Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts
6.3.2. What are Variables 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 si 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
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, ..., Rk, be the range of k samples. The average range is
Then an estimate of can be computed as
So, if we use
(or a given target) as an estimator of
/d2 as an estimator of
the parameters of the
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
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/c4||
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:
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.