TABLE OF CONTENTS
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
(XBar and RCharts)

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: XBar and RCharts 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 PCharts or UCharts.

PCharts and UCharts measure integers (whole numbers).

Time a task: PCharts might measure the percent of times a task is completed in ten minutes or less.
 Time a process: PCharts might measure the percent of times a process takes more than two days.
 Track "quit" times: PCharts might measure what proportion of the time people work overtime.
 Track physical measurements: PCharts might measure what percent of time something was too heavy, too hot, too long, etc.
NOTE: Don't use XBar Charts or RCharts for any tracking of defects or defectives. Instead, use UCharts or PCharts.
 
NOTE: the more formal jargon of Quality Assurance professionals defines a "defect" as being any nonconformance 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
XBar Charts 
Range Charts
RCharts 
Track the average for a subgroup.
 210 individual measurements
constitute a subgroup
 Find the average of the
subgroup. This is the "xbar" 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 (bellshaped 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 xbar 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. XBar Charts therefore track the averages of individual subgroups.
DATA FOR XBAR CHART AND RCHART 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 
37.0 
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 fourday 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 SigmaChart 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 XBar 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
XBAR CHART AND THE RANGE CHART
Central Tendencies
A. The central tendency for the Xbar 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 RChart 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 xbar chart is 37.3.
B. The central tendency for the Rchart is 8.7.
 
STEP #4: FIND THE A_{2} CONSTANT WHICH IS USED FOR FINDING THE CONTROL LINES OF THE XBAR CHART
Size of the
subgroup
2
3
4
5
6
7
8
9
10

The A_{2} 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 A_{2} Value to use would be 0.577. For the four day week, the A_{2} 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 A_{2} 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 PCharts and UCharts 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 XBar Chart.
 
STEP #6: FIND THE ONESIGMA LINES, WARNING LINES, AND CONTROL LINES FOR THE XBAR (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 OneSigma Line is 39.0 and the Lower OneSigma Line is 35.7. For the four day weeks, the OneSigma 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 D_{4} CONSTANT FOR THE RANGE
CHART
Size of the
subgroup
2
3
4
5
6
7
8
9
10

D_{4} constant
3.267
2.575
2.282
2.115
2.004
1.924
1.864
1.816
1.777

The D_{4} 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. OneSigma 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 XBAR 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 onesigma lines.
C. The weekly average is drawn in black.

STEP #10: CREATE AN RCHART

The Range is drawn in black. The Upper Control Line is drawn in red. For range charts, onesigma 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 XBAR OR
RANGE CHART
On the XBar 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 XBar 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 "nonspecial" weeks to add to our data.

This is the new XBar 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, XBar Charts and RCharts are the probably the most used kinds of SPC charts. For the service industry, PCharts and UCharts are more common, and XBar and and RCharts 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.
TABLE OF CONTENTS