MEDIAN AND RANGE CHART | Six Sigma and Beyond: Statistical Process Control, Volume IV

The median and range chart is a special-purpose control chart for variables data. This control chart technique is similar to the Xbar and R chart and offers the following advantages:

Median and range charts do not require daily calculations. They are constructed quickly and simply. Simplicity aids the acceptance of control charts by operation personnel.
Median and range charts provide descriptions of location and spread with a single chart. This helps people understand the relationship between the distribution of medians and the distribution of individual measurements. The median and range chart is a useful educational device.
Median and range charts record individual measurements on the control chart and may be used to compare the output from several process streams (e.g., cavity-to- cavity or fixture-to-fixture differences).

The median and range chart also has disadvantages:

Median and range charts are used less frequently than are Xbar and R charts. They do not resemble other control charts. People may require a period of training or adjustment before they feel comfortable using the technique.
The median is a less efficient estimator than the mean. Given the same data, the standard deviation for the median is larger than the standard deviation for the mean ( ). Therefore, control limits for the same data will be wider for the median chart than for the Xbar chart. This makes the median chart less sensitive to changes in the process than is the Xbar chart.

STEPS FOR CONSTRUCTING A MEDIAN AND RANGE CHART

1. Determine the Sampling Frequency and Sample Size

Collect data from the process by sampling sets of three or more measurements. An odd sample size ( n ) should be used for this control chart because it simplifies the identification of medians. Take the sample subgroups at logical and regular intervals. For this charting method, the sample size ( n ) must remain constant for all sample groups. The parts that comprise the sample groups must also come from only one process stream (i.e., a single tool, head, die cavity, etc.).

Although many control charts are constructed for characteristics for a product, process parameter may also be measured and used to control the process. These measurements (speeds, feeds, cycles, times, temperatures , etc.) are often very useful for the prevention of quality problems.

2. Record Measurements on the Control Chart

Because a chart for location and spread must be used for variables data, both of these chart forms and a data table are usually placed on the same sheet of paper. The vertical scales of the charts refer to the location of the median and range values, and the horizontal scales indicate the sequence of sample groups across time, date, and/or production shifts. The data values, time index, and plotted points should be aligned vertically.

The measured values may be plotted directly on the control chart. This saves the time needed to record the data and helps identify the median of the subgroups. (Step 6 should be completed before the data are plotted on the control chart. To make data recording easier, the scale for the control chart may match the divisions of the measurement device.)

3. Calculate the Median and Range for Each Sample

Identify the median for each sample group . This sample statistic serves as an estimate of central tendency for the process.

The range indicates the process short- term variability and should be calculated for each subgroup .

R = X _highest - X _lowest

When the measurements are plotted directly on the control chart, the scale for the control chart may be used to measure the range of each subgroup.

4. Calculate the Process Center Line

The average range (Rbar) and the mean of the sample medians serve as the process center lines. These values are typically calculated only after 25 or more sample groups have been collected from the process. Because the range chart represents piece-to-piece variation and information from this chart is used when calculating all control limits, the range chart is calculated and interpreted first.

where k is the number of subgroups, R ₁ and are the range and median of the first subgroup, R ₂ and are from the second subgroup, etc.

5. Calculate Control Limits for the Charts

Control limits indicate the amount that sample ranges and medians would vary if only common causes of variation were operating. The control limits are based on the size of the sample groups and the amount of variation found within the subgroups. Control limits reflect the natural variability of the process and are not specification limits or objectives. The within-sample (piece-to-piece) variation is reflected by the sample ranges. The formulas for calculating control limits for both the median and range charts are listed in Table 10.1. Constant values that vary by the sample size ( n ) are typically used for calculating control limits. The constants used for calculating control limits are presented in Table 10.1.

Table 10.1: Table of Constants and Formulas for Median and Range Charts
Chart for Medians ( X )		Chart for Ranges (R)
Subgroup Size	Factors for Control Limits	Divisors Estimate of Standard Deviation	Factors for Control Limits
n	A ₂	d ₂	D ₃	D ₄
2	1.880	1.128	”	3.267
3	1.023	1.693	”	2.574
4	0.729	2.059	”	2.282
5	0.577	2.326	”	2.114
6	0.483	2.534	”	2.004
7	0.419	2.704	0.076	1.924
8	0.373	2.847	0.136	1.864
9	0.337	2.970	0.184	1.816
10	0.308	3.078	0.0223	1.777

where d ₂ , D ₄ , D ₃ , and A ₂ are constants varying by sample size, with values for sample sizes from 2 to 10.

