124 million people who work outside of their homes. This rounding is a common phenomenon when collecting data from people.

Histogram of travel time, US 2000 census. Area under the curve equals the total number of cases. This diagram uses Q/width from the table.

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. In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x The area under the curve represents the total number of cases (124 million). This type of histogram shows absolute numbers.

Histogram of travel time, US 2000 census. Area under the curve equals 1. This diagram uses Q/total/width from the table.

This histogram differs from the first only in the vertical scale. In Astronomy, Geography, Geometry and related sciences and contexts a direction passing by a given point is said to be vertical if The height of each bar is the decimal percentage of the total that each category represents, and the total area of all the bars is equal to 1, the decimal equivalent of 100%. The curve displayed is a simple density estimate. In Probability and Statistics, density estimation is the construction of an estimate based on observed Data, of an unobservable underlying Probability This version shows proportions, and is also known as a unit area histogram.

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. They only place the bars together to make it easier to compare data.

Activities and demonstrations

The SOCR resource pages contain a number of hands-on interactive activities demonstrating the concept of a histogram, histogram construction and manipulation using Java applets and charts. The Statistics Online Computational Resource (SOCR is a suite of online tools and interactive aids for hands-on learning and teaching concepts in statistical analyses and

Mathematical definition

In a more general mathematical sense, a histogram is a mapping m_i that counts the number of observations that fall into various 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_i meets the following conditions:

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 M_i of a histogram m_i is defined as:

Number of bins and width

There is no "best" number of bins, and different bin sizes can reveal different features of the data. Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. You should always experiment with bin widths before choosing one (or more) that illustrate the salient features in your data.

The number of bins k can be calculated directly, or from a suggested bin width h:

The braces indicate the ceiling function. In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The

Sturges' formula^[1]

which implicitly bases the bin sizes on the range of the data, and can perform poorly if n < 30.

Scott's choice^[2]

where h is the common bin width, and s is the sample standard deviation. In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values

Freedman-Diaconis' choice^[3]

which is based on the interquartile range

Continuous data

The idea of a histogram can be generalized to continuous data. In Descriptive statistics, the interquartile range (IQR, also called the midspread, middle fifty and middle of the #s, is a measure of Let (see Lebesgue space), then the cumulative histogram operator H can be defined by:

H(f)(y) = with only finitely many intervals of monotony this can be rewritten as

. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

h(f)(y) is undefined if y is the value of a stationary point. In Mathematics, particularly in Calculus, a stationary point is an input to a function where the Derivative is zero (equivalently the

See also

• Density estimation
• Freedman-Diaconis rule
• Image histogram
• Kernel density estimation , another method of visualizing probability density functions that can be preferred to histograms. In Probability and Statistics, density estimation is the construction of an estimate based on observed Data, of an unobservable underlying Probability In Statistics, the Freedman-Diaconis rule can be used to select the size of the bins to be used in a Histogram. An image histogram is type of Histogram which acts as a graphical representation of the tonal distribution in a Digital image. In Statistics, kernel density estimation (or Parzen window method named after Emanuel Parzen) is a non-parametric way of estimating

References

1. ^ Sturges, H. A. (1926). "The choice of a class interval". J. American Statistical Association: 65–66.  
2. ^ Scott, David W. (1979). "On optimal and data-based histograms". Biometrika 66 (3): 605–610. doi:10.1093/biomet/66.3.605. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  
3. ^ Freedman, David; Diaconis, P. (1981). "On the histogram as a density estimator: L₂ theory". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57 (4): 453–476. doi:10.1007/BF01025868. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  

• Webster's Third New International Dictionary, Merriam-Webster; Ind Una edition (June 2002). Webster's Dictionary is the name given to a common type of English language dictionary in the United States.

• Lancaster, H. O. An Introduction to Medical Statistics. John Wiley and Sons. 1974. ISBN 0 471 51250-8

External links

• Journey To Work and Place Of Work (location of census document cited in example)
• Understanding histograms in digital photography
• Histograms: Construction, Analysis and Understanding with external links and an application to particle Physics.
• A Method for Selecting the Bin Size of a Histogram