# Calculating Variance for Business: Approaches & Examples

Instructor: Bob Bruner

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

In statistics, variance is a calculation that measures the variability, or dispersion, of individual points within a dataset. Variance calculations provide a simple but effective analysis tool in many areas of business and finance.

## Going Beyond the Average

Calculating an average, or mean, value can be a useful method to quickly summarize large amounts of data. However, average values do not help us understand the variability that may be contained within that data. For example, you might have gotten a 3.0 grade point average during your last semester in school, but if you had actually failed one of your classes, your average grade doesn't quite tell the whole story.

Take a quick look at these two different strings of random numbers, along with their calculated averages:

(41, 53, 60, 57, 66, 34, 50 52, 35, 55, 40, 55, 50, 48, 47): Average value = 49.5

(41, 60, 23, 44, 64, 31, 88, 36, 26, 28, 49, 87, 75, 42, 49): Average value = 49.5

Given that these datasets have the exact same average, and the fact that they start and end with relatively the same values, we might conclude that they are basically similar in nature.

However, suppose these values represented a critical quality assurance metric used by a manufacturing company, such as the count of failures found in production. Can we trust that the same level of product quality is shown by each set of values? If not, how can we quantify the differences hidden in the individual values?

## Variance

In statistics, variance is a direct measure of data variability. It reflects how spread out individual data points are from their average value. Variance is calculated by computing the average of the squared differences of each individual value from the mean value, according to the following formula:

Let's follow through on our quality assurance example. Suppose each dataset represents actual failure counts per 100,000 production units. As input to production, we have used raw materials provided by two different suppliers, giving us two different sets of numbers. We can compute separate variance values for each dataset using the following calculations from the formula.

• For each data point, we first calculate the difference between each value and the average value of 49.5, resulting in the following sets of values:

(-8.5, 3.5, 10.5, 7.5, 16.5, -15.5, 0.5, -9.5, 5.5, 0.5. -1.5, -2.5)

(-8.5, 10.5, -26.5, -5.5, 38.5, -18.5, 14.5, -13.5, -23.5, -21.5, -0.5, 34.5, 25.5, -7.5, -0.5)

• Square each of the individual values:

(72.25, 12.25, 110.25, 56.25, 272.25, 240.25, 0.25, 6.25, 210.25, 30.25, 90.25, 30.25, 0.25, 2.25, 6.25)

(72.25, 110.25, 702.25, 30.25, 1482.25, 342.25, 210.25, 182.25, 552.25, 462.25, 0.25, 1190.25, 650.25, 56.25, 0.25)

• Finally, compute the variance as the average of the new sets of squared differences:

Supplier 1 Variance = 76.0

Supplier 2 Variance = 402.9

From these calculations, we can surmise that while the average of production defects related to the two suppliers is the same, the spread of individual values is much greater in the second case. Assuming that a small tolerance, or range of failures about the average is useful, we would decide to use Supplier 1 rather than Supplier 2.

## Using Variance in the Stock Market

Most investors want to understand not only the expected returns on their investments, but also the risk associated with those returns. Variance is a tool we can use to quantify that risk. The greater the variance, the greater the risk is that individual data points will deviate from the average.

For example, purchasing stocks is considered to be riskier than purchasing bonds. We can confirm this by comparing the variance of the yearly returns associated with stock and bond indices. The table below has 15 years of returns for a large stock index compared to the returns for treasury bonds. We will use a spreadsheet to perform the same variance calculations as laid out in our previous example.

Note again that we first obtain the average of each set of data, calculate differences to the average and their squared values, and finally average the squared difference column to obtain the variance. The stock variance is 234.88, while the bond variance is only 63.37. While the overall stock index provided about twice the average return over 15 years compared to the bonds index, it did so with a greater risk of losing value in any particular year.

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