Histogram | |
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One of the Seven Basic Tools of Quality | |

First described by | Karl Pearson |

Purpose | To roughly assess the probability distribution of a given variable by depicting the frequencies of observations occurring in certain ranges of values. |

A **histogram** is an approximate representation of the distribution of numerical data. It was first introduced by Karl Pearson.^{ [1] } To construct a histogram, the first step is to "bin" (or "bucket") the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) must be adjacent and are often (but not required to be) of equal size.^{ [2] }

- Examples
- Mathematical definitions
- Cumulative histogram
- Number of bins and width
- Applications
- See also
- References
- Further reading
- External links

If the bins are of equal size, a rectangle is erected over the bin with height proportional to the frequency—the number of cases in each bin. A histogram may also be normalized to display "relative" frequencies. It then shows the proportion of cases that fall into each of several categories, with the sum of the heights equaling 1.

However, bins need not be of equal width; in that case, the erected rectangle is defined to have its *area* proportional to the frequency of cases in the bin.^{ [3] } The vertical axis is then not the frequency but *frequency density*—the number of cases per unit of the variable on the horizontal axis. Examples of variable bin width are displayed on Census bureau data below.

As the adjacent bins leave no gaps, the rectangles of a histogram touch each other to indicate that the original variable is continuous.^{ [4] }

Histograms give a rough sense of the density of the underlying distribution of the 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.

A histogram can be thought of as a simplistic kernel density estimation, which uses a kernel to smooth frequencies over the bins. This yields a smoother probability density function, which will in general more accurately reflect distribution of the underlying variable. The density estimate could be plotted as an alternative to the histogram, and is usually drawn as a curve rather than a set of boxes. Histograms are nevertheless preferred in applications, when their statistical properties need to be modeled. The correlated variation of a kernel density estimate is very difficult to describe mathematically, while it is simple for a histogram where each bin varies independently.

An alternative to kernel density estimation is the average shifted histogram,^{ [5] } which is fast to compute and gives a smooth curve estimate of the density without using kernels.

The histogram is one of the seven basic tools of quality control.^{ [6] }

Histograms are sometimes confused with bar charts. A histogram is used for continuous data, where the bins represent ranges of data, while a bar chart is a plot of categorical variables. Some authors recommend that bar charts have gaps between the rectangles to clarify the distinction.^{ [7] }^{ [8] }

This is the data for the histogram to the right, using 500 items:

Bin/Interval | Count/Frequency |
---|---|

−3.5 to −2.51 | 9 |

−2.5 to −1.51 | 32 |

−1.5 to −0.51 | 109 |

−0.5 to 0.49 | 180 |

0.5 to 1.49 | 132 |

1.5 to 2.49 | 34 |

2.5 to 3.49 | 4 |

The words used to describe the patterns in a histogram are: "symmetric", "skewed left" or "right", "unimodal", "bimodal" or "multimodal".

- Symmetric, unimodal
- Bimodal
- Multimodal
- Symmetric

It is a good idea to plot the data using several different bin widths to learn more about it. Here is an example on tips given in a restaurant.

- Tips using a $1 bin width, skewed right, unimodal
- Tips using a 10c bin width, still skewed right, multimodal with modes at $ and 50c amounts, indicates rounding, also some outliers

The U.S. Census Bureau found that there were 124 million people who work outside of their homes.^{ [9] } Using their data on the time occupied by travel to work, the table 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 ]} The problem of reporting values as somewhat arbitrarily rounded numbers is a common phenomenon when collecting data from people.^{[ citation needed ]}

Data by absolute numbers Interval Width Quantity Quantity/width 0 5 4180 836 5 5 13687 2737 10 5 18618 3723 15 5 19634 3926 20 5 17981 3596 25 5 7190 1438 30 5 16369 3273 35 5 3212 642 40 5 4122 824 45 15 9200 613 60 30 6461 215 90 60 3435 57

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.

