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histogram

 
Dictionary: his·to·gram   (hĭs'tə-grăm') pronunciation

n.
A bar graph of a frequency distribution in which the widths of the bars are proportional to the classes into which the variable has been divided and the heights of the bars are proportional to the class frequencies.

[Greek histos, mast, web + -GRAM.]


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Statistics Dictionary: histogram
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A diagram that uses rectangles to represent frequency. It differs from the bar chart in that the rectangles may have differing widths, but the key feature is that, for each rectangle, the area is proportional to the frequency represented. The term 'histogram' was introduced by Karl Pearson in his lectures prior to 1895.




Histogram. This histogram represents data on the cross-sectional area of 30 erratics (boulders left behind by retreating glaciers). Note the use of wider intervals for the classes corresponding to the scarcer larger boulders. In a histogram, area is proportional to frequency.




Graph using vertical or horizontal bars whose lengths indicate quantities. Along with the pie chart, the histogram is the most common format for representing statistical data. Its advantage is that it not only clearly shows the largest and smallest categories but gives an immediate impression of the distribution of the data. In fact, a histogram is a representation of a frequency distribution.

For more information on histogram, visit Britannica.com.

A bar graph that uses the width of the bars to represent the various classes and the height of the bars to represent their relative frequencies.

Camera Histograms

Digital camera histograms show the image's overall exposure. Using 256 vertical bars to represent brightness levels from 0 to 255, the leftmost bar is the darkest pixel level (0), and the rightmost bar is the lightest (255). The height of the bars represents the total number of pixels at that brightness level.

What is of most interest to the photographer is how the bars spread horizontally from left to right. For example, if there are no bars on the left, there are no black pixels in the image.

Light Pixels
Histograms appear on the same LCD screen used to preview the image. In this daylight example, there is an absence of dark shadows because the histogram shows no pixels on the left side (dark side).

Dark Pixels
In this example, the camera was pointed into a totally dark room, and the bars are confined entirely to the leftmost side (dark side).

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Investment Dictionary: Histogram
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1. A graphical representation, similar to a bar chart in structure, that organizes a group of data points into user-specified ranges. The histogram condenses a data series into an easily interpreted visual by taking many data points and grouping them into logical ranges or bins.



2. The MACD histogram is a very common technical indicator that illustrates the difference between the MACD line and the trigger line. This difference is then plotted on a chart in the form of a histogram to make it easy for a trader to determine a specific asset's momentum.

Investopedia Says:
1. Histograms are commonly used in statistics to demonstrate how many of a certain type of variable occurs within a specific range. For example, a census focused on the demography of a country may use a histogram of how many people there are between the ages of 0 and 10, 11 and 20, 21 and 30, 31 and 40, 41 and 50 etc. This histogram would look similar to the graph above.

2. MACD histograms are a popular tool used in technical analysis to gauge the strength of an asset's momentum. An increasing MACD histogram signals an increase in upward momentum while a decreasing histogram is used to signal downward momentum.

Related Links:
This method, which works particularly well for currency traders, helps avoid stop-order triggers before the real reversal. Trading The MACD Divergence
The use of the simple MACD histogram could change how you trade currency pairs for good. Keep An Eye On Momentum
This straightforward histogram can help you analyze the buying and selling interest in a stock. Gauging Support And Resistance With Price By Volume
Read the case against this well-established indicator. Candle Sheds More Light Than The MACD


Business Dictionary: Histogram
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A bar graph in which the frequency of occurrence for each class of data is represented by the relative height of the bars.

Dental Dictionary: histogram
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n

A bar graph; a graphic representation of a frequency distribution.

Geography Dictionary: histogram
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A graph which uses bars (rectangles) to show the frequency of certain classes of values within a dataset. Classes can be descriptive, as in a histogram showing numbers of voters for different parties, or numerical, so that the numbers, or percentages of a population in different age groups (0-4, 5-9, 10-14, and so on) are illustrated by a rectangle (bar). The widths of the rectangles should be proportional to the class intervals just as the heights are proportional to the frequencies of occurrence (numbers, or percentages) within each class.

In digital photography, an electronic bar chart showing the distribution of tones in an image, from completely dark (on the left) to completely light (on the right). In advanced cameras it can be displayed ‘live’ as an exposure aid in the viewfinder, to indicate the dynamic (tonal) range of the image being taken.

— Robin Lenman

A graph used in statistics in which frequency distributions of interval-level data are represented by contiguous rectangles. In a histogram, the area of each rectangle is directly proportional to the frequency of each class interval represented. Compare bar chart.

Veterinary Dictionary: histogram
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A graph in which values found in a statistical study are represented by lines or symbols placed horizontally or vertically, to indicate frequency distribution.

Wikipedia: 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

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]

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 30 but less than 35 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. 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.

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}.

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}.

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.

See also

Density estimation

Notes

  1. ^ Howitt, D. and Cramer, D. (2008) "Statistics in Psychology". Prentice Hall
  2. ^ 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. 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. 

References

External links


Translations: Histogram
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Dansk (Danish)
n. - histogram, blokdiagram

Nederlands (Dutch)
histogram

Français (French)
n. - histogramme

Deutsch (German)
n. - Histogramm, Balkendiagramm

Ελληνική (Greek)
n. - (στατιστικό) ιστόγραμμα

Italiano (Italian)
istogramma

Português (Portuguese)
n. - histograma (m)

Русский (Russian)
гистограмма

Español (Spanish)
n. - histograma

Svenska (Swedish)
n. - histogram, stapeldiagram

中文(简体)(Chinese (Simplified))
柱状图

中文(繁體)(Chinese (Traditional))
n. - 柱狀圖

한국어 (Korean)
n. - 막대 그래프

日本語 (Japanese)
n. - 柱状図表, 柱状グラフ

العربيه (Arabic)
‏(الاسم) الرسم البياني النسيجي, رسم بياني مؤلف من سلسله من المستطيلات في علم الاحصاء‏

עברית (Hebrew)
n. - ‮תרשים עמודות, היסטוגרמה‬


Best of the Web: histogram
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Some good "histogram" pages on the web:


Math
mathworld.wolfram.com
 
 
 

 

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