Control Charts for Moving Averages and R charts track the performance of processes that have long production runs or repeated services. Sometimes, there may be insufficient number of sample measurements to create a traditional and R chart. For example, only one sample may be taken from a process. Rather than plotting each individual reading, it may be more appropriate to use moving average and moving range charts to combine n number of individual values to create an average.

Control Charts for Moving Averages When a new individual reading is taken, the oldest value forming the previous average is discarded. The new reading is combined with the remaining values from the previous average to form a new average. This is quite common in continuous process chemical industry, where only one reading is possible at a time.

Control Charts for Moving Averages By combining individual values produced over time, moving averages smooth out short term variations and provide the trends in the data. For this reason, moving average charts are frequently used for seasonal products.

Control Charts for Moving Averages Interpretation: a point outside control limits interpretation is same as before - process is out of control runs above or below the central line or control limits interpretation is not the same as before - the successive points are not independent of one another

Example: Eighteen successive heats of a steel alloy are tested for RC hardness. The resulting data are shown below. Set up control limits for the moving-average and moving-range chart for a sample size of n=3. Heat Hardness Average Range Heat Hardness Average Range 1 0.806 10 0.809 2 0.814 11 0.808 3 0.810 12 0.810 4 0.820 13 0.812 5 0.819 14 0.810 6 0.815 15 0.809 7 0.817 16 0.807 8 0.810 17 0.807 9 0.811 18 0.800

Example: Eighteen successive heats of a steel alloy are tested for RC hardness. The resulting data are shown below. Set up control limits for the moving-average and moving-range chart for a sample size of n=3. Heat Hardness Average Range Heat Hardness Average Range 1 0.806 10 0.809 0.810 0.002 2 0.814 11 0.808 0.809 0.003 3 0.810 0.810 0.008 12 0.810 0.809 0.002 4 0.820 0.815 0.010 13 0.812 0.810 0.004 5 0.819 0.816 0.010 14 0.810 0.811 0.002 6 0.815 0.818 0.005 15 0.809 0.810 0.003 7 0.817 0.817 0.004 16 0.807 0.809 0.003 8 0.810 0.814 0.007 17 0.807 0.808 0.002 9 0.811 0.813 0.007 18 0.800 0.805 0.007

Exponentially Weighted Moving Average (EWMA) The EWMA values are obtained as follows: Control limits are set at Where

Chart with a Linear Trend As the tool or die wears a gradual change in the average is expected and considered to be normal the measurement gradually increases the R chart is likely to remain in control - the estimate of  may not change. The difference between upper and lower specifications limits is usually set substantially greater than 6 , to provide some margin of safety against the production of defective products

Chart with a Linear Trend Step 1: A trend line is obtained for the chart. A simplified formula is available if there are an odd number of subgroups subgroups are taken at a regular interval and the origin is assumed at the middle subgroup

Chart with a Linear Trend Step 2: For each subgroup a separate pair of control limits is obtained above and below the trend line (so, the control limits are sloping lines parallel to the trend line)

Chart with a Linear Trend Step 3: Estimate . For k = 3, 4 etc. the initial aimed-at mean value, is set k above the lower specification limit and the process is stopped for readjustment (a new setup is made, tool/die is changed) when the observed mean value reaches k below the upper specification limit.

Text Problem 10.25: A certain manufacturing process has exhibited a linear increasing trend. Sample averages and ranges for the past 15 subgroups, taken every 15 minute in subgroup of 5 items, are given in the following table. Fit the linear trend line to these data, and plot a trended control chart with 3-sigma limits.

Text Problem 10. 26: Specifications on the process in Problem 10 Text Problem 10.26: Specifications on the process in Problem 10.25 are 20030. The process may be stopped at any time and readjusted. If on readjustment the mean is to be set exactly 4 above the lower specification and the process is to be stopped for readjustment when the mean reaches a level exactly 4 below the upper specification: (a) Calculate the aimed-at starting and stopping values of (b) Estimate the duration of a run between adjustments

Reading and Exercises Chapter 10 (moving average and linear trend): pp. 382-391 (Sections 10.6-7) 10.24, 10.27