Data by proportion Interval Width Quantity (Q) Q/total/width 0 5 4180 0.0067 5 5 13687 0.0221 10 5 18618 0.0300 15 5 19634 0.0316 20 5 17981 0.0290 25 5 7190 0.0116 30 5 16369 0.0264 35 5 3212 0.0052 40 5 4122 0.0066 45 15 9200 0.0049 60 30 6461 0.0017 90 60 3435 0.0005

This histogram differs from the first only in the 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 estimate. 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: 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.)^{ [10] }

The data used to construct a histogram are generated via a function *m*_{i} that counts the number of observations that fall into each of the disjoint categories (known as *bins*). Thus, if we let *n* be the total number of observations and *k* be the total number of bins, the histogram data *m*_{i} meet the following conditions:

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*_{j} is defined as:

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 Graunt's work in the 17th century, but no systematic guidelines were given^{ [11] } until Sturges' work in 1926.^{ [12] }

Using wider bins where the density of the underlying data points 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.

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.^{ [13] }

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

The braces indicate the ceiling function.

which takes the square root of the number of data points in the sample (used by Excel's Analysis Toolpak histograms and many other) and rounds to the next integer.^{ [14] }

Sturges' formula^{ [12] } is derived from a binomial distribution and implicitly assumes an approximately normal distribution.

Sturges' formula implicitly bases bin sizes on the range of the data, and can perform poorly if *n* < 30, because the number of bins will be small—less than seven—and unlikely to show trends in the data well. On the other extreme, Sturges' formula may overestimate bin width for very large datasets, resulting in oversmoothed histograms.^{ [15] } It may also perform poorly if the data are not normally distributed.

When compared to Scott's rule and the Terrell-Scott rule, two other widely accepted formulas for histogram bins, the output of Sturges' formula is closest when *n* ≈ 100.^{ [15] }

The Rice Rule ^{ [16] } is presented as a simple alternative to Sturges' rule.

Doane's formula^{ [17] } is a modification of Sturges' formula which attempts to improve its performance with non-normal data.

where is the estimated 3rd-moment-skewness of the distribution and

Bin width is given by

where is the sample standard deviation. Scott's normal reference rule^{ [18] } is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.^{ [11] }

The Freedman–Diaconis rule gives bin width as:^{ [19] }^{ [11] }

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.

This approach of minimizing integrated mean squared error from Scott's rule can be generalized beyond normal distributions, by using leave-one out cross validation:^{ [20] }^{ [21] }

Here, is the number of datapoints in the *k*th bin, and choosing the value of *h* that minimizes *J* will minimize integrated mean squared error.

The choice is based on minimization of an estimated *L*^{2} risk function ^{ [22] }

where and are mean and biased variance of a histogram with bin-width , and .

Rather than choosing evenly spaced bins, for some applications it is preferable to vary the bin width. This avoids bins with low counts. A common case is to choose *equiprobable bins*, where the number of samples in each bin is expected to be approximately equal. The bins may be chosen according to some known distribution or may be chosen based on the data so that each bin has samples. When plotting the histogram, the *frequency density* is used for the dependent axis. While all bins have approximately equal area, the heights of the histogram approximate the density distribution.

For equiprobable bins, the following rule for the number of bins is suggested:^{ [23] }

This choice of bins is motivated by maximizing the power of a Pearson chi-squared test testing whether the bins do contain equal numbers of samples. More specifically, for a given confidence interval it is recommended to choose between 1/2 and 1 times the following equation:^{ [24] }

Where is the probit function. Following this rule for would give between and ; the coefficient of 2 is chosen as an easy-to-remember value from this broad optimum.

A good reason why the number of bins should be proportional to is the following: suppose that the data are obtained as independent realizations of a bounded probability distribution with smooth density. Then the histogram remains equally "rugged" as tends to infinity. If is the "width" of the distribution (e. g., the standard deviation or the inter-quartile range), then the number of units in a bin (the frequency) is of order and the *relative* standard error is of order . Comparing to the next bin, the relative change of the frequency is of order provided that the derivative of the density is non-zero. These two are of the same order if is of order , so that is of order . This simple cubic root choice can also be applied to bins with non-constant width.

