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# Understanding Variation in Manufacturing

Updated: Dec 2, 2021

Variation is defined as “a change or slight difference in condition, amount, or level, typically within certain limits.” In this article we are concerned with variation within a manufacturing environment.

Every manufacturing process will have some level of variation. By this we mean that if we take measurements of an output of a process, we will observe that not all the measurements will be the same. The variation might be the length of a critical dimension, the resistivity of a material, or the force needed to fracture a material as a few examples.

If we take our output data from all the measurements, we could arrange them as a collection of values, distributed around a central (mean) value. From a statistical perspective, this is what we call spread or variability of the process. We can represent this variability using a numerical calculation, known as the variance.

If you have the measurement data for the full population (e.g. every part made) you can calculate the variance using the below formula:

This looks complicated at first, so let's break it down. The calculation can be described as:

1. Find the mean of the set of data.

2. Subtract the mean from each number in the dataset.

3. Square the result.

5. Divide by the total number in the population

However, in reality, it is very rare that you would have every single data point. It is much more common that you would take a sample of data from a data set. For example, you may measure 100 random parts on a production line of thousands.

In this instance, the formula to measure the variance of a sample is:

Notice how the calculation is almost identical to the population variance, except we now divide by ‘n-1’, i.e. the sample size minus 1. It is not in the scope of this article to explain this slight difference, there are plenty of good online resources if you want to understand the underlying mathematics behind this. simply do an online search for Population vs Sample variance if you’re interested, but it is not too important for this article.

## Standard Deviation

If this formula looks a little familiar to you, well done! The reason is because, if we take the square root of the variance you get the standard deviation. Standard variation is a very important measure throughout statistical process control.

As a reminder, the formula for standard deviation is:

The standard deviation (square root of the variance) gives us information about the spread of the data, and therefore how good a manufacturing process is. This can be shown graphically through the Empirical Rule.

The empirical rule tells us that, for a process with a normal distribution: