Control Charts Statistical Process Control The objective of

Control Charts

Statistical Process Control The objective of a process control system is to provide a statistical signal when assignable causes of variation are present

Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

Steps In Creating Control Charts 1. Take samples from the population and compute the appropriate sample statistic 2. Use the sample statistic to calculate control limits and draw the control chart 3. Plot sample results on the control chart and determine the state of the process (in or out of control) 4. Investigate possible assignable causes and take any indicated actions 5. Continue sampling from the process and reset the control limits when necessary

Control Charts for Variables § For variables that have continuous dimensions § Weight, speed, length, strength, etc. § x-charts are to control the central tendency of the process § R-charts are to control the dispersion of the process § These two charts must be used together

x-charts For x-Charts when we know s Upper control limit (UCL) = x + zsx Lower control limit (LCL) = x - zsx where x = mean of the sample means or a target value set for the process z = number of normal standard deviations sx = standard deviation of the sample means = s/ n s = population standard deviation

x-charts - Example The weights of boxes of Oat Flakes within a large production lot are sampled each hour. 12 different samples where selected and weighted and the average of each sample is presented in the following table. Each sample contains 9 boxes and the standard deviation of the population is 1. Managers want to set control limits that include 99. 73% of the sample mean. Hou r Sample average of 9 boxes Hour Sample average of 9 boxes 1 16. 1 7 15. 2 2 16. 8 8 16. 4 3 15. 5 9 16. 3 4 16. 5 10 14. 8 5 16. 5 11 14. 2 6 16. 4 12 17. 3

x-charts - Example For 99. 73% control limits, z = 3 UCLx = x + zsx = 16 + 3(1/3) = 17 ounces LCLx = x - zsx = 16 - 3(1/3) = 15 ounces

Patterns in Control Charts Upper control limit Target Lower control limit Normal behavior. Process is “in control. ”

Patterns in Control Charts Upper control limit Target Lower control limit One plot out above (or below). Investigate for cause. Process is “out of control. ”

Patterns in Control Charts Upper control limit Target Lower control limit Trends in either direction, 5 plots. Investigate for cause of progressive change.

Patterns in Control Charts Upper control limit Target Lower control limit Two plots very near lower (or upper) control. Investigate for cause.

Patterns in Control Charts Upper control limit Target Lower control limit Run of 5 above (or below) central line. Investigate for cause.

Patterns in Control Charts Upper control limit Target Lower control limit Erratic behavior. Investigate.

x-charts For x-Charts when we don’t know s Upper control limit (UCL) = x + A 2 R Lower control limit (LCL) = x - A 2 R where R = average range of the samples A 2 = control chart factor found in the Table in the next slide x = mean of the sample means

Control Chart Factors (3 sigma) Range Sample Size Mean Factor n 2 A 1. 880 2 D 3. 268 4 D 03 3 4 5 6 7 8 9 10 12 1. 023. 729. 577. 483. 419. 373. 337. 308. 266 2. 574 2. 282 2. 115 2. 004 1. 924 1. 864 1. 816 1. 777 1. 716 0 0 0. 076 0. 136 0. 184 0. 223 0. 284 Upper Range Lower

x-charts - Example Super Cola bottles soft drinks labeled “net weight 12 ounces”. Indeed, an overall process average of 12 ounces has been found by taking many samples, in which each sample contained 5 bottles. The average range of the process is 0. 25 ounces. We want to determine the upper and lower control limits for averages in this process.

x-charts - Example Process average x = 12 ounces Average range R =. 25 Sample size n = 5 UCLx = x + A 2 R = 12 + (. 577)(. 25) = 12 +. 144 = 12. 144 ounces UCL = 12. 144 LCLx = x - A 2 R = 12 -. 144 = 11. 857 ounces LCL = 11. 857 Mean = 12

R–Chart § Type of variables control chart § Shows sample ranges over time § Difference between smallest and largest values in sample § Monitors process variability § Independent from process mean

R–Chart For R-Charts Upper control limit (UCLR) = D 4 R Lower control limit (LCLR) = D 3 R where R = average range of the samples D 3 and D 4 = control chart factors from the previous Table - Control Chart Factors (3 sigma)

R–Chart - Example The average range of a product at the National Manufacturing Co. is 5. 3 pounds. With a sample size of 5, the owners want to determine the upper and lower control chart limits for the range UCLR = D 4 R = (2. 115)(5. 3) = 11. 2 pounds LCLR = D 3 R = (0)(5. 3) = 0 pounds UCL = 11. 2 Mean = 5. 3 LCL = 0

Mean and Range Charts (a) (Sampling mean is shifting upward but range is consistent) These sampling distributions result in the charts below UCL x-chart (x-chart detects shift in central tendency) LCL UCL R-chart LCL (R-chart does not detect change in mean)

Mean and Range Charts (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) UCL x-chart (x-chart does not detect the increase in dispersion) LCL UCL R-chart LCL (R-chart detects increase in dispersion)