6. Determine the Vertical Scale for the Median and Range Charts

It is important to determine the vertical scale for a control chart after calculating the process center lines and the control limits. If the scales are determined before these calculations, the process center lines may be far from their expected positions at the center of the charts. In addition, careless selection of the chart scales may force the control limits beyond the range of the charts' scales. The vertical increments should be selected so that the plotting of the sample points is relatively simple and easily understood . Sometimes, the divisions of the measuring device are used as the scale increments of the control chart.

The Rbar value should be located about one third to one half of the distance from the bottom of the vertical scale of the R chart. The UCL _R should be positioned approximately two thirds to three quarters of the distance up the scale. When the sample size is six (6) or fewer, there is no lower control limit for the R chart.

The mean median value should be located at about the center of the scale on the median chart. Center the control limits around about the mean median value and space them about one half to two thirds of the distance between the mean median value and the ends of the chart scale.

7. Plot the Range and Median for Each Subgroup

The range should be plotted as a solid dot on the range chart. Each individual measurement should be plotted in the median chart as a solid dot. The median should be identified by drawing a circle around the middle measurement for each subgroup. A solid line should connect the plotted data. This line indicates the time order of the data and aids the analysis of patterns and trends in the process.

8. Draw the Process Center Lines and Limits on the Charts

Draw the mean median and Rbar lines as solid horizontal lines. The control limits are usually drawn as dashed, bold, horizontal lines. All lines should be labeled and dated. During the initial study phase, the lines are considered trial center lines and control limits. These may require recalculation as the process is modified and improved.

9. Analyze the Range Chart for Process Control

Because the control limits for both the range and averages charts are based on the value of Rbar, the R chart must be analyzed first. If the piece-to-piece variation is not stable, the control limits for the averages chart may be inappropriate and lead to false conclusions. Analyze the data points by comparing them to the control limits and examining them for unusual patterns or trends. Evaluate the process against the five criteria for process control before labeling it as stable and in control.

Are there points beyond the control limits?
Are there runs of seven (7) or more points?
Are there trends of seven (7) or more points?
Are there cycles of points?
Is there unusual variation (the 1/3 or 2/3 rule)?

If all of the plotted data in a range chart are within the control limits and randomly distributed around the center line, the spread of the process is stable and predictable (see Figure 10.1). This condition is a prerequisite for calculating ongoing control limits, studying capability, and generalizing experimental studies to longer and larger production runs.

Figure 10.1: A stable and predictable process.

If any of the plotted data are above the upper control limit (see Figure 10.2), this indicates one or more of the following:

The control limits have been miscalculated or have been drawn incorrectly.
The plotted sample that is above the control limit has been miscalculated or plotted improperly.
The within-sample (piece-to-piece) variation has increased from that of the past (the nature of the excessive variation must be investigated).
The measurement system has changed.
Control limits calculated from these data are questionable (they are probably too small).

Figure 10.2: A process with a point outside of UCL.

When any of the plotted data are below the lower control limit (see Figure 10.3), this indicates one or more of the following:

The sample size ( n ) must be equal to or greater than seven (7) for there to be a lower control limit for this control chart.
The control limits have been miscalculated or have been drawn incorrectly.
The plotted sample that is below the control limit has been miscalculated or plotted improperly.
The process spread has decreased from that of the past (the nature of the reduced variation should be investigated and made a part of the manufacturing routine).
The measurement system has changed.
Control limits calculated from these data may be questionable (they are probably too large).

Figure 10.3: A process with a point out of LCL.

If there is a run of seven (7) or more points above the Rbar (see Figure 10.4), this indicates any or all of the following:

A sustained decrease in the process capability
A change in the measurement system
That all control limits may need to be revised because the process spread has changed

Figure 10.4: A process with a run above Rbar.

A run of seven (7) or more points below the Rbar (see Figure 10.5) means any or all of the following:

The process spread has decreased (the cause should be identified so that the improved process can be replicated).
The measurement system has changed.
The control limits may be questionable (they are probably too large or too wide).

Figure 10.5: A process with a run below the Rbar.

An upward trend of seven (7) or more data points (see Figure 10.6) indicates that the process spread is increasing. This should be investigated immediately because process capability is degrading.

Figure 10.6: A process with degrading variation.

