 ## CHAPTER FOUR: AVERAGES (X-BAR) AND RANGE CHARTS

from
How to Recognize When Special Causes Exist,
A Guide to Statistical Process Control

© Ends of the Earth Learning Group 1998

by
Linda Turner and Ron Turner

AVERAGES AND RANGE CHARTS

(X-Bar and R-Charts)

 Average and Range Charts measure continuous data that can have fractions in the values. Time a task: time a task five times a day, find the average for the day, and find the range between high and low. Time a process: Time the length of time a customer takes from entering your business to leaving. Time how long a bill takes to get paid. Time how long it takes to repair a broken computer. In each case, find the average and range of small subgroups. Track the "quit times": Track the times of day when the last person leaves. Track the number of overtime hours each person works every week. Track physical measures: weights, temperatures, volumes, lengths, mass, widths, etc. NOTE: X-Bar and R-Charts are not for values that have whole numbers only. Data must be continuous (meaning it can be fractional such as 32.6 minutes.) When counting errors (which are whole numbers, such as 2 or 3, use either P-Charts or U-Charts. P-Charts and U-Charts measure integers (whole numbers). Time a task: P-Charts might measure the percent of times a task is completed in ten minutes or less. Time a process: P-Charts might measure the percent of times a process takes more than two days. Track "quit" times: P-Charts might measure what proportion of the time people work overtime. Track physical measurements: P-Charts might measure what percent of time something was too heavy, too hot, too long, etc. NOTE: Don't use X-Bar Charts or R-Charts for any tracking of defects or defectives. Instead, use U-Charts or P-Charts.

NOTE: the more formal jargon of Quality Assurance professionals defines a "defect" as being any non-conformance with standards (in other words, a defect is a "mistake") and a "defective" as being any product or service that has one or more defects (in other words, a defective is a "failure").

 Average Charts X-Bar Charts Range Charts R-Charts Track the average for a subgroup. 2-10 individual measurements constitute a subgroup Find the average of the subgroup. This is the "x-bar" value that goes into the Average Chart. The average of the averages is the central tendency. The averages for the subgroup will distribute themselves into a normal curve (bell-shaped distribution.) This means there will be roughly the same number of high averages as low. Special averages will be those that are outside the control lines. Track the range of the subgroup. Use the same subgroup you used for the x-bar charts Find the range between the highest and lowest values in the subgroup. This is the "R" that goes into the Range Chart. The average of the ranges is the central tendency. The Lower Control Line will be zero. When values exceed the Upper Control Line, that indicates one of the elements in that subgroup was special.

Note: when you draw a "bar" for horizontal line above a variable like x, that means you are using the average value of a subgroup. X-Bar Charts therefore track the averages of individual subgroups.

DATA FOR X-BAR CHART AND R-CHART EXERCISES
The number of minutes of overtime worked every day.
Values in blue are calculated from the raw data.
 WEEK DAY 1 DAY 2 DAY 3 DAY 4 DAY 5 AVERAGE RANGE 1 34 40 35 42 43 38.8 9 2 32 32 37 42 41 40.6 10 3 36 35 30 36 37 34.8 7 4 37 38 36 43 37 38.2 7 5 40 35 33 32 54 38.8 22 6 -- 44 45 48 48 46.3 4 7 48 37 34 33 43 39.0 15 8 40 34 35 38 39 37.2 6 9 36 41 35 37 35 36.8 6 10 36 42 38 36 44 39.2 8 11 37 34 37 42 38 37.6 8 12 35 30 38 33 36 34.4 8 13 40 38 37 35 36 37.2 5 14 44 36 35 25 37 35.4 19 15 44 42 38 37 40 40.2 7 16 37 33 42 35 33 36.0 9 17 40 36 38 39 35 37.6 5 18 37 35 33 34 38 35.4 5 19 33 32 31 37 35 33.6 6 20 31 38 37 36 33 35.0 7

