Also called: averages and range chart
Description The –
X (pronounced X-bar) and R chart is a pair of control charts used to study variable data. It is especially useful for data that does not form a normal distribution, although it can be used with normal data as well. Data are subgrouped, and averages and ranges for each subgroup are plotted on separate charts.
When to Use
• When you have variable data, and . . .
• When data are generated frequently, and . . .
• When you want to detect small process changes
• Especially in manufacturing, where a sample of four or five pieces may be used to represent production of several hundred or thousand pieces
Procedure Construction
1. Determine the appropriate time period for collecting and plotting data. Determine the number of data points per subgroup (n). Collect at least 20n data points to start the chart. For example, with subgroup size of three, 60 data points will be needed to create 20 subgroups.
2. If the raw data do not form a normal distribution, check whether the averages of the subgroups form a normal distribution. (The normality test can be used.) If not, increase the subgroup size.
3. Calculate––
X (X-double-bar),–
R (R-bar), and the control limits using the worksheet for the –
X and R chart or moving average–moving range chart (Figure 5.22) and the chart for –
X and R or moving average–moving range (Figure 5.23).
4. On the “Average” part of the chart, mark the numerical scale, plot the subgroup averages, and draw lines for the average of the averages and for the control limits for X. On the “Range” part of the chart, mark the numerical scale, plot the ranges, and draw lines for the average range and for the control limits for R.
5. Continue to follow steps 4, 5, and 6 of the basic procedure.
Analysis
1. Check the R chart for out-of-control signals. All the signals listed on page 157 can be used.
2. If the R chart is in control, check the –
X chart for out-of-control signals. All the signals listed on page 157 can be used.
control chart 161
Process: _____________________________ Calculated by: _________________________
Data dates: __________________________ Calculation date: ______________________
Step 1. Calculate average X _
and range R (the difference between the highest and lowest values) for each subgroup. Record on chart.
Number of values in each subgroup = n = _______
Number of subgroups to be used = k = _______
Step 2. Look up control limit factors.
n A2 D3 D4
Step 3. Calculate averages (X __
Average of the averages = X __
= ΣX _
÷ k
= _______ ÷ _______ = _______
Sum of the ranges = ΣR = _______
Average of the ranges = R _
= ΣR ÷ k
= _______ ÷ _______ = _______
Step 4. Calculate control limits.
3sestimate for X _
chart = 3sX– = A2 × R
_
= _______ × _______ = _______
Upper control limit for X _
chart = UCLX– = X __
+ 3sX–
= _______ + _______ = _______
Lower control limit for X _
chart = LCLX– = X __
– 3sX–
= _______ – _______ = _______
Upper control limit for R chart = UCLR = D4 × R _
= _______ × _______ = _______
Lower control limit for R chart = LCLR = D3 × R _
= _______ × _______ = _______
Figure 5.22 X and R chart or moving average–moving range chart worksheet.–
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control chart 163
Process: Date #1 #2 #3 #4 #5 #6 Sum Avg. Range Average Range
Variable:Units: X:R:
Limits set by: UCLR:
Date: UCLx:LCLx: Figure 5.23
– Xand Rchart or moving average–moving range chart.
Example
The ZZ-400 team collected a set of 40 values, arranged in time sequence, for product purity. The histogram in Figure 5.24 indicates that the data are slightly skewed, as data that are being pushed toward 100 percent or zero percent often are. The team will try subgrouping the data by twos. Figure 5.25 shows the subgrouped data.
164 control chart
Figure 5.24 Histogram of–
X and R chart example.
X and R or Moving Average–Moving Range Chart Process:
ZZ-400 Purity % PW 3/29/05
99.88
Figure 5.25 X and R chart example.–
The first subgroup contains values one and two: 99.7 and 99.6. Their average is 99.65 and their range is 0.1. The second subgroup contains values three and four: 99.7 and 99.4. Their average is 99.55 and their range is 0.3. The calculations are continued for all the data (see Figure 5.25). At this point, the team checks the distribution of subgroup averages with a normal probability plot. The distribution is approximately normal, so subgroup size of two is acceptable.
Now, using the worksheet, averages and control limits can be calculated. See the completed worksheet (Figure 5.26) for the calculations.
The subgroup averages and ranges are plotted, and the averages and control limit lines are drawn on the chart. Figure 5.25 shows the control chart. There are no out-of-control signals.
This example is part of the ZZ-400 improvement story on page 80.
Considerations
• This is the most commonly used type of control chart.
• The subgroup size and sampling method should be chosen to minimize the chance of variations occurring within the subgroup and to maximize the chance of variations occurring between the subgroups. However, each measurement should be independent of the others.
• The larger the subgroup size, the better the control chart will detect changes in the average.
• For most nonnormal distributions, a subgroup size of two or three will be adequate to have the averages form a normal distribution and, therefore, to have the chart calculations be accurate. If the data are highly nonnormal, a subgroup size of four or five might be needed.
• For subgroup sizes smaller than seven, LCLRwill be zero. Therefore, there will be no out-of- control signals below LCLR.
• An out-of-control signal on the –
X chart indicates that the center of the process distribution has changed. An out-of-control signal on the R chart indicates that the width of the process distribution has changed, although the process averages (the –
X values) may show no unusual variation. Think of it like shooting at a target: R changes when your shots form a tighter or looser pattern around the bull’s-eye. –
X changes when the pattern is no longer centered around the bull’s-eye.
• When the chart is first drawn, if the R chart is out of control, the control limits calculated for the –
X chart will not be valid. Find and eliminate the source of variation in the R chart, then start over to establish new control limits.
control chart 165
Process: _____________________________ Calculated by: _________________________
Data dates: __________________________ Calculation date: ______________________
Step 1. Calculate average X _
and range R (the difference between the highest and lowest values) for each subgroup. Record on chart.
Number of values in each subgroup = n = _______
Number of subgroups to be used = k = _______
Step 2. Look up control limit factors.
n A2 D3 D4
Step 3. Calculate averages (X __
Average of the averages = X __
= ΣX _
÷ k
= _______ ÷ _______ = _______
Sum of the ranges = ΣR = _______
Average of the ranges = R _
= ΣR ÷ k
= _______ ÷ _______ = _______
Step 4. Calculate control limits.
3sestimate for X _
chart = 3sX– = A2 × R
_
= _______ × _______ = _______
Upper control limit for X _
chart = UCLX– = X __
+ 3sX–
= _______ + _______ = _______
Lower control limit for X _
chart = LCLX– = X __
– 3sX–
= _______ – _______ = _______
Upper control limit for R chart = UCLR = D4 × R _
= _______ × _______ = _______
Lower control limit for R chart = LCLR = D3 × R _
= _______ × _______ = _______
Figure 5.26 X and R chart example worksheet.–
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