A downward trend of seven (7) or more data points (see Figure 10.7) means that the process spread has decreased. The process is improving and the control limits may need to be recalculated. The process capability analysis should also be reassessed after the process is made stable and predictable at a new level of variation.

Figure 10.7: A process with an improvement in variation.

Cycles (see Figure 10.8) indicate repeating patterns of increasing and decreasing process spreads . These are opportunities for process improvement.

Figure 10.8: A process with cycles.

Unusual variation (see Figure 10.9) in the sample ranges means one or more of the following:

The control limits have been miscalculated.
The process or sampling plan is stratified.
The data have been edited and changed from the original readings .

Figure 10.9: A process with unusual variation.

10. Analyze the Median Chart for Process Control

Use the five criteria for process control to evaluate the median chart. Identify and note all out-of-control conditions and factors that contribute to instability. The time, shift, date, material lot number, material vendor, process parameters, and production personnel included in the manufacturing operation during the out-of-control period should be noted in the process log. The problem-solving techniques included in Chapter 2 will help the identification of special causes of variation.

Are there one (or more) point(s) beyond the control limits?
Are there runs of seven (7) or more points?
Are there trends of seven (7) or more points?
Are there cycles of points?
Is there unusual variation (the 1/3 or 2/3 rule)?

If all of the plotted data on the Xbar chart are within the control limits and randomly distributed around the center line, the location of the process is stable and predictable (see Figure 10.10). The process is in control. This condition is needed for calculating ongoing control limits, studying process capability, and generalizing experimental studies to longer and larger production runs.

Figure 10.10: A stable and predictable process.

Control limits on this chart are meant for the circled data points (medians), not for individual data points. Individual data points (not circled) beyond the control limits do not mean that the process is out of control.

If any of the plotted medians (X) are outside of the control limits (see Figure 10.11), this indicates any or all of the following:

The control limits have been miscalculated or have been drawn incorrectly.
The plotted sample average that is beyond the control limit has been miscalculated or plotted improperly.
The process average has changed significantly from that of the past (the cause for the process change must be identified and investigated).
The measurement system has changed.
The process center line calculated from these data is questionable because the center line is influenced by the extreme variation.

Figure 10.11: A process with points outside of the control limits.

If there is a run of seven (7) or more averages on one side of the process average line (see Figure 10.12), this means any or all of the following:

The process location has changed (or is still different) from that of the past.
The measurement system has changed.
The control limits calculated from these data are questionable and may require recalculation.

Figure 10.12: A process with a run below the median.

A trend (up or down) of seven or more sample averages (see Figure 10.13) means that the process average is gradually changing or drifting. The cause of the change must be identified and assessed. Undesirable trends must be eliminated from the manufacturing process. Trends due to process improvement efforts must be stopped at an optimal setting. The process must be made stable and predictable at a new level. The process center line, control limits, and capability analyses must be recalculated after the process is controlled at a new level.

Figure 10.13: A process with a downward trend.

Cycles indicate repeating patterns of high and low process locations (see Figure 10.14). These are opportunities for improvement because a number of process settings have been observed . The factors that cause the different levels of the cycle and the optimal setting must be identified.

Figure 10.14: A process with cycles.

Unusual variation (see Figure 10.15) in the sample averages means any or all of the following:

The control limits have been miscalculated.
The center line for the range chart has been miscalculated.
The process or sampling plan is stratified.
The data have been edited or changed from the original readings (excessive rounding errors).

Figure 10.15: A process with unusual variation.

11. Identify and Eliminate the Special Causes of Variation

How people react to control chart signals is the most critical aspect of the SPC program. If special variation is to be identified and eliminated, personnel must analyze the operation and all of the resources used during the actual process. In addition to the control chart, fishbone diagrams, Pareto charts, process flow charts, and controlled experiments may be needed to identify and resolve factors that create instability in the process. Promptness of reaction, elimination , and prevention of special causes of variation can never be overemphasized. A production department's reaction to out-of-control conditions indicates its commitment to understanding and controlling the process. The control chart provides signals to help production departments maintain good quality.

12. Extend the Control Limits for Ongoing Process Control

When the data from the process (at least 25 subgroups) are consistently within the control limits and in control, the control limits may be extended to future samples of data. The process operators and supervisors should closely monitor the charts and act promptly to correct special causes of variation. If the process is improved, or if extreme sources of variation are eliminated, the process center lines and control limits may no longer be appropriate. The changes made to the process may make it necessary to recalculate the process center lines and control limits. After the process is stabilized and made predictable, the median and range technique may be simplified. The range chart is discontinued and range information is read from the median chart instead. The range is read as the distance between the highest and lowest points of each subgroup. The control limits from the range chart may be superimposed on the median chart (using a note card or a sheet of clear plastic).

