Quality Control Charts - Computational Details
This topic describes the computations of the plot points, center lines, and control limits for standard control charts for a single part. It also describes computational details for CuSum, moving range, moving average, and EWMA charts. For additional computational details for short run charts for multiple parts, refer to Short Run Transformations.
Plot points
The plotted points in control charts indicate characteristics of the samples. For example, in X-bar charts the plot points are the sample means. Likewise, for R and S charts with ns of at least 2 for all samples, the plot points are the sample ranges, standard deviations, and variances, respectively.
For all plots from individual observations, moving ranges of adjacent observations are computed. For C, U, and Np charts, the plot points are counts of the number of defects for each sample, and for P charts, the plot points are the percentages of defects for each sample.
If an R, S, or S2 chart has been requested and all samples have an n of 1, a moving range chart is automatically produced. The plot points for each sample j (except for the first sample) in moving range charts with an n of 1 are computed using the formula:
rj = absj( x - xj-1 )
where rj is the moving range for each sample j (for j > 1), and xj is the value of the measurement for each sample j. The moving averages plotted for each sample (after the first Npoint - 1 samples) in moving average charts are computed as:
aj = ( mj-Npoint+1+ ... + mj ) / Npoint, if j > Npoint - 1
where aj is the moving average for each sample j after the first Npoint samples, mj is the mean for each sample j, and Npoint is the span, or number of terms of the moving average. The exponentially weighted moving average plot points for each sample j in exponentially weighted moving average (EWMA) charts are recursively computed as:
Ej = w * mj + ( 1 - w ) *Ej-1
where Ej is the moving average for each sample j, mj is the mean for each sample j, and w is the weight parameter (lambda), which has a default value of .10. Note that for first sample (j =0) in each set (see The Nature of Sets) E0 will be equal to the respective centerline value (which, by default, is equal to the mean).
Center lines
The center lines in control charts indicate overall characteristics of the samples. For example, in X-bar charts the center line is the expected value of the sample means, which is estimated by the grand mean, or weighted average of the samples means.
With equal sample ns, the X-bar chart center line is therefore simply the average of the sample means. Similarly, for R and S charts with ns of at least 2 for at least some of the samples, the center lines are estimates of the expected values of the sample ranges or standard deviations, which, when the sample ns are equal, will be the average sample range or the average sample standard deviation.
With equal sample ns, the center line in C, U, and Np charts is the average of the sample counts, and for P charts the center line is the average of the sample percents. For moving range, moving average and EWMA charts with equal sample ns, the center lines are the weighted averages of the plot points for the respective charts.
When there are unequal sample ns and the Separate limits option is selected, the computed center line for the jth sample in R and S charts will be sample size dependent multiples (for d2, and c4; see, Montgomery, 1996; note that for fractional values of N interpolated values for d2 and c2 will be computed) of the process sigma, which is described below. Similarly, for U, Np, and P charts with unequal sample sizes, the Separate limits method will produce sample center lines that vary with sample size.
It is important to note, however, that the Separate limits method, when applied to equal sample sizes, produces center lines that correspond to the overall characteristics of the samples described above. The Separate limits option with unequal n simply takes sample size into account when estimating the expected values of the sample ranges, standard deviations, proportions, etc.
Control limits: Separate limits method
The control limits represent extreme values, beyond which a plot point is considered an outlier and out of control. The upper and lower control limits for a chart are based on the dispersion of the sample measurements. This section gives details for the computation oftrol limits using the Separate limits method, which when sample sizes are equal, yields constant control limits across samples, and which when sample sizes are unequal, yields stepped control limits across samples.
Two other methods for dealing with unequal ns are the default Average N method for unequal sample sizes and the optional Normalized limits method. These methods are described in the Alternative computational methods for unequal ns section below.
The formula used for the process sigma based upon samples in the data file is the average R divided by the d2 based upon the average n (if n is not an integer value, then interpolated values are computed).
X-bar and R charts
Using the Separate limits method for X-bar and R charts, where dispersion is estimated from sample ranges, the process sigma, or the standard deviation of the population of measurements, is defined as:
process sigma = ( r1 / d2 + ... + rk / d2 ) / k
for each of the k samples with ns greater than 1, where r1 ... rk are the ranges for each of the k samples, and d2 is a sample size specific constant (see Montgomery, 1991). Note that each d2 value is determined based upon the respective n for the sample. The standard error of the mean, sigmamean, is then computed as:
sigmamean = process sigma / ( ( n1 + ... + nk ) / k ) 1/2
and the standard error of the range, sigmarange, is computed as:
sigmarange = ( d3 ( Avg. N ) )* process sigma
where d3 is a sample size specific constant (see Montgomery, 1996), and Avg. N is the rounded average sample size (interpolated values for d3 are computed for non-integer values of N). Using the Separate limits method, the upper control limit (UCL) and the lower control limit (LCL) for each of the j samples in the X-bar chart follow the formulas:
LCLj = M - ( ( q * process sigma ) / nj1/2)
UCLj = M + ( ( q * process sigma ) / nj1/2)
where M is the weighted average of the sample means and q is a multiple with a default value of 3.