- In hydrology the histogram and estimated density function of rainfall and river discharge data, analysed with a probability distribution, are used to gain insight in their behaviour and frequency of occurrence.
^{ [26] }An example is shown in the blue figure. - In many Digital image processing programs there is an histogram tool, which show you the distribution of the contrast / brightness of the pixels.

Wikimedia Commons has media related to Histograms . |

- Data binning
- Density estimation
- Kernel density estimation, a smoother but more complex method of density estimation

- Entropy estimation
- Freedman–Diaconis rule
- Image histogram
- Pareto chart
- Seven basic tools of quality
- V-optimal histograms

In statistics and probability theory, the **median** is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result.

In statistics, the **standard deviation** is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In probability theory and statistics, **skewness** is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.

In probability theory and statistics, the **beta distribution** is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by *α* and *β*, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.

**Pearson's chi-squared test** is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is the most widely used of many chi-squared tests – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900. In contexts where it is important to improve a distinction between the test statistic and its distribution, names similar to *Pearson χ-squared* test or statistic are used.

In statistics, a **frequency distribution** is a list, table or graph that displays the frequency of various outcomes in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval.

In probability theory and statistics, the **Rayleigh distribution** is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.

In statistics, a **bimodal****distribution** is a probability distribution with two different modes, which may also be referred to as a bimodal distribution. These appear as distinct peaks in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form bimodal distributions.

The **mode** is the value that appears most often in a set of data values. If **X** is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled.

In probability theory and statistics, the **continuous uniform distribution** or **rectangular distribution** is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, *a* and *b*, which are the minimum and maximum values. The interval can either be closed or open. Therefore, the distribution is often abbreviated *U*, where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable *X* under no constraint other than that it is contained in the distribution's support.

**Sample size determination** is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for example, in a stratified survey there would be different sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group.

The **Pearson distribution** is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.

In statistics, the **Freedman–Diaconis rule** can be used to select the width of the bins to be used in a histogram. It is named after David A. Freedman and Persi Diaconis.

In statistics, **kernel density estimation** (**KDE**) is a non-parametric way to estimate the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the **Parzen–Rosenblatt window** method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy.

The following is a glossary of terms used in the mathematical sciences statistics and probability.

In statistics, a **binomial proportion confidence interval** is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments. In other words, a binomial proportion confidence interval is an interval estimate of a success probability *p* when only the number of experiments *n* and the number of successes *n _{S}* are known.

In statistics, the **frequency** of an event is the number of times the observation occurred/recorded in an experiment or study. These frequencies are often graphically represented in histograms.

In probability theory and statistics, the **skew normal distribution** is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

The **generalized normal distribution** or **generalized Gaussian distribution (GGD)** is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To distinguish the two families, they are referred to below as "version 1" and "version 2". However this is not a standard nomenclature.

Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density estimation with improved statistical properties. Apart from histograms, other types of density estimators include parametric, spline, wavelet and Fourier series. Kernel density estimators were first introduced in the scientific literature for univariate data in the 1950s and 1960s and subsequently have been widely adopted. It was soon recognised that analogous estimators for multivariate data would be an important addition to multivariate statistics. Based on research carried out in the 1990s and 2000s, **multivariate kernel density estimation** has reached a level of maturity comparable to its univariate counterparts.