Although this simplified technique takes less time, it is harder to recognize all forms of special variation. It is impossible to search for runs, trends, cycles, and unusual variation. Therefore, the simplification of median and range charts should be approached with caution.

13. Analyze the Control Chart Data for Process Capability

After the manufacturing process has been stabilized and is operating with only common sources of variation, assess the capability of the process. This analysis requires using a distribution to estimate the process population. The population is then compared with and evaluated against the product specifications.

STANDARDIZED CHARTS FOR ATTRIBUTE DATA

Frequently, sample sizes on attribute charts, in general, and proportion-defective ( p ) and yield ( q ) charts, in particular, will vary to a significant extent. This often results from those situations in which attribute analysis is employed for daily, weekly, or monthly analysis in a final test context. In such cases, sample sizes will often vary by more than ±25% from the average sample size ( n ) used to calculate control limits. In these or similar instances, one of the following options may be employed:

Calculate a separate set of control limits for each sample.
Calculate a set of control limits for those individual ( i ) samples where the sample size ( n _i ) is within ±25% of n and separate limits for those samples that fall outside this range.
Calculate three sets of control limits, each corresponding to the minimum, average, and maximum expected and observed sample sizes.

(Opinions regarding the percentage variability of any n from vary considerably among quality professionals. In fact, many conservative statisticians employ ±10% variability from nbar as a determinant for employing options other than the standard p chart.)

Although all of these approaches are statistically sound, all present problems from the standpoint of the day-to-day interpretation of the plotted data. Varying control limits for all or some samples frequently tends to make all but highly trained statisticians uncomfortable. The calculation of limits each time that a new sample is drawn becomes tedious , and the proper identification of runs, trends, and cycles becomes difficult when sample sizes (and therefore limits) vary widely.

An additional problem is frequently encountered in the interpretation of attribute charts in those cases in which the following are both true:

Sample sizes periodically are very large.
Proportion defective and defect rates are very low.

In these cases, the day-to day problems noted previously are compounded, in that the following are true:

Extreme changes in sample sizes will cause points to be plotted off the chart or so closely together (i.e., tightly) that interpretation of point-to-point drift is difficult.
Rounding error, even when four places to the right of the decimal point are employed, will cause plotted values to appear on or inside control limits when in fact they are outside of the limit.
Meaningful changes in process defective rates or yield levels appear to be insignificant as they occur within the context of the combination of large sample sizes and already-small defective rates.

For example, let us assume that two days of final inspection data yield the following results:

n	np	p	q
101256	47	.00046	.99954
83411	40	.00047	.99953

where

p	=	.0005
q	=	.9995
n	=	sample size
np	=	number of defective parts
p	=	Proportion of defective parts
q	=	1 - p ; yield; or the proportion of good parts

In most cases, these p or q values would be plotted as the same value; even if they were not, the "feel" for the magnitude of difference between the values would commonly be lost. One might suggest that this problem could be solved by scaling the control chart with a finer definition; perhaps converting the entire range of the scale from possibly .0010 (or 0.9990) to 0 (or 1.000). The problem with this procedure would become evident when a third day's data resulted in the following:

N	=	4,376	p	=	.00046
np	=	2	q	=	.99943

Recalling that the control limits are a function of the sample size, the control limits for the three samples would be calculated as follows (the calculations below are presented for a p chart; yield charts results would be similar):

Sample	n	np	p	UCL _p	LCL _p
1	101,256	47	.00046	.00071	.00029
2	83,411	40	.00047	.00073	.00027
3	4,376	2	.00046	.00151	-.00051(0)

As shown above by these data, the UCL for the chart (or the LCL for a yield chart) would be off the chart if the scale were adjusted to show differences between plotted values of, for example, .00046 and .00047. Note that this would result even though the p values for samples 1 and 3 are equal, where

Finally, let us return to the issue of rounding error under these conditions. Using standard rounding procedure, the following values would be obtained at varying sample sizes for a q (yield) chart where qbar = .9995, the observed yield was .9994, and values were to be rounded to four places:

n	Calculated LCL	Rounded LCL	Interpreted Control Condition
460,000	.999401	.9994	In control
485,000	.999404	.9994	In control
510,000	.999406	.9994	In control
535,000	.999408	.9994	In control
560,000	.999410	.9994	In control
585,000	.999412	.9994	In control
610,000	.999414	.9994	In control
635,000	.999416	.9994	In control