The lower control limit (LCL) and the upper control limit (UCL) for the R-bar chart for each of the j samples follow the formulas:
LCLj = max( ( d2( nj) * process sigma - ( q * d3( nj) * process sigma ) ), 0 )
UCLj = d2( nj) * process sigma + ( q * d3( nj) * process sigma )
X-bar and S charts
Using the Separate limits method for X-bar and S charts, where dispersion is estimated from sample standard deviations, the process sigma, or the standard deviation of the population of measurements, is defined as:
process sigma = ( s1/ c4( n1) + ... + sk/ c4( nk) / k
for each of the k samples with ns greater than 1, where s1 ... sk are the standard deviations for each of the k samples, and c4 is a sample size specific constant (see Montgomery, 1996). The standard error of the mean, sigmamean, is then computed as:
sigmamean = process sigma / ( ( n1+ ... + nk) / k )1/2
and the standard error of the standard deviation, sigmastd.dev., is computed as:
sigmastd.dev= c5( Avg. N ) * process sigma
where c5 is defined as the square root of (1-c42). Using the Separate limits method, the lower control limit (LCL) and the upper control limit (UCL) for each of the j samples in the X-bar chart follow the formulas:
LCLj= M - ( ( q * process sigma ) / nj1/2)
UCLj= M + ( ( q * process sigma ) / nj1/2)
where M is the weighted average of the j sample means and q is a multiple with a default value of 3.
The lower control limit (LCL) and the upper control limit (UCL) for the S chart for each of the j samples follow the formulas:
LCLj= max( ( c4( nj) * process sigma - ( q * c5( nj) * process sigma ) ), 0 )
UCLj= c4( nj) * process sigma + ( q * c5( nj) * process sigma )
Control limits for C, U, Np, and P charts
The Separate limits method for C charts always produces constant control limits across samples. The LCL and UCL for the C chart are:
LCL = max( ( M - ( q * M1/2) ), 0 )
UCL = M + ( q * M1/2)
where M is the average of the sample counts of defects, and q is a multiple with a default value of 3. For U charts, the LCL and UCL for each of the j samples are:
LCLj= max( (M - ( q * ( M / nj)1/2) ), 0 )
UCLj= M + ( q * ( M / nj)1/2)
where M is the weighted average of the sample counts, q is a multiple with a default value of 3, and nj is the sample size. For Np charts, the LCL and UCL for each of the j samples are:
LCLj= max( ( nj * p - ( q * ( nj * p * ( 1 - p ) )1/2) ), 0 )
UCLj= min( ( nj * p - ( q * ( nj * p * ( 1 - p ) )1/2) ), nj)
where p is the weighted average of the sample percentages of defects. For P charts, the LCL and UCL for each of the j samples are:
LCL = max( ( p - ( q * ( p * ( 1 - p ) / nj)1/2) ), 0 )
UCLj = min( ( p - ( q * ( p * ( 1 - p ) / nj)1/2) ), 1 )
Control limits for moving range, moving average, and EWMA charts
The LCL and UCL for moving range charts are computed using the same formulas as for the control limits for R charts, except that instead of N=1, N=2 is used (as moving ranges are always computed from 2 adjacent observations). The LCL and UCL for moving average charts are:
LCLj= M - ( q * ( process sigma / min( j, Npoint) )
* ( ( 1 / n1+max(j-Npoint, 0) ) + ... + ( 1 / nj ) )1/2
UCLj = M + ( q * ( process sigma / min( j, Npoint) )
* ( ( 1 / n1+max(j-Npoint, 0) ) + ... + ( 1 / nj ) )1/2
where M is the weighted average of the sample means, sigma is the process sigma estimated from the sample standard deviations, and Npoint is the span for the moving average. The LCL and UCL for exponentially weighted moving average charts are:
LCLj = M - ( q * process sigma * w * cfactor1/2 )
UCLj = M + ( q * process sigma * w * cfactor1/2 )
where w is the weight parameter (lambda) and cfactor is a correction factor computed as the sum from k=0 to j-1 of the expression:
( 1 - w )2k / nj-k
Alternative computational methods for unequal ns
In addition to the Separate limits method, two other methods for computing control limits for control charts are available for some charts when sample sizes are unequal. These methods are the Average n method, which is the default method for dealing with unequal ns, and the optional Normalized limits method. The advantage of the Average n method is that, like the Separate limits method with equal n, the Average n method will produce control charts with constant control limits across samples, even though sample sizes may vary.
This can simplify the control charts when there are, perhaps, several different sizes of samples, but all the samples are roughly the same size. The Average n method works by computing the average size of the samples from which the process sigma is computed ( all samples with ns greater than 1 for R, S, or S2 charts, all samples for X-bar charts). The average n value is then substituted for the sample ns in the formulas for control limits described above, thus producing control charts with limits that do not vary with the size of samples.
The Normalized limits method works by standardizing the quantity to be controlled (mean, proportion, etc.) according to units of sigma. The control limits can then be expressed in straight lines, while the location of the sample points in the plot depend not only on the characteristic to be controlled, but also on the respective sample ns.
The disadvantage of this procedure is that the values on the vertical (Y) axis in the control chart are in terms of sigma rather than the original units of measurement, and therefore, those numbers cannot be taken at face value (for example, a sample with a value of 3 is 3 times sigma away from specifications; in order to express the value of this sample in terms of the original units of measurement, we need to perform some computations to convert this number back).
CuSum chart
The cumulative sum (CuSum) chart is constructed following the recommendations found in, for example, Montgomery (1996, Chapter 7). Statistica computes the recommended tabular or algorithmic CuSum chart, and not the "old-style" V-mask control limits that were commonly in use when these charts were (literally) made by hand.
Specifically, the points plotted in the chart are computed by choosing for each sample i one of the values Ci+ and Ci-, depending which of two quantities Ci+ and Ci- is more extreme, where Ci+ and Ci- are computed as:
Ci+ = Max[0,xi - (m0+K)+Ci -1 +]
Ci- = Min[0,xi - (m0- K)+Ci -1 -]
where C0+ and C0- are equal to zero. The value of K is usually called the reference value (or allowance or slack value), and is related to the shift parameter delta expressed in standard deviation units (i.e., delta=|mu1-mu0|/sigma, where mu1 is the shift against you want to protect):
K = delta/2 * sigma
See also, Unbiasing Constants c4, c5, d2, d3, d4.