- ↑ Pearson, K. (1895). "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**186**: 343–414. Bibcode:1895RSPTA.186..343P. doi: 10.1098/rsta.1895.0010 . - ↑ Howitt, D.; Cramer, D. (2008).
*Introduction to Statistics in Psychology*(Fourth ed.). Prentice Hall. ISBN 978-0-13-205161-3. - ↑ Freedman, D.; Pisani, R.; Purves, R. (1998).
*Statistics*(Third ed.). W. W. Norton. ISBN 978-0-393-97083-8. - ↑ Charles Stangor (2011) "Research Methods For The Behavioral Sciences". Wadsworth, Cengage Learning. ISBN 9780840031976.
- ↑ David W. Scott (December 2009). "Averaged shifted histogram".
*Wiley Interdisciplinary Reviews: Computational Statistics*.**2:2**(2): 160–164. doi:10.1002/wics.54. - ↑ Nancy R. Tague (2004). "Seven Basic Quality Tools".
*The Quality Toolbox*. Milwaukee, Wisconsin: American Society Quality. p. 15. Retrieved 2010-02-05. - ↑ Naomi, Robbins. "A Histogram is NOT a Bar Chart".
*Forbes*. Retrieved 31 July 2018. - ↑ M. Eileen Magnello (December 2006). "Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician".
*The New Zealand Journal for the History and Philosophy of Science and Technology*. 1 volume. OCLC 682200824. - ↑ US 2000 census.
- ↑ Dean, S., & Illowsky, B. (2009, February 19). Descriptive Statistics: Histogram. Retrieved from the Connexions Web site: http://cnx.org/content/m16298/1.11/
- 1 2 3 Scott, David W. (1992).
*Multivariate Density Estimation: Theory, Practice, and Visualization*. New York: John Wiley. - 1 2 Sturges, H. A. (1926). "The choice of a class interval".
*Journal of the American Statistical Association*.**21**(153): 65–66. doi:10.1080/01621459.1926.10502161. JSTOR 2965501. - ↑
*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. - ↑ "EXCEL Univariate: Histogram".
- 1 2 Scott, David W. (2009). "Sturges' rule".
*WIREs Computational Statistics*.**1**(3). doi:10.1002/wics.35. - ↑ Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University (chapter 2 "Graphing Distributions", section "Histograms")
- ↑ Doane DP (1976) Aesthetic frequency classification. American Statistician, 30: 181–183
- ↑ Scott, David W. (1979). "On optimal and data-based histograms".
*Biometrika*.**66**(3): 605–610. doi:10.1093/biomet/66.3.605. - ↑ Freedman, David; Diaconis, P. (1981). "On the histogram as a density estimator:
*L*_{2}theory" (PDF).*Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*.**57**(4): 453–476. CiteSeerX 10.1.1.650.2473 . doi:10.1007/BF01025868. S2CID 14437088. - ↑ Wasserman, Larry (2004).
*All of Statistics*. New York: Springer. p. 310. ISBN 978-1-4419-2322-6. - ↑ Stone, Charles J. (1984). "An asymptotically optimal histogram selection rule" (PDF).
*Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer*. - ↑ Shimazaki, H.; Shinomoto, S. (2007). "A method for selecting the bin size of a time histogram".
*Neural Computation*.**19**(6): 1503–1527. CiteSeerX 10.1.1.304.6404 . doi:10.1162/neco.2007.19.6.1503. PMID 17444758. S2CID 7781236. - ↑ Jack Prins; Don McCormack; Di Michelson; Karen Horrell. "Chi-square goodness-of-fit test".
*NIST/SEMATECH e-Handbook of Statistical Methods*. NIST/SEMATECH. p. 7.2.1.1. Retrieved 29 March 2019. - ↑ Moore, David (1986). "3". In D'Agostino, Ralph; Stephens, Michael (eds.).
*Goodness-of-Fit Techniques*. New York, NY, USA: Marcel Dekker Inc. p. 70. ISBN 0-8247-7487-6. - ↑ A calculator for probability distributions and density functions
- ↑ An illustration of histograms and probability density functions

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

Wikimedia Commons has media related to Histogram . |

Look up in Wiktionary, the free dictionary. histogram |

- Exploring Histograms, an essay by Aran Lunzer and Amelia McNamara
- Journey To Work and Place Of Work
*(location of census document cited in example)* - Smooth histogram for signals and images from a few samples
- Histograms: Construction, Analysis and Understanding with external links and an application to particle Physics.
- A Method for Selecting the Bin Size of a Histogram
- Histograms: Theory and Practice, some great illustrations of some of the Bin Width concepts derived above.
- Histograms the Right Way
- Interactive histogram generator
- Matlab function to plot nice histograms
- Dynamic Histogram in MS Excel
- Histogram construction and manipulation using Java applets, and charts on SOCR
- Toolbox for constructing the best histograms

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