Part 2. MM4XL Tools > 1. Strategic Tools > Quality Manager > 2. SPC Variable Charts > X-Range Control Charts

Quality Manager

X-Range Control Charts

These two charts monitor the location and the variation of a process, respectively. The Xbar chart shows how the process changes according to a central measure of dispersion, the average, and the Range chart shows when the variation of the process changes. For example, they could be used to monitor whether a satisfactory cleanliness level is maintained in 5 restaurants of the same chain throughout the day, or to monitor the sales trend of a product for 4 sales representatives, or to monitor visits to a website with and without pay-per-click advertising. These charts, however, should be used only when the rate of data collection is slow. In all other cases, the X-Sigma charts with larger samples are preferrable because the sigma value is more accurate than the range value, due to the fact that the latter is found using only two values of a sample, the largest and smallest one, while sigma uses all values in the range.

The picture below shows X-R charts drawn with MM4XL's Quality Manager tool. If an input range was not selected in window 1, the X-R charts will not be available in the list of chart types and the right side of the window below will be blank. After the desired chart type is selected, the charts will display in the right side of the window as shown below. The result can, of course, be printed in a worksheet.

Total Quality Management Control Charts Excel Add-In Software

Technical notes

The control concept of the X-R charts is based on the following assumptions:

  • The input data has at least 2 observations in each sample.
  • The size of the samples is equal for all groups.
  • The data are normally distributed or approximate normality. This implies that the data is collected in a short time and there are enough measurements. A common rule of thumb suggests using at least 20 samples and 100 points. If this doesn't approximate normality you should increase the sample size (use the Process Capability tool to verify whether a process approximates normality). When the sample size exceeds 5 units some authors suggest using the X-Sigma charts instead of the X-Range charts.
  • All groups have equal weight.
  • Observations are collected independently, in order to avoid using autocorrelated data.
  • For the Range chart only, between-group (sample) variation must be due to special causes, which implies a correct functioning of the process.

Unstable (or out of control) processes run outside of control limits and/or present random patterns of variation, which must be stabilized in order to be correctly analyzed with control charts. Stabilizing a process may require collecting new data.

In order to detect a change, the average and the range of the input data are shown in two charts within boundaries as in the picture below. In both charts control limits are placed three standard deviations above and below the central line. Measurements falling outside control limits indicate a change in the process. In practice, 99.7% of normally distributed observations fall within the three standard deviation boundaries, and there are only 27 chances in every 10,000 that it falls outside. Therefore, it is reasonable to conclude that observations outside of the limits show nonconformity in the process, and the analyst should explain why this occurred.

Total Quality Management Control Charts Excel Add-In Software

Total Quality Management Control Charts Excel Add-In Software

When an item goes beyond limits a change has occurred. The change can be bad or good depending on the measurement data. For instance, if the data refer to sales levels falling below the central line, this represents negative performance. When outside the LCL the change is bad and the source of change should be identified and removed from the process. Above the central line the change is good and the source of the change should be made common practice in the process.

In general, the rules governing the normal distribution can be used to interpret control charts:

  • Randomness of data
  • Symmetry of the distribution
  • 99.7% of the observations lie within the 3 standard deviations
  • 95.5% of the observations lie within the 2 standard deviations

Total Quality Management Control Charts Excel Add-In Software

Other rules of thumb to identify variation in the data suggest paying attention to data showing:

  • 7 successive observations on one side of the central line (there is a probability equal to 0.57 or 0.78% of finding such a distribution, and it is reasonable to believe it may be due to a process out of control)
  • 7 successive observations either increasing or decreasing
  • 2 successive points placed very close to one of the limits (the probability of two successive normally distributed points lying between two and three standard deviations on one side of the central bar is 0.05%)

    Input data

    The input data for the X-R charts require two or more columns of data. The picture below shows a suitable input data in the range B1:F26, mind the hidden rows.

Total Quality Management Control Charts Excel Add-In Software

Output results

The output from the X-R charts is made up of two charts and two tables, in accordance with the user selection in the third Quality Manager window (see Introduction to Quality Manager). The first table, shown below, contains basic indexes describing the process. Xdbar is the overall process mean computed on all observations. Rbar is the average Range value of ranges for all groups of observations. StDevBar is the average standard deviation value of standard deviations for all groups of observations.

Total Quality Management Control Charts Excel Add-In Software

For the sake of brevity, the second table is not shown here. In 10 columns it shows the details of the chart limits by item. The second to fifth columns are used to draw the Range chart and the remaining columns are used to draw the Xbar chart.

Total Quality Management Control Charts Excel Add-In Software

The X-Range Chart shown below refers to an input variable with all observations within confidence limits. Although there has been a slight change in the range chart between the seventh and eleventh sample, this has not altered the system. Also the 5 sequential points in the lower half starting at sample 20 tend to verify that the process is stable and can be used for the purpose of control.

Total Quality Management Control Charts Excel Add-In Software

The Xbar Chart below confirms a change in average for sample number 10, and also shows a slight negative bump for samples 19-21.

Total Quality Management Control Charts Excel Add-In Software

A joint reading of the two charts helps us to monitor that a given process performs as expected. This implies a thorough knowledge of the process in analysis, in order to explain any cause of variation detected by the charts.
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