- Chris Butterworth

# Industrial Problem Solving - Sum of Variances

Too much variation is a common cause of defects in a manufacturing company. This variation causes some items to be over or under the specification limit. All manufacturers lose products due to excess variation. Many food manufacturers struggle with overweight / underweight products.

It can be challenging to come up with a root cause when there are so many candidates. But if you understand a simple statistical law, you can solve such problems.

The idea is this; the variance of the final product is equal to the sum of the variances of the components. Mathematically, this can be written as

... which shows that the variance of the total (final product, in this case) is equal to the sum of the variance of each component that makes up the final product.

Note that the variance is the square of the standard deviation.

var = sd^2

This works when the components are all independent of one another. If they are not independent, then the math is a little different. Two components would be correlated when the amount of one component was dependent on how much of another component was already introduced. This would not be a common situation but it is good to be aware of this fact.

Now, each component does not contribute equally. There will be one component that is responsible for the largest share of this variation. Once we know which component this is, we can apply process controls to this one item and it will bring about a significant reduction in final product variability.

This works with any variable of course, but for the purpose of this brief post, I will walk through an example where the variable is weight. This is an easy example that helps you to understand the method. Applying this to a measurement other than weight may be more difficult but can be done.

Consider a product that is composed of 15 different items and that this product has a wide distribution of weight as depicted in the histogram below.

With weight specification of 25.0 ± 1.0 grams, there is a lot of product being rejected from this process. The standard deviation of the product distribution shown above is 500 mg. Note that variance is the square of the standard deviation so in this case, variance = 250,000 mg^2.

The next task is to determine which of the 15 components is the largest contributor. With a decent sample size of component A we calculate a variance of 400 mg^2. Component A is not a very significant driver of variability. Component B yields a variance of 160,000 mg^2. This is the surely largest source of variation. At this point you wouldn't bother to measure the other components unless you had reason to think your estimates were way off.

The pareto chart below shows all 15 components rank ordered by their measure of variance. Clearly, any effort on any component other than the dominant one will have very little effect on the variability of the final product.

This is a clue generating technique known as the Sum of Variances - it lets us know which component out of many is the main driver of variation. It's our job to figure out how to control this variable. It would be wise to start with a time series plot of the data to uncover any patterns or trends that we could exploit. Statistical process control (SPC) charts could also help us in controlling this component.

This simple technique revealed valuable insights that we can use to solve a Quality problem. Empirical evidence paired with analytical techniques is the smart path.

About the Author

I am a freelance Master Black Belt with a 30 year career in manufacturing. I have managed many projects in industry and have been successful with solving complex problems through the use of analytics.

Through my business, __Belfield Consulting & Training__, I offer contract services as well as on-site training workshops for you and your team.

Problem solving is my business and I am continually developing these skills.