D2 values for the distribution of the average range appear in the following table.
The columns and rows represent the subgroup size (n) and number of subgroups (k). For a given subgroup size, say n=2, notice that the value of d2 changes as the number of subgroups, k, increases. As an example, notice that d2=1.150 for n=2 and k=15. When k is infinite, the value of d2=1.128. Notice the difference in the value of d2 is small.
When we use control charts, such as the average and range chart, we often wait to collect k=20 to 30 subgroups. The reason we do so is simple. The value of d2 when k=30 is close to the value of d2 derived from a continuous distribution of subgroups having the same subgroup size. This is why the table of d2 values is published up to k =15.
Control charts use range statistics and d2 values to estimate the standard deviation to compute control limits. The average and range chart is a perfect example. Such a chart is often used to track the behavior of a product feature during production.
Sample Parts made Under Like Conditions
Often production personnel sample n=5 consecutive parts from a process and plot the average and range. Sampling in this manner often assures we have parts made under like conditions. Once we have many subgroups we can compute control limits about these averages. When the subgroup average or range falls outside a control limit we should question if the conditions changed. So the calculation of control limits based on a subgroup of parts made under like conditions is an important concept.
How many subgroups do we need to compute control limits?
To compute the control limits, we need to collect a critical number of subgroups. Often we use 30 or more subgroups. Once we compute the range for each subgroup we then calculate the average range. To compute the average range we add the ranges and divide by the number of range values. We use the following expression to compute the average range.
Once we know the average range we need to look up the correct d2 constant. As an example, suppose we collected k=30 subgroups where each subgroup contains n=5 parts. Also assume that the average range is 10.82. As such, the appropriate d2 value for n=5 is 2.326. Using the expression below we can estimate the standard deviation for a subgroup of parts. This standard deviation is a often referred to as measure of within subgroup variation.
So there you have it. This is how we estimate the within subgroup standard deviation for a collection of parts made under like conditions. In another post, I’ll discuss how we can compute the control limits for the average and range charts using within subgroup variation. If you want to learn more about range statistics then click on the following links.
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