6. Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts
6.3.2. What are Variables Control Charts?

6.3.2.1.

Shewhart X-bar and R and S Control Charts

and S Charts

and S Shewhart Control Charts

We begin with and s charts. We should use the s chart first to determine if the distribution for the process characteristic is stable.

Let us consider the case where we have to estimate by analyzing past data. Suppose we have m preliminary samples at our disposition, each of size n, and let s_i be the standard deviation of the ith sample. Then the average of the m standard deviations is

Control Limits for and S Control Charts

We make use of the factor c₄ described on the previous page.

The statistic is an unbiased estimator of . Therefore, the parameters of the S chart would be

Similarly, the parameters of the chart would be

, the "grand" mean is the average of all the observations.

It is often convenient to plot the and s charts on one page.

and R Control Charts

and R control charts

If the sample size is relatively small (say equal to or less than 10), we can use the range instead of the standard deviation of a sample to construct control charts on and the range, R. The range of a sample is simply the difference between the largest and smallest observation.

There is a statistical relationship (Patnaik, 1946) between the mean range for data from a normal distribution and , the standard deviation of that distribution. This relationship depends only on the sample size, n. The mean of R is d₂ , where the value of d₂ is also a function of n. An estimator of is therefore R /d₂.

Armed with this background we can now develop the and R control chart.

Let R₁, R₂, ..., R_k, be the range of k samples. The average range is

Then an estimate of can be computed as

control charts

So, if we use (or a given target) as an estimator of and /d₂ as an estimator of , then the parameters of the chart are

The simplest way to describe the limits is to define the factor and the construction of the becomes

The factor A₂ depends only on n, and is tabled below.

The R chart

R control charts

This chart controls the process variability since the sample range is related to the process standard deviation. The center line of the R chart is the average range.

To compute the control limits we need an estimate of the true, but unknown standard deviation W = R/ . This can be found from the distribution of W = R/ (assuming that the items that we measure follow a normal distribution). The standard deviation of W is d₃, and is a known function of the sample size, n. It is tabulated in many textbooks on statistical quality control.

Therefore since R = W , the standard deviation of R is _R = d₃ . But since the true is unknown, we may estimate _R by

As a result, the parameters of the R chart with the customary 3-sigma control limits are

As was the case with the control chart parameters for the subgroup averages, defining another set of factors will ease the computations, namely:

D₃ = 1 - 3 d₃ / d₂ and D₄ = 1 + 3 d₃ / d₂. These yield

The factors D₃ and D₄ depend only on n, and are tabled below.

Factors for Calculating Limits for and R Charts

n	A2	D3	D4
2	1.880	0	3.267
3	1.023	0	2.575
4	0.729	0	2.282
5	0.577	0	2.115
6	0.483	0	2.004
7	0.419	0.076	1.924
8	0.373	0.136	1.864
9	0.337	0.184	1.816
10	0.308	0.223	1.777

In general, the range approach is quite satisfactory for sample sizes up to around 10. For larger sample sizes, using subgroup standard deviations is preferable. For small sample sizes, the relative efficiency of using the range approach as opposed to using standard deviations is shown in the following table.

Efficiency of R versus S

n	Relative Efficiency
2	1.000
3	0.992
4	0.975
5	0.955
6	0.930
10	0.850

A typical sample size is 4 or 5, so not much is lost by using the range for such sample sizes.

Time To Detection or Average Run Length (ARL)

Waiting time to signal "out of control"

Two important questions when dealing with control charts are:

1. How often will there be false alarms where we look for an assignable cause but nothing has changed?
2. How quickly will we detect certain kinds of systematic changes, such as mean shifts?

The ARL tells us, for a given situation, how long on the average we will plot successive control charts points before we detect a point beyond the control limits.

For an chart, with no change in the process, we wait on the average 1/p points before a false alarm takes place, with p denoting the probability of an observation plotting outside the control limits. For a normal distribution, p = .0027 and the ARL is approximately 371.

A table comparing Shewhart chart ARL's to Cumulative Sum (CUSUM) ARL's for various mean shifts is given later in this section.

There is also (currently) a web site developed by Galit Shmueli that will do ARL calculations interactively with the user, for Shewhart charts with or without additional (Western Electric) rules added.