The plot below shows 3 subgroups of size 8 for each of two different processes. For Process 1 the 3 subgroups look similar, while for Process 2 subgroup 2 has lower readings than subgroups 1 and 3.

*Data from Dr. Donald J. Wheeler's SPC Workbook (1994).*

**Three Estimates of Standard Deviation**

For each process, there are three ways we can obtain an estimate of the standard deviation of the population that generated this data. **Method 1** consists of computing a global estimate the standard deviation using all the 8x3 = 24 observations. The standard deviation of Process 2 is almost twice as large the standard deviation of Process 1.

In **Method 2** we first calculate the range of each of the 3 subgroups, compute the average of the 3 ranges, and then compute an estimate of standard deviation using Rbar/d2, where d2 is a correction factor that depends on the subgroup size. For subgroups of size 8 d2 = 2.847. This is the local estimate from an R chart that is used to compute the control limits for an Xbar chart.

Since for each process the 3 subgroups have the same ranges (5, 5, and 3), they have the same Rbar = 4.3333, giving the same estimate of standard deviation, 4.3333/2.847 = 1.5221.

Finally, for **Method 3** we first compute the standard deviation of the 3 subgroup averages,

and then scale up the resulting standard deviation by the square root of the number of observations per subgroup, √8 = 2.8284. For Process 1 the estimate is given by 0.5774×√8 = 1.7322, while for Process 2, 3×√8 = 8.485.

The table below shows the Methods 1, 2, and 3 standard deviation estimates for Process 1 and 2. Readers familiar with ANalysis Of VAriance (ANOVA) will recognize Method 2 as the estimate based on the within sum-of-squares, while Method 3 is the estimate coming from the between sum-of-squares.

You can quickly see that for Process 1 all 3 estimates are similar in magnitude. This is a consequence of Process 1 being stable or in a state of statistical control. Process 2, on the other hand, is out-of-control and therefore the 3 estimates are quite different.

In SPC an R chart answers the question "Is the * within* subgroup variation consistent across subgroups?" While the XBar chart answers the question “Allowing for the amount of variation within subgroups, are there detectable differences

*the subgroup averages?”. In an ANOVA the signal-to-noise ratio, F ratio, is a function of Method 3/Method 2, and signals are detected whenever the F ratio is statistically significant. As you can see there is a one-to-one correspondence between an XBar-R chart and the oneway ANOVA.*

__between__

A process that is in a state of statistical control is a process with no signals from the ANOVA point of view.

In an upcoming post Brenda will talk about how we can use Method 1 and Method 2 to evaluate process stability.

## No comments:

## Post a Comment