Histogram
From Freepedia
- For the histogram used in digital image processing, see Color_histogram.
In statistics, a histogram is a graphical display of tabulated frequencies. That is, a histogram is the graphical version of a table which shows what proportion of cases fall into each of several or many specified categories. The categories are usually specified as nonoverlapping intervals of some variable.
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Examples
There are many different ways to display the same table, and two kinds of histograms are shown below. 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 census found that there were 124 million people who work outside of their homes. People were asked how long it takes them to get to work, and their responses were divided into categories: less than 5 minutes, more than 5 minutes and less than 10, more than 10 minutes and less than 15, and so on. The tables shows the numbers of people per category in thousands, so that 4,180 means 4,180,000.
The data in the tables are displayed graphically by the following diagrams. An interesting feature of both diagrams is the spike in the 30 to 35 minutes category. It seems likely that this is an artifact: half an hour is a common unit of informal time measurement, so people whose travel times were perhaps a little less than or a little greater than 30 minutes might be inclined to answer "30 minutes".
Data by absolute numbers
Image:Travel time histogram total n.png
| Interval | Width | Quantity | Quantity/width |
| 0 | 5 | 4,180 | 836 |
| 5 | 5 | 13,687 | 2,737 |
| 10 | 5 | 18,618 | 3,723 |
| 15 | 5 | 19,634 | 3,926 |
| 20 | 5 | 17,981 | 3,596 |
| 25 | 5 | 7,190 | 1,438 |
| 30 | 5 | 16,369 | 3,273 |
| 35 | 5 | 3,212 | 642 |
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 is ideal for an overview of absolute numbers.
Data by proportion
Image:Travel time histogram total 1.png
| Interval | Width | Quantity (Q) | Q/total/width |
| 0 | 5 | 4,180 | 0.0067 |
| 5 | 5 | 13,687 | 0.0220 |
| 10 | 5 | 18,618 | 0.0300 |
| 15 | 5 | 19,634 | 0.0316 |
| 20 | 5 | 17,981 | 0.0289 |
| 25 | 5 | 7,190 | 0.0115 |
| 30 | 5 | 16,369 | 0.0263 |
| 35 | 5 | 3,212 | 0.0051 |
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 height of all the bars is equal to 1, the decimal equivalent of 100%. This version is ideal for comparing proportions.
Mathematical Definition
In a more general mathematical sense, a histogram is simply a mapping that counts the number of observations that fall into various disjoint categories (known as bins), whereas the graph of a histogram, which is often taught at high-school, is merely one way to represent a histogram. Thus, if we let N be the total number of observations and n be the total number of bins, the histogram <math>h_k</math> meets the following conditions:
<math>N = \sum_{k=1}^n{h_k}</math>
where k is an index over the bins.
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 <math>H_k</math> of a histogram <math>h_k</math> is defined as:
<math>H_k = \sum_{k\prime=1}^k{h_{k\prime}}</math>
External links
- Journey To Work and Place Of Work (location of census document cited in example)
- Teach/Me Data Analysis
- Understanding histograms in digital photography
