Bob Luttman, Robert Luttman & Associates
Average/Range Control Charts
- Use
A variables chart used to monitor process center and spread using equal sized subsamples.
- Construction
1. Collect at least 20 samples with an equal sample size (N= number of samples)
2. Calculate and plot (on separate graphs) the average and range of each sample
3. Calculate and plot the control limits:
Range:R-bar = average of ranges = Sum(Ri) / N
UCLR = D4R-bar
LCLR = D3R-bar
Average:
Grand Average = Sum(X-bari) / N
UCLX-bar = Grand Average + A2R-bar
LCLX-bar = Grand Average - A2R-bar
NOTE: The Average Control Chart is meaningless unless the Range Control Chart is in control!
Average/Range - Why TWO Control Charts?
Parsing Variance
When a series of samples is analyzed and charted, variation comes from two sources:
- Variation between the individual samples as reflected by differences in the means.
- Variation within the individual samples as reflected in the standard deviation or range.
For Example:
Statistical control requires that both sources are in control. AND, since the variance of the mean is dependent on the variance within the sample, within sample variance (either the Range or the Standard Deviation) must be in control first.
Average/Range - Example
You are monitoring volume (cases/day) in the General Surgery Operating Rooms (ni = 5 Rooms). You have collected data for one month:
The Range chart is in control so you may now plot the Xbar chart:
The Xbar chart is not in control. Two points, near the end of the month, indicate a significant rise is volume.
Average / Range - Interpretation
- Special Cause - Same as other control charts.
- Common Cause -
1. Average bad / Range good - The process is stable and precise but performing poorly. Probably only a small number of causes are negatively impacting the process.
2. Average bad / Range bad - The worst case scenario. A wildly fluctuating process (in fact, perhaps a "non-process") with a high degree of process variation. The process is changing frequently. First examine and reduce process variation - define and standardize one method.
3. Average good / Range bad - An unusual event. Could indicate an outlier in the data, inconsistent but adequate performance, or increasing process variation. Likely to occur when a large number of "processors" (either people or machines) are involved and standardization is not emphasized.
4. Average good / range good - Quality nirvana. A stable, predictable, and high performance process. Do NOT forget continuous improvement, though.
Constants for Xbar and R Charts
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Average / Standard Deviation
- Use
A variables chart that monitors process center and spread using subsamples of unequal size.
- Construction
1. Collect at least 20 samples (N= number of samples, ni = size of sample i)
2. Calculate and plot (on separate graphs) the average and standard deviation of each sample
3. Calculate and plot the control limits:
Standard Deviation:
S-bar = average of standard deviations =
UCLS = B4S-bar
LCLS = B3S-bar
Average:
Grand Average = Sum(X-bari) / N
UCLX-bar = Grand Average + A3S-bar
LCLX-bar = Grand Average - A3S-bar
NOTE: The Average Control Chart is meaningless unless the Standard Deviation Control Chart is in control!
Constants for X-bar and S Charts
Sample Size (ni) A3 B3 B4 2 2.659 0 3.267 3 1.954 0 2.568 4 1.628 0 2.266 5 1.427 0 2.089 6 1.287 0.030 1.970 7 1.182 0.118 1.882 8 1.099 0.185 1.815 9 1.032 0.239 1.761 10 0.975 0.284 1.716 11 0.927 0.321 1.679 12 0.886 0.354 1.646 13 0.850 0.382 1.618 14 0.817 0.406 1.594 15 0.789 0.428 1.572 16 0.763 0.448 1.552 17 0.739 0.466 1.534 18 0.718 0.482 1.518 19 0.698 0.497 1.503 20 0.680 0.510 1.490 21 0.663 0.523 1.477 22 0.647 0.534 1.466 23 0.633 0.545 1.455 24 0.619 0.555 1.455 25 0.606 0.565 1.435
For ni > 25
Average / Standard Deviation - Example
You are trying to reduce turnaround time in the Operating Rooms. You collect data for one month. First, you plot an S chart to check the process variability:
The S chart is not in control. Assume, though , that despite diligent effort, you cannot find the reason for the special cause variation. You leave the point in the data and plot the Xbar chart.
The Xbar chart is in control. Therefore, any effort to reduce turnaround time must focus on common cause variation.
Average / Standard Deviation - Interpretation
- Special Cause - Same as other control charts.
- Common Cause -
1. Average bad / Standard Deviation good - The process is stable and precise but performing poorly. Probably only a small number of causes are negatively impacting the process.
2. Average bad / Standard Deviation bad - The worst case scenario. A wildly fluctuating process (in fact, perhaps a "non-process") with a high degree of process variation. The process is changing frequently. First examine and reduce process variation - define and standardize one method.
3. Average good / Standard Deviation bad - An unusual event. Could indicate an outlier in the data, inconsistent but adequate performance, or increasing process variation. Likely to occur when a large number of "processors" (either people or machines) are involved and standardization is not emphasized.
4. Average good / Standard Deviation good - Quality nirvana. A stable, predictable, and high performance process. Do NOT forget continuous improvement, though.
