Histogram

Histogram
Histogram of arrivals per minute.svg
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

In statistics, a histogram is a graphical display of tabular frequencies, shown as adjacent rectangles. Each rectangle is erected over an interval, 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 based on relative frequencies. It then shows the proportion of cases that fall into each of several categories, with the total area equaling 1. The categories are usually specified as consecutive, non-overlapping intervals of a variable. The categories (intervals) must be adjacent, and often are chosen to be 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.[2]

Contents

Etymology

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

The etymology of the word histogram is uncertain. Sometimes it is said to be 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'. It is also said that Karl Pearson, who introduced the term in 1895, derived the name from "historical diagram".

[3]

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. An interesting feature of this graph is that the number recorded for "at least 15 but less than 20 minutes" is higher than for the bands on either side. This is likely to have arisen from people rounding their reported journey time. 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 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 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. The intervals are placed together in order to show that the data represented by the histogram, while exclusive, is also continuous. (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.)[4]

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.

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 function 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}.

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 mj is defined as:

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

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. A good discussion of rules for choice of bin widths is in Modern Applied Statistics with S, § 5.6: Density Estimation.[5]

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

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

The braces indicate the ceiling function.

Sturges' formula[6]
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[7]
h = \frac{3.5 \sigma}{n^{1/3}},

where \sigma is the sample standard deviation.

Square-Root Choice
k = \sqrt{n},

which takes the square root of the number of data points in the sample (used by Excel histograms and many others)

Freedman–Diaconis' choice[8]
h = 2 \frac{\operatorname{IQR}(x)}{n^{1/3}},

which is based on the interquartile range.

Choice based on minimization of an estimated L2 risk function[9] 
\underset{h}{\operatorname{arg\,min}} \frac{ 2 \bar{m} - v } {h^2}

where \textstyle \bar{m} and \textstyle v are mean and biased variance of a histogram with bin-width \textstyle h, \textstyle \bar{m}=\frac{1}{k} \sum_{i=1}^{k} m_i and \textstyle v= \frac{1}{k} \sum_{i=1}^{k} (m_i - \bar{m})^2 .

See also

References

  1. ↑ Howitt, D. and Cramer, D. (2008) Statistics in Psychology. Prentice Hall
  2. ↑ Nancy R. Tague (2004). "Seven Basic Quality Tools". The Quality Toolbox. Milwaukee, Wisconsin: American Society for Quality. p. 15. http://www.asq.org/learn-about-quality/seven-basic-quality-tools/overview/overview.html. Retrieved 2010-02-05. 
  3. ↑ M. Eileen Magnello (December 1856). "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. ISSN 1177–1380. http://www.rutherfordjournal.org/article010107.html. 
  4. ↑ Dean, S., & Illowsky, B. (2009, February 19). Descriptive Statistics: Histogram. Retrieved from the Connexions Web site: http://cnx.org/content/m16298/1.11/
  5. ↑ W. N. Venables and B. D. Ripley: "Modern Applied Statistics with S", Springer, in (4thedition) section 5.6: Density Estimation
  6. ↑ Sturges, H. A. (1926). "The choice of a class interval". J. American Statistical Association: 65–66. 
  7. ↑ Scott, David W. (1979). "On optimal and data-based histograms". Biometrika 66 (3): 605–610. doi:10.1093/biomet/66.3.605. 
  8. ↑ 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. 
  9. ↑ [|Shimazaki, H.]; Shinomoto, S. (2007). "A method for selecting the bin size of a time histogram". Neural Computation 19 (6): 1503–152. doi:10.1162/neco.2007.19.6.1503. PMID 17444758. http://www.mitpressjournals.org/doi/abs/10.1162/neco.2007.19.6.1503. 

Further reading

External links

Statistics
 
Descriptive statistics
Continuous data
Location
Mean (Arithmetic, Geometric, Harmonic) · Median · Mode
Dispersion
Range  · Standard deviation  · Coefficient of variation  · Percentile  · Interquartile range
Shape
Variance · Skewness · Kurtosis · Moments · L-moments
Count data
Index of dispersion
Summary tables
Grouped data  · Frequency distribution · Contingency table
Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Partial correlation · Scatter plot
Statistical graphics
Bar chart · Biplot · Box plot · Control chart · Correlogram · Forest plot · Histogram · Q-Q plot · Run chart · Scatter plot · Stemplot · Radar chart
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