Two aspects of these data are of some significance. First, under many circumstances, all of the observed yield values from this example would probably have been plotted on the LCL and would not have been considered to be out-of-control points or values. Second, and perhaps more important, note that although the sample sizes shown are significantly different, the reflected discrepancies between the observed q value (.9994) and the calculated LCL values reflect very little difference in terms of a perceived context, which is true regardless of the control condition decision. The typical individual responsible for maintaining control charts would simply not believe that a change of difference of .000004 in yield (where n = 535,000 and 585,000) was very significant.

A useful option to the varying-limits chart, which will also obviate many of the problems previously noted, is the standardized-value attribute chart. Recall that, for example, the standard p chart control limits are calculated as:

where pbar (yield) = 1 - pbar. It is important to remember that the formula component

represents the standard error of the sampling process; that is, a value corresponding to the expected sampling error at any sample of size n. Control chart convention places the control limits at ±3 standard error increments above and below the process average (pbar), which is the origin of the constant (3).

In a standardized-value chart, we are interested in transforming each observed value ( p, for example) into a standardized value, that is, a value that will represent the number of standard errors that the observed value falls away from the process average. To calculate a standardized value for a p chart, for example, we would employ the following formula:

where

p _s	=	the standardized p value
P	=	the observed sample proportion defective
pbar	=	the process average proportion defective
q	=	1 - p (yield)
n	=	sample size

As may be observed by closely examining the formula, the resultant standardized value will represent the deviation of p from pbar in terms of standard error units. Values of p equal to pbar will result in a standardized value of 0; values of p that would fall exactly on the control limit(s) would reflect standardized values of 3 (or -3). Therefore, the standardized-value chart allows for a control chart in which the process average will always have a scale value of 0, with control limits of +3 and -3 ( assuming that tighter control is not desired). For example, let us assume the following:

n = 59,593 pbar = .0026 p = .0031

Then

The standardized value would be plotted directly on the control chart and obviously would fall between the process average of 0 and the upper control limit of +3. The advantages of this form of chart are as follows:

The process average value and control limits never have to be recalculated in terms of the control chart. The p value employed in the calculation of the P _s values would, of course, have to be modified as appropriate, but the chart configuration and scaling would remain constant.

The standardized values will reflect meaningful differences between plotted values when sample sizes are relatively large and process-defective rates are fairly small. For example, returning to the yield data previously presented, the standardized values would appear as follows:

n	q	standardized p	LCL _s
460,000	.9994	-3.03	-3.0
485,000	.9994	-3.12	-3.0
510,000	.9994	-3.19	-3.0
535,000	.9994	-3.27	-3.0
560,000	.9994	-3.35	-3.0
585,000	.9994	-3.42	-3.0
610,000	.9994	-3.49	-3.0
635,000	.9994	-3.56	-3.0

Note that in this case, the changes in the chart at different sample sizes for equal yield would be reflected by plotted point ( q _s ) differences of .03 to .56. This compares with a standard chart in which the plotted points remain the same but the control limits vary; for this data, the control limits varied from .999401 to .999416, or only .000015. In most cases, the individual using this chart would have a much easier time interpreting the standardized chart than the typical varying-limits chart. This assertion is based on the fact that

The standardized value provides a more accentuated degree of difference; that is, the "appearance" of change will be more noticeable.
Operators have generally been trained to describe out-of-control conditions, such as runs and trends in the context of the plotted points, not control limit changes. Many operators, and supervisors for that matter, would not have recognized that the eight identical yield values (.9994) actually represented a trend condition. A quick glance at the standardized values, however, shows not only that the points are outside of the control limits, but that the process-defective values (were they occurring in a sequential process) were, in fact, trending farther and farther away from the process average. Standard trends and runs (which occur within the control limits) would be just as obvious when displayed by standardized values, and just as difficult to observe with a varying-limits chart.

Finally, the standardized-value approach may be used with any of the common attribute charts. For a yield ( q ) chart, the standardized-value formula would appear as

For a u chart, the formula would appear as follows:

Although this same procedure could be employed with np and c charts, it is unlikely that this need would arise given that these charts require equal sample sizes. This condition would occur relatively infrequently within an associated context of large sample sizes and small discrepancy values.