 AVE. 37.7 36.6 36.2 37 41.1 37.3 8.7

STEP #1: DO YOU HAVE ENOUGH DATA TO CREATE A CHART?
 Requirements A. You need at least 20 subgroups B. There must be at least 100 elements in the subgroups. (At least 100 days within the 20 weeks worth of data.) C. Subgroups must have at least two, but no more than ten elements. Use a sigma chart when more elements are in the subgroups. Answers A. There are twenty weeks of data so there are sufficient subgroups. B. Five days per week times 20 weeks equals 100 data elements in total. If each week had four days worth of data, then we would need a total of 25 subgroups. Note that in this sample, we have one four-day week so that our total is 99 and not 100 elements. It's okay to use the data in this case, but the numbers will improve as you add more weeks. C. Subgroups have five raw data elements each, except for week six which has four days in it. If a subgroup had more than ten data elements, then a Sigma-Chart should be used meaning the standard deviation instead of the range would be used. (We don't discuss Sigma Charts in this book.)

STEP #2: FIND THE AVERAGE AND RANGE OF EACH SUBGROUP
 The "blue" columns include the averages and ranges for each subgroup. For Week #1, the average was 38.8 found by adding the five day totals together and dividing by five. The average for each week goes on the X-Bar or Averages Chart. For Week #1, the range was nine because the high day of 43 minus the low day of 34 is nine. The range for each week goes on the Range Chart.

STEP #3: FIND THE CENTRAL TENDENCY FOR THE X-BAR CHART AND THE RANGE CHART
 Central Tendencies A. The central tendency for the X-bar chart is the average of the total data base. Add all the daily scores and then divide by the number of the days. This is sometimes called the "average of the averages." B. The central tendency for the R-Chart is the average of the ranges (add all the ranges together and divide by the number of weeks). Answers: A. The central tendency for the x-bar chart is 37.3. B. The central tendency for the R-chart is 8.7.

STEP #4: FIND THE A2 CONSTANT WHICH IS USED FOR FINDING THE CONTROL LINES OF THE X-BAR CHART
 Size of the subgroup 2 3 4 5 6 7 8 9 10 The A2 constant is: 1.880 1.023 0.729 0.577 0.483 0.419 0.373 0.337 0.308 For the weeks with five days in a subgroup, the appropriate A2 Value to use would be 0.577. For the four day week, the A2 value would be 0.729.

The A2 Constant is used to approximate the standard deviation from the range. It is not quite as accurate as using the actual standard deviation, but before computers it was widely used because it was much easier to calculate ranges than standard deviations.

STEP #5: FIND THE CONTROL LINE CONSTANT
 The Control Line Constant is found by multiplying the average of the ranges by the A2 constant. The Control Line Constant is equivalent to 3 sigma. When Walter Shewhart first developed these tables, he did not use the entire set of 8 Rules for Identifying Specialness, but rather relied only on Rule #1, "Any points outside the Control Lines." For this sample data, the Control Line Constant for the five day weeks would be (.577)(8.65) which would be 4.99. For the four day week, it would be (.729)(8.65) or 6.31. Notice that the Control Line Constant gets bigger with less data in the same way that control lines were wider apart for P-Charts and U-Charts when "n" the size of the subgroup was smaller. To get One Sigma from these numbers, simply divide them by three, which will yield 1.66 for five days and 2.10 for four days. These sigma are used to find the One Sigma Lines, Warning Lines, and Control Lines for the X-Bar Chart.

STEP #6: FIND THE ONE-SIGMA LINES, WARNING LINES, AND CONTROL LINES FOR THE X-BAR (AVERAGES) CHART
 The One Sigma Lines equal the average of the averages plus or minus one sigma. In this case, for the five day weeks, that would be 37.3 plus or minus 1.7. The Upper One-Sigma Line is 39.0 and the Lower One-Sigma Line is 35.7. For the four day weeks, the One-Sigma Lines would be 39.4 and 35.2. The Warning Lines equal the average plus or minus two sigma. In this case for the five day weeks, the Upper Warning Line is 40.7 and the Lower Warning Line is 34.0 For the four day weeks, the Warning Lines would be 41.5 and 33.1. The Control Lines equal the average plus or minus three sigma. In this case, for the five day weeks, the Upper Control Line would be 42.3 and the Lower Control Line would be 32.3. For the four day weeks, the Control Lines would be 43.6 and 31.0.

