Control Chart Constants

This section is optional reading and is provided for readers interested in understanding how the control chart constants were derived.

To assist in understanding the control constants presented in tabular form, the following passage has been adapted from Wheeler [1]. The statistical method for developing control limits will be explained. The data used to make control charts can be displayed in histogram form, as shown below in Figure A-1.



Figure A-1: Histograms for individuals, averages, and ranges of an arbitrary set of data. (source Wheeler)

As shown in Figure A-1, the histograms can show data grouped as individual values, subgroup averages and subgroup ranges. To construct control limits we are interested in knowing how to come up with bounds on all three of these charts. First consider the theoretical models for individuals, averages, and ranges as shown in Figure A-2. These graphs should be familiar to readers from basic statistics.



Figure A-2: Theoretical Models for Individuals, Averages, and Ranges for a stable process (source Wheeler)

From this theoretical models displayed in Figure A-2, the following values are known:

MEAN(X) = the location parameter for the distribution of Individual Values MEAN(Xbar) = the location parameter for the distribution of Subgroup Averages MEAN(R) = the location parameter for the distribution of Subgroup Ranges SD(X) = the dispersion parameter for the distribution of Individual Values SD(Xbar) = the dispersion parameter for the distribution of Subgroup Averages SD(R) = the dispersion parameter for the distribution of Subgroup Ranges

These values are shown in Figure A-3 below.



Figure A-3: Parameters for theoretical models for individuals, averages, and ranges for a stable process (source Wheeler)

Since the six parameters are related, the parameters of the distribution of Subgroup Averages may be expressed in terms of the distribution of X, and the paramters for the distribution of Subgroup Ranges may be expressed by SD(X), as shown in Figure A-4.



Figure A-4: Relationships between parameters for the theoretical models for individuals, averages, and ranges for a stable process (source Wheeler)

With these relationships in mind, consider the two common measures of location and the balance point (averages) and the 50th percentile (medians). For the Subgroup Averages, the balance point is the Grand Average:

Balance Point for Xbar Histogram = Grand Average = Xbar value

The histogram of the Subgroup Ranges will have a balance point that is the Average Range:

Balance Point for Range Histogram = Average Range = Rbar value

Then, as shown in Figure A-4, the MEAN(Xbar) = MEAN(X). From this relationship, the histogram of Individual Values will also have a balance point defined by the Grand Average:

Balance Point for Individuals Histogram = Grand Average = Xbar value.

From this, the two average will summarize the location of all three histograms. These balance points are shown in Figure A-5.



Figure A-5: Balance points for histograms of Individuals, Averages, and ranges. (source Wheeler)

Now the dispersion of the histograms needs to be shown. We use the information in Figure A-4 to develop this. First from Figure A-5,

Balance Point for Range Histogram = Rbar

Then from Figure A-4, MEAN(R) = $$d_2$$*SD(X). These two relationships can be combined to obtain

Sigma(X) = $$\frac{Rbar}{d_2}$$ 							(Eq - 1)

which is a measure of the dispersion of the Individual Values.

We also know from Figure A-4 that SD(Xbar) = $$\frac{SD(X)}{\sqrt{n}}$$

This relation is combined with the measure of the dispersion (Eq - 1) to produce the measure of dispersion for Subgroup Averages:

Sigma(Xbar) = $$\frac{Rbar}{d_2 \sqrt{n}}$$

Finally, from Figure A-4 we know that SD(R) = $$d_3 SD(X)$$. This relation combined with (Eq - 1) results in the measure of dispersion for Subgroup Ranges:

Sigma(R) = $$\frac{d_3Rbar}{d_2}$$

These dispersion measurements are shown in Figure A-6.



Figure A-6: Balance points and measures of dispersion for histograms of Individuals, Averages, and Ranges. (source Wheeler)

These measures of location and dispersion can then be combined into the following formulas for three-sigma limits. The three Sigma(X) values are used to determine the Natural Process Limits:

Natural Process Limits for $$X = Xbar \pm3 \frac{Rbar}{d_2}$$

Control Limits for $$Xbar = Xbar \pm \frac{3Rbar}{d_2 \sqrt{n}}$$

Control Limits for $$R = Rbar \pm \frac{3 d_3 Rbar}{d_2}$$

From this it can be see that the three-sigma limits is an approach and not a fixed distance.



Figure A-7: Three Sigma Limits for Individuals, Averages, and Ranges (source Wheeler)

Then, these values can be related to the values shown in Table A. From this, the structure of the control chart constants for Average and Range charts are as follows

$$A_2 = \frac{3}{d_2 \sqrt{n}}$$

$$D_3 = 1 - \frac{3 d_3}{d_2}$$

$$D_4 = 1 + \frac{3 d_3}{d_2}$$

These formulas are then used to develop the control chart constants for Average and Range Charts in terms of the quantities $$d_2$$, $$d_3$$ and the subgroup of size n, as shown in Figure A-8.

Figure A-8: Distributions for Standardized Ranges. (source Wheeler)

So, for a given subgroup size, n, the constants $$d_2$$ and $$d_3$$ are the mean and standard deviation of the theoretical distribution for the standardized ranges.