Statistics for Maintenance Modelling: Basic Probability Concepts

# Constructing a Histogram¤

The histogram of drill bit diameters was formed using the most common form of the histogram. The plot is obtained by splitting the range of the data into equal-sized bins (called classes). Then for each bin, the number of points from the data set that fall into each bin are counted. That is:

- Vertical axis: Frequency (i.e. counts for each bin)
- Horizontal axis: Response variable

The classes can either be defined arbitrarily by the user or via some systematic rule.

The histogram has a variant which is called the relative histogram, where the counts are replaced by normalized counts. There are two common methods used to normalize the counts.

1. The first method is where the normalized count is the count in a class divided by the total number of observations. In this case the relative counts are normalized to sum to one (or 100 if a percentage scale is used). This is the intuitive case where the height of the histogram bar represents the proportion of the data in each class.
2. The second method is where the normalized count is the count in the class divided by the number of observations times the class width. For this normalization, the area (or integral) under the histogram is equal to one. From a probabilistic point of view, this normalization results in a relative histogram that is most akin to the probability density function. If you want to overlay a probability density on top of the histogram, use this normalization. Although this normalization is less intuitive (relative frequencies greater than 1 are quite permissible), it is the appropriate normalization if you are using the histogram to model a probability density function.