Histograms

History

Histograms were introduced into the context of statistics as a columnar representation of frequency distributions arranged along the x axis. Karl Pearson defined histograms as an estimate of the probability distribution of a given variable by depicting the frequencies of observations occurring in certain ranges of values, also known as continuous variables.

The graphic display of a histogram is an important aspect of measuring the distribution. Although the graphic display of the histogram can show many visual patterns, many agree that a histogram should always display information succinctly. Charles Joseph Minard created an influential histogram showing the losses suffered by Napoleon’s army in the Russian campaign of 1812 (see Figure 1). This histogram is notable for the two-dimensional representation of six types of data: the number of Napoleon’s troops, [Page 781]distance, temperature, the latitude and longitude, direction of travel, and location relative to specific dates.

Figure 1 Charles Joseph Minard’s map of Napoleon’s Russian campaign of 1812

Statistics

The benefit of histograms as a visual presentation of frequency distribution is summarized through five indicators that provide strong evidence for the proper distributional model:

• 1.
  The center, that is, the location of the data distribution
• 2.
  Spread, that is, the scale of the range of the data
• 3.
  Skew of data
• 4.
  Presence of outliers
• 5.
  Presence of multiple models extends scope of the data.

Under a frequency distribution, the data are arranged into numerically ordered class groupings. When developing a frequency distribution, each class grouping should have the same width. In order to determine the width of a class interval, the range of the data is divided by the number of desired class groupings.

Data

In cases of large data sets, it is easier to present and handle the data by grouping the values into class intervals, which are sometimes known as bin widths. Sturges’s rule states that the data range should be split into k equally spaced classes where k = [1 + log10n] and where Log10(N) is the log base 10 of the number of observations. According to this rule, 1,000 observations would be graphed with 11 class intervals because 10 is the closest integer to Log2(1,000). The ceiling operator takes the closest integer above the calculated value. However, if the data are not normally distributed, additional classes may be required. The idea of skewness of the distribution is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative or even undefined. The formula is written as k = [+ log2n] where it is estimated the third moment of the skewness of the distribution, where it is derived from a binomial distribution and implicitly assumes an approximately normal distribution.

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