Here’s an example of a control limit calculation using an X-bar and a R chart:

Assume we are measuring the weight of a product as it leaves an assembly line. Every hour, we take a sample of 5 products and weigh them. These samples are used to compute the X-bar and R values for each hour.

To obtain the X-bar values, we first compute the average of the five product weights for each hour.

Hour 1: X-bar = (20 + 21 + 22 + 20 + 19) / 5 = 20.4

Hour 2: X-bar = (19 + 21 + 22 + 21 + 20) / 5 = 20.4

Hour 3: X-bar = (21 + 22 + 23 + 22 + 21) / 5 = 21.8

Hour 4: X-bar = (22 + 23 + 24 + 23 + 22) / 5 = 22.8

Hour 5: X-bar = (21 + 22 + 23 + 22 + 21) / 5 = 21.8

Next, we calculate the range (R) of the product weights for each hour. The range is the difference between the largest and smallest value in the sample.

Hour 1: R = 22 – 19 = 3

Hour 2: R = 21 – 19 = 2

Hour 3: R = 23 – 21 = 2

Hour 4: R = 24 – 22 = 2

Hour 5: R = 23 – 21 = 2

Now we can calculate the control limits for the X-bar chart using the following formulas:

UCL (upper control limit) = X-bar + A2Rbar LCL (lower control limit) = X-bar – A2Rbar

Where X-bar is the average of the subgroup means, Rbar is the average of the subgroup ranges, and A2 is a constant that depends on the sample size.

To calculate Rbar, we take the average of all R values Rbar = (3 + 2 + 2 + 2 + 2) / 5 = 2.2

We need to use a table of A2 constant values to find the right value based on the sample size. For a sample of 5, A2=2.66.

Now we can use the above formulas to calculate the control limits for the X-bar chart

UCL = 20.4 + (2.66 * 2.2) = 25.74 LCL = 20.4 – (2.66 * 2.2) = 15.06

So the control limits for the X-bar chart are 25.74 and 15.06. Any data points that fall outside of these limits would indicate that a special cause variation is present and further investigation is needed.

It is important to note that this is only an example calculation; in practise, calculating control limits would typically involve more data points and a longer period of time to ensure the process’s stability. Furthermore, the control limits are recalculated on a regular basis to ensure that they are accurate and reflect the current state of the process.

It’s also worth noting that this example only looks at calculating control limits for the X-bar chart. Control limits for other chart types, such as R charts, P charts, and C charts, must also be calculated in order to fully utilise a control chart. Each chart type is used for a specific type of data, and the appropriate chart must be selected.

It’s also critical to understand that control limits are calculated under the assumption that the process is stable and under statistical control. When there are special cause variations, the control limits may not accurately reflect the expected range of variation. As a result, control charts, along with other statistical tools and techniques, are essential for understanding the process and identifying and addressing any special cause variations that may exist.