The Construction of a Control Chart


The construction of a Control Chart follows some simple general rules. The first thing to plot is the Center Line. Usually, this is a mean or average of the data being charted, but not always. Next we need an estimate of Sigma calculated for each subgroup. The Control Limits are then plotted at 3-sigma above and below the Center Line for each subgroup. In this discussion, we won’t get into the various rules which are then applied that indicate whether we have an out of control situation. Simple, but the devil is in the details.

Calculating the Center Line

i chart (also called X chart, individual chart, Item chart)
For the i chart, where each point of the data is plotted individually, the Center Line is the mean. The calculation of the Center Line (CL) is simply the average of the data being plotted: sum(data) / number of data points.

Attribute Charts (for count or pass/fail type data)
Since the c chart is simply plotting counts, the Center Line calculation is similar with the CL being the average of the counts being plotted.

The p chart and the u chart calculate the mean as the sum of the numerator divided by the sum of the denominator. That is, the average fraction rejected for the p chart (pcap) and total nonconformities divided by the total units (ucap) for the c chart.

Variable Charts (for measurement type data)
The Xbar chart is a little more complex. With the Xbar chart we are plotting the mean (Xbar) values of the subgroups. If all the subgroups are the same size, then the CL is XBarBar, the mean of the means. So Xbar is the mean of each of those subgroups, one for each subgroup. XBarBar is the mean of the XBars.

If we have varying subgroups, then we need to weight them by subgroup size. To calculate the estimated sigma we sum the products of the Xbar and Subgroup size and divide that by the sum of the subgroup sizes:

(n_1*Xbar_1+n_2*Xbar_2+…+n_N*Xbar_N) / (n_1+n_2+…+n_N)

where n_i is the subgroup size of the ith subgroup and Xbar_i is the mean of the data in the ith subgroup.

The R Chart uses the average of the Subgroup Ranges for the Center Line. If, however, we have varying subgroup sizes, then we need to use SigmaCap *d2_i. SigmaCap is explained in calculations of Sigma for Xbar and R Charts below. (Notice that this can mean that the CL can vary by subgroup as shown in the example Range Chart Construction.)

The Median Chart has two methods for calculating the CL depending on whether we are using Mean of the medians (/mean option) or median of the medians. We start by calculating the medians of each subgroup, then taking either the average or the median of these to arrive at the Center Line.

Estimating Sigma

We next need to estimate sigma for each subgroup. We do this because we are trying to find if there is a difference in the subgroup variation between subgroups. The estimated sigma for each subgroup is the within subgroup variation. Once we have this estimated sigma, the upper control limits is the CL + 3*sigma and the lower control limit is CL – 3*sigma.

i Chart
The i chart uses moving ranges to calculate the estimated sigma. In the command gichart width by 2, the 2 indicates the number of points used to calculate the moving range. The first moving range is between points 1 and 2, the second between points 2 and 3, etc. The estimated sigma is calculated using the average of these moving ranges. Sigma, then, is this average divided by d2.

d2 is a published constant that is dependent on the subgroup size, in this case 2. For an i chart that uses 2 as the subgroup size, this value is 1.128. Many control charts use some constant such as this, each dependent on the subgroup size.

We’ve set up some illustrations of these calculations on our Statit e-QC Live Demo. Log into live.statit.com and choose the Chart Construction category. I Chart Construction will demonstrate the calculations for the i chart we have discussed.

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Attribute Charts
The attribute charts are exceptions in that they do not need the constant. For example, the p chart estimates sigma with sqrt(pcap(1-pcap)/n_i) where pcap is the CL calculated above and n_i is the size of the subgroup for that point. Since the p chart calculates the fraction or percent of rejects, the n-i is the denominator for each subgroup. It is important to note that the magnitude of sigma is inversely proportional to the square root of the number of date points making up a subgroup. Thus, your control limits are tighter for a plotted point that has larger denominator.

These calculations are illustrated in P Chart Construction in the category Chart Construction at http://live.statit.com

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The u chart is similar with the calculation for sigma being sqrt(ucap/ni), where ucap is the total nonconformities divided by total units.

Variable Charts
On the Variable Charts, things are a bit more complex. The Xbar chart can estimate sigma based on either subgroup ranges or subgroup standard deviations. When based on subgroup ranges and different subgroup sizes, the estimated Sigmacap is calculated as:

(R_1*f_1/d2_1+R_2*f_2/d2_2 + … + R_N*f_N/d2_N)/(f_1+f_2+…+f_N)

where f_i and d2_i are a ratio of constants and a constant, respectively, that are subgroup size dependent. The Sigmacap is an overall process sigma with the above formula calculating one number. However, the subgroup sizes come into play with the weighting that the f and d2 constants give. Then, to finally calculate the subgroup sigma, the number from the above formula is divided by the subgroup size, n_i. This produces the inverse relationship between sigma and the subgroup size, so that a larger subgroup will have a smaller sigma.

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There is a similar formula for subgroup standard deviation calculations.

For the Range Chart, the same formula for Sigmacap is used, but to calculate the subgroup sigma, the subgroup d3 constant is multiplied by the Sigmacap. Again, the value of d3 is dependent on the subgroup size. The list of d2 and d3 values are available for subgroup sizes with the “d2 d3 Table” macro in the Chart Construction category on http://live.statit.com. Notice that the Lower Control Limit cannot go below zero. If the calculation yields a negative value, zero is used.

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Conclusion

In conclusion, we can identify the common components of a control chart. First is the Center Line, which may be an average of some values or the median. The calculation of the Center Line depends on the type of chart.

The Control Limits are calculated as CL + 3*sigma and CL – 3*sigma. The calculation of the sigma value is dependent on the chart type and, perhaps, on the options chosen for the chart. The control limits on an I chart are constant while those for the Xbar, p chart, u chart and most other charts are dependent on the subgroup size.

Many constants are used in calculation of the control limits for variable charts. The constants are, as well, dependent on the subgroup size.

One thing we cannot do is calculate a standard deviation of the data and assume that that is the value of sigma we want to use. Sigma and, thus, Control Limits depend on the subgroup size of the chart, and usually need to be calculated subgroup by subgroup.

References:

Statit Custom QC Help -> Quality Control -> QC Formulas
Burr, I.W. (1969), "Control Charts for Measurements with Varying Sample Sizes", Journal of Quality Technology, 1:163:7.

If you would like additional information, please send email to statit.support@acs-inc.com.