STEP #7: FIND THE D4 CONSTANT FOR THE RANGE CHART

 Size of the subgroup 2 3 4 5 6 7 8 9 10 D4 constant 3.267 2.575 2.282 2.115 2.004 1.924 1.864 1.816 1.777 The D4 constant for the five day weeks is 2.115 and for the four day week is 2.282.

STEP #8: IDENTIFY THE UPPER CONTROL LINE FOR THE RANGE CHART
 Multiply the D4 constant times the average range in order to find the Upper Control Line for the Range Chart. The Lower Control Line is always zero. One-Sigma and Warning Lines are not used with range Charts. In this case, for the five day weeks, 2.115 would be multiplied by an average range of 8.65 to yield 18.3. For the four day weeks, 2.282 would be multiplied by the average range to yield 19.8. Any ranges below these values are considered normal.

STEP #9: CREATE AN X-BAR CHART A. The vertical axis measures "Average Minutes Per Day". The horizontal axis tracks data by the week. B. The red lines are the control lines. The green lines are the warning lines. The blue lines are the one-sigma lines. C. The weekly average is drawn in black.

STEP #10: CREATE AN R-CHART The Range is drawn in black. The Upper Control Line is drawn in red. For range charts, one-sigma lines and warning lines are not used. The Lower Control Line is always zero.

STEP #11 IDENTIFY ANY POINTS THAT ARE OUTSIDE THE CONTROL LINES OF EITHER THE X-BAR OR RANGE CHART
 On the X-Bar Chart, Week 6 is clearly "out of control" and special. This was a four day week, and probably the extra minutes worked on average that week have to do with the fact that Monday (Day 1) was a holiday. On the Range Chart, Weeks Five and Fourteen appear special. Usually, range charts are more sensitive to specialness than are X-Bar Charts. Something about these weeks led to extreme differences in daily workload. Friday (Day 5) of Week 5 peaked at 54 minutes, well above any other days. This probably had to do with the fact that it was a Friday before a holiday. Thursday (Day 4) of Week 14 was very low. If this day was investigated, it will usually be discovered that something was obviously different. That "something" would have been the special cause.

STEP #12: REMOVE ANY SUBGROUPS THAT HAVE "SPECIAL CAUSES" AND RECALCULATE THE CENTRAL TENDENCY AND SIGMA.

Subgroups that have special causes are by definition not part of a stable process. Remove them from your data base and recalculate the central tendency and control lines. Usually this causes the control lines to come closer together. It is conceivable that after recalculating them, more subgroups will fall into the "special" category in which case you would repeat step #12.

For our sample data in this chapter, we would remove the three "special" weeks. This would cause us to now have too few subgroups. We would typically still calculate the numbers, but be a little more skeptical about results until we had gotten three more "non-special" weeks to add to our data. This is the new X-Bar Chart which recalculated the central tendency (now 36.9 instead of the original 37.3) and the control lines (now at 41.3 and 32.6, which is the closer than the original 42.3 and 32.3.) No other subgroups now appear special. The new range is drawn in black at 7.5 in place of the original range at 8.7. The new Upper Control Line is drawn in red and has dropped from the original 19.3 to 16.9. No new data points have become "special."

For manufacturers, X-Bar Charts and R-Charts are the probably the most used kinds of SPC charts. For the service industry, P-Charts and U-Charts are more common, and X-Bar and and R-Charts are typically used only for "time" variables. When first starting data collection, most processes will be wildly "out of control" meaning there will be many "special" data points. As you get your "special causes" under control, processes will stablilize and it will become rare for "special" days to occur. When this happens, you need to focus your improvement efforts on "system" or "common" causes instead of "special fixes."

For tips on how make your SPC charts more sensitive, see Appendix A of Chapter Two. 