# Using the Central Limit Theorem in Business

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.

The central limit theorem can be used to help evaluate data from various distribution patterns. Using this theorem we can apply statistical methods that would otherwise only apply to normal distributions of data.

## Normal Distributions

Many statistical methods rely upon the analysis of data that is normally distributed. The properties of this well-known bell curve distribution pattern allow us to make a number of inferences about the basic characteristics of the data, and simplify the mathematics behind many associated statistical calculations.

But what if your data is not normally distributed? In this case, the central limit theorem provides a useful way to apply the mathematics of normal distributions to almost any data that has been sufficiently sampled.

## The Central Limit Theorem (CLT)

The central limit theorem states that, given multiple samples taken from a population, the mean of those samples will converge on the actual population mean. More importantly, these mean or average samples will form a normal distribution pattern. This is true regardless of the actual population itself, which can have most any distribution, including being unknown or undefined.

Here is an example of a distribution of 100 data points that does not follow a normal distribution profile:

And here we have taken 30 random samples from that data in 30 different trials, each time recording the average of the samples. We see that the average data begins to form a normal distribution pattern, as predicted by the theorem:

Another useful aspect of the theorem states that when multiple independent variables are added together, the sum of those variables, when normalized, will be normally distributed.

In order for the central limit theorem to hold true, a sufficiently large sample size must be created. As a general rule, sample sizes of at least 30 are required. The greater the number of samples used, the more valid the statistical approximation will be.

Now we will discuss areas where CLT can be useful:

### Financial Market

Suppose that we are assembling a portfolio of stocks or other financial holdings and want to balance the overall risk against the possible rewards. We can help make that assessment by using the central limit theorem and our knowledge of the patterns found in normal distributions.

One way to approach this issue would be to consider how individual sectors of the economy have performed during various business cycles, and incorporate that information into our investment model. We know that the returns for each sector might be quite variable over time. However, using the historical data, we can repeatedly take samples from each sector during various overall market cycles, and find the associated mean value for each sector.

In doing so, the central limit theorem states that we will create a good approximation of each sector's average return, along with a normal distribution of those sample averages. Each sector's normal distribution pattern will be narrower or wider, reflecting the extremes and risk inherent in that sector. All of this can be quantified using the normal distribution patterns, and their standard deviations and associated confidence levels.

### Quality Control

Quality control is an important consideration in many businesses. In some production systems the number of variables that can affect the overall quality of a product may seem overwhelming. How can we control for temperature, pressure, machine tolerance, differences in raw material, human error, or any number of possible issues related to a particular product, all at the same time?

According to the central limit theorem, if the individual contributions to a quality measurement have the same relative effect, then their cumulative effect will follow a normally distributed pattern. Knowing that distribution, we can confidently elect to control or monitor only the factors that have the most significant effect on overall quality.

The central limit theorem can also help us when setting up sampling guidelines for the quality control measurements that are required. Suppose, for example, that we are creating special widgets using multiple machines and operators that have the same relative effect widget production. Then, according to the theorem, the sum of the individual machine variances and individual operator skills should be normally distributed. Given that fact, we may not need to sample each individual batch extensively.

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