Histogram

A histogram graph is a plot that lets one represent data points that lie in a range of values called classes or bins creating a frequency distribution of a continuous set of data. This allows for the inspection of the data for its underlying distribution (e.g. normal distribution), outliers, skewness, etc. Being one of the seven basic tools of quality control it is one of the most widely used ways to represent any data for statistical analysis.

Quick details

What: Discover Change, Distribution

Why: Determine the consistency of your process with statistical analysis

History of Histogram

Karl Pearson introduced several now-commonplace statistical tools. One of these was the histogram, a diagram similar to a bar chart. The use of a histogram in statistics is to represent a set of continuous, rather than discrete, data. For this reason, Pearson explained that it could be employed as a tool in the study of history, for example, to chart historical time periods, and coined the name ‘histogram’ in 1891 to convey its use as a ‘historical diagram’.

Bar Graph vs Histogram: A bar graph represents categories of variables on the x-axis. While a histogram represents continuous non-overlapping numerical intervals in a progression, hence the bins(rectangles) are consecutive.

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When to Use a Histogram?

1Compare the frequency of occurrence of quantitative data – Compare the height of bars

Use a histogram in data visualization when an entire range of values of continuous numerical data can be bucketed into a series of intervals—and then how many values fall into each interval can be counted. The bins (or intervals) must be adjacent and are often (but not required to be) of equal size. When these intervals are of equal width then the height of the bars is proportional to the frequency and can be used to compare the data. 

Frequency of employees in different age ranges – Equal bin width Histogram

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2Compare the frequency of occurrence of quantitative data – Compare bar area when intervals are unequal

In a histogram, it is the area of the bar that indicates the frequency of occurrences for each bin. This means that the height of the bar does not necessarily indicate the correct frequency, but the product of height multiplied by the width of the bin indicates the frequency of occurrences within that bin. When the bars are not equally spaced the height of the bin does not reflect the frequency and should not be used as criteria for comparison.

Frequency density = Frequency/class width; Variable bin width histograms
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3Get an overview of statistical anomalies in data

The use of a histogram in statistics is defined by the need to check the consistency of your process by understanding the spread of the data and discovering the outliers. They are also used to estimate where values are concentrated, what the extremes are, and identify any gaps or unusual values in your data distribution. Determine the mode of the distribution by finding the peak of the histogram, as the value which is most frequently occurring or has the largest probability of occurrence. For many phenomena, it is quite common for the distribution of the response values to cluster around a single mode (unimodal- normal distribution) and then distribute themselves with lesser frequency out into the tails. Similarly, discover for bi-modal or multi-modal datasets. This can help to diagnose problems such as the non-uniformity of data and study the cause of outliers.

Histograms representing different distribution patterns around the mode
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4Represent and discover probability occurrences

Histograms are useful for giving a rough view of the probability distribution and are used to provide insight into their behavior and frequency of occurrence. For instance, In hydrology, the estimated density function of rainfall and river discharge data are analyzed using a probability distribution histogram graph.

Use histograms to give a rough sense of the density of the underlying distribution of the data for density estimation: when estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. However, only nonnegative numbers can be used for the scale that gives us the height of a given bar of the histogram.

Histogram representing probability distributions
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Types of Histograms

1. Equal bin width Histogram

If the bins are of equal size, a rectangle is erected over the bin with height proportional to the frequency—representing the equal bin width histogram.

2. Variable bin width histograms

When bins are not of equal width, the erected rectangle is defined to have its area proportional to the frequency of cases in the bin. The vertical axis is then not the frequency but frequency density—the frequency per unit of the class width on the horizontal axis.

3. Normalized or cumulative histograms

A histogram may also be normalized to display “relative” frequencies. It then shows the proportion of cases that fall into each of several categories, with the sum of the heights equaling 1.

 

When Not to Use a histogram?

1When you need to show distribution against non-numerical categories

Do not use a histogram graph to plot the frequency of score occurrences in a non-continuous data set. Use bar charts for other types of variables including ordinal and nominal data sets since it’s a graph of categorical variables. The bar charts have gaps between the rectangles to clarify this distinction. 

2When you need to represent and discover correlations between two variables

Use a scatter plot when correlations between x and y-axis quantities are needed rather than to represent and gain an understanding of the distribution of a single variable across different intervals. Ask if you need to determine the way one variable changes with respect to the change in the other. In that case, you can use various correlation charts like line graphs, scatter plots, etc.

 

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