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Histogram

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A histogram is a visual tool used to represent the distribution of quantitative data. It allows researchers to see how values are spread across a range. By using rectangles of varying heights, a histogram reveals the density of the underlying data. This makes it easier to estimate a probability density function, which is a mathematical way to describe how likely certain values are to occur.

Example histogram
Example histogram

While they look similar to bar charts, histograms serve a very different purpose. A bar chart compares different categories, such as different types of fruit. In contrast, a histogram illustrates how a single variable is distributed across continuous intervals.

To construct a histogram, you must first perform a process called "binning." This involves taking the entire range of values and dividing it into a series of intervals called bins. These bins are typically consecutive and non-overlapping. They are usually designed to be of equal size, though this is not strictly required. Once the bins are set, you count how many data points fall into each specific interval. The height of each bar is then determined by this count or frequency.

Black cherry tree histogram
Black cherry tree histogram

If the intervals on the x-axis all have a length of one, the histogram becomes identical to a relative frequency plot.

There are different ways to interpret the vertical scale of a histogram. One common method uses absolute numbers, where the area of each block represents the total count of cases. For example, a histogram of U.S. Census Bureau data regarding travel times to work shows the total number of people in each time interval.

Travel time histogram total n Stata
Travel time histogram total n Stata

Another method is the unit area histogram, which shows proportions. In this version, the total area of all the bars is normalized to exactly 1. This represents the entire dataset as a whole. The height of each block in this version represents the frequency density, or the percentage per horizontal unit.

Travel time histogram total 1 Stata
Travel time histogram total 1 Stata

Researchers often use specific terms to describe the patterns found in a histogram. A distribution might be described as "symmetric" if both sides look similar. It can also be described as "skewed left" or "skewed right" if the data leans toward one side. If a histogram has one clear peak, it is called unimodal. If it has two peaks, it is bimodal, and more than two peaks make it multimodal.

Cumulative vs normal histogram
Cumulative vs normal histogram

You can also create a cumulative histogram. This type of graph maps the cumulative number of observations in all the bins up to a specific point.

Determining the number of bins is a critical step in data analysis. There is no single "best" number of bins, as different widths can reveal different features. Using wider bins can reduce noise caused by sampling randomness. Using narrower bins can provide greater precision in areas where data density is high.

Wikipedia bincounts
Wikipedia bincounts

Several mathematical rules exist to help choose the number of bins. Sturges's formula, developed in 1926, is based on a binomial distribution and assumes a normal distribution. The square-root choice is another method often used in software. Other options include the Rice rule and Doane's formula, which attempts to improve performance for non-normal data. Scott's normal reference rule is also widely used and is the default in Microsoft Excel.

The history of the histogram involves several important figures in statistics. While the technique of using bars to represent measurements was devised by the Scottish economist William Playfair in 1786, the name "histogram" is newer. Karl Pearson, the founder of mathematical statistics, introduced the term in 1892 during lectures at University College London.

Example histogram
Example histogram

Although some mistakenly believe the name comes from Greek words for "history" or "drawing," Pearson actually derived it from a different Greek root. He chose a word meaning "something set upright" or "mast" to refer to the vertical bars in the graph. He viewed the term as part of a series of new names, similar to "stigmogram" or "radiogram."

Histograms are deeply connected to more advanced statistical concepts like kernel density estimation. A histogram can be viewed as a simplistic version of this estimation, which uses a "kernel" to smooth frequencies over the bins. While density estimates are often drawn as smooth curves, histograms are often preferred in practical applications. This is because the statistical properties of a histogram are easier to model mathematically. In a histogram, each bin can vary independently, whereas the variations in a kernel density estimate are much more difficult to describe.

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🖼️ Images & Media (8)
File:Black_cherry_tree_histogram.svg
Black_cherry_tree_histogram.svg
File:Example histogram.png
Example histogram.png
File:Travel time histogram total n Stata.png
Travel time histogram total n Stata.png
File:Travel time histogram total 1 Stata.png
Travel time histogram total 1 Stata.png
File:Cumulative vs normal histogram.svg
Cumulative vs normal histogram.svg
File:Wikipedia bincounts.svg
Wikipedia bincounts.svg
File:Gumbel distribtion.png
Gumbel distribtion.png
File:Daisy histogram.jpg
Daisy histogram.jpg
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