Histogram

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An example histogram of the heights of 31 Black Cherry trees.

In statistics, a histogram is a graphical display of tabulated frequencies, shown as bars. It shows what proportion of cases fall into each of several categories: it is a form of data binning. The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent. The intervals (or bands, or bins) are generally of the same size.[1]

Histograms are used to plot 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.

An alternative to the histogram is kernel density estimation, which uses a kernel to smooth samples. This will construct a smooth probability density function, which will in general more accurately reflect the underlying variable.

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.

Contents

[edit] Etymology

The word histogram derived from the 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'. The term was introduced by Karl Pearson in 1895.[2]

[edit] Examples

As an example we consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, [1], Table 2). 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.
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 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 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.
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 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. 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.

[edit] 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.

[edit] Mathematical definition

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.

In a more general mathematical sense, a histogram is a mapping mi 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 mi meets the following conditions:

n = \sum_{i=1}^k{m_i}.

[edit] 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 Mi of a histogram mi is defined as:

M_i = \sum_{j=1}^i{m_j}.

[edit] 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:

k = \left \lceil \frac{\max x - \min x}{h} \right \rceil.

The braces indicate the ceiling function.

Sturges' formula[3]
k = \lceil \log_2 n + 1 \rceil ,

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

Scott's choice[4]
h = \frac{3.5 \sigma}{n^{1/3}} ,

where σ is the sample standard deviation.

Freedman-Diaconis' choice[5]
h = 2 \frac{\operatorname{IQR}(x)}{n^{1/3}} ,

which is based on the interquartile range.

[edit] Continuous data

The idea of a histogram can be generalized to continuous data. Let f \in L^1(R) (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
h(f)(y) = \sum_{\xi\in\{x : f(x)=y\}} \frac{1}{|f'(\xi)|}.

h(f)(y) is undefined if y is the value of a stationary point.

[edit] See also

[edit] Density estimation

[edit] Notes

  1. Howitt, D. and Cramer, D. (2008) "Statistics in Psychology". Prentice Hall
  2. M. Eileen Magnello (December 2005). "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 Volume 1. ISSN 1177-1380. http://www.rutherfordjournal.org/article010107.html. 
  3. Sturges, H. A. (1926). "The choice of a class interval". J. American Statistical Association: 65–66. 
  4. Scott, David W. (1979). "On optimal and data-based histograms". Biometrika 66 (3): 605–610. doi:10.1093/biomet/66.3.605. 
  5. 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. 

[edit] References

[edit] External links

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