Range Statistics and the d2 Constant Used in Statistical Process Control Charts
Range statistics are often used in statistical process control charting. One type of statistical process control chart is the average and range chart. Another type is the individual and moving range chart. To calculate control limits for each SPC chart requires we estimate the standard deviation. This estimate of the standard deviation depends on the sampling program. Click on link to learn more about statistical process control limits and how they’re derived.
Range Statistics for the Average and Range Charts
When using an average and range chart a typical sample size is n=5 consecutive sample specimens and we call this a subgroup. Each sample specimen, in a subgroup, should be produced under homogeneous or like conditions. When produced under like conditions we say the samples specimens came from a system of common causes. The time between subgroups should be constant, if possible. In case of a special cause event, more subgroups may be needed to confirm the event is real.
Range Statistics for the Individuals and Moving Range Charts
When using an individual and moving range chart there are n=1 data points per sampling period. It is a subgroup of n=1 sample. The time between data points should be constant, when possible. Since the change between two consecutive data points captures the smallest time frame we use a moving range of n=2 consecutive data points. For example, the first moving range is the absolute value of the difference between the second and first data point. Likewise the second moving range is the absolute value of the difference between the third and second data point and so on.
How Do We Calculate the Standard Deviation From these Two Range Statistics?
To calculate the standard deviation for these two range statistics we use the following expressions. The first is for the average and range charts and the second is for the individual and moving range charts.
Range Statistics and the d2 Constant
Where did the d2 constant come from? They’re found in statistical process control tables. But how were they derived? It’s better explained by numerical example and it’s something that you can also figure out on your own using Excel or Minitab.
In Minitab, I created a column of 100,000 normally distributed numbers having a mean of 0 and standard deviation of 1. In the next column I created another column of 100,000 normally distributed numbers having the same mean and standard deviation. In the third column I computed the range statistics. To compute the range statistics I subtracted the smallest from the largest value for each row. This yields a column of 100,000 range values. Shown in the figure below is a histogram for the range statistics for n=2.
When I compute the average for the histogram of range statistics for n=2 we have d2=1.13. Notice this value is the same value found in any statistical process control table. Let see if we get the same value for d2, found in statistical process control tables for n=5.
When I compute the average for the histogram of range statistics for n=5 we get d2=2.33. Notice this value is the same value reported in any statistical process control table. Note the d2 constant is reported to two decimal places.
Things to keep in Mind about Range Statistics
Notice that I used normally distributed values to compute the range statistics in Minitab. Also notice the shape of the distribution becomes normally distributed as the sample size increases from n=2 to n=5. This is one reason why n=5 is a common subgroup sample size. Also notice we used histograms to model an infinite curve. If we knew the probability distribution function for n=2 and n=5 we could precisely compute the d2 constants using calculus. That I will leave for another post.
The d2 Constant Explained!
So there you have it. If you ever wondered where these d2 values came from – well now you know! Now I need you to let me know if you appreciate this post by leaving your comments below. I look forward to hearing from you.