# Six Sigma Control Limits

by Elliott Taylor; Updated September 26, 2017The six sigma quality system relies heavily on statistical process control, or SCP, and statistical analysis. Control limits are statistical process control tools which allow you to determine whether your process is stable and in control, or trending towards increased variability which could lead to defects in the end product.

## Understanding Control Limits

Control limits are split into upper control limits and lower control limits. The upper control limit, or UCL is typically set at three standard deviations, or sigma, above the process mean, and the lower control limit, LCL, would be set three sigma below the mean. Since approximately 99 percent of normal process variability will occur within plus or minus three sigma, if your process is in control it should roughly approximate a normal distribution around the mean, and all data points should be inside the upper and lower control limits.

## How to Calculate Control Limits

In order to calculate control limits, you must first know your process mean. Start with a sample of 30 or more process observations, for example the height of a solder bump on a circuit board, measured in thousandths of an inch. Calculate the mean by adding all the values and dividing by the number of observations. If your sample size is 30 and the sum of your observed values is 173, the formula would be 173/30 = 5.8.

The standard deviation is easiest to calculate using the STDEV function in a spreadsheet program or the automated standard deviation calculator in a statistical analysis program. Check the resources section for an easy standard deviation calculator. For this example, let's assume the standard deviation is 1.8.

The formula to calculate the upper control limit is (Process Mean)+(3_Standard Deviation) = UCL. In our example, this would be 5.8+(3_1.8) = 11.3. The lower control limit would be calculated as (Process Mean)-(3_Standard Deviation) = LCL. Going back to our example, this would be 5.8-(3_1.8) = 0.3.

To sum up, our process mean for this sample would be 5.8, and would be exactly centered between the upper control limit of 11.3 and the lower control limit of 0.3. These values will be used in the next section to generate control charts

## Generating Control Charts

A control chart is simply a line chart showing sequential measurements of a process characteristic, such as the width of a machined part, with lines added to show the upper and lower control limits. Statistical analysis software packages will have automated control chart functions.

In a spreadsheet program, a simple control chart may be created as follows: Put the actual measurements from your sample in the first column and label it "Measurement". Put the process mean value in the cells in the next column and label it "Center". Insert the upper control limit value in the third column and label it "UCL". Finally, enter the lower control limit value in the last column and label it "LCL".

Select all the data in those four columns and create a line chart based on that data. Your output should be a zigzag line in the middle with your actual observations, crossing and re-crossing the straight center line showing the process mean, with the upper control limit as a horizontal line above it and the lower control limit as a horizontal line below it.

## Interpreting Control Charts

When you evaluate a control chart, you are looking for signals that the process may be out of control or trending towards being out of control. According to the American Society for Quality, the following indicators could signal a process that is out of control:

A single point that is outside either of the control limits; two out of three points in a row that are on the same side of the center line and two sigma or greater away from it; four of five successive points on one side of the center line and greater than one sigma away from it; and finally eight or more points in a row that are trending the same direction.

If any of these warning signs are present, your process may be out of control or about to be out of control. While your measurements may still be within acceptable ranges, if your process is not in control, it's already time to take action because you will soon see defective units produced by the process.

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