Statistics Quality: Control - Statistical Process Control and Using Control Charts Processes Processing an application for admission to a university and deciding whether or not to admit the student. Reviewing an employee’s expense report for a business trip and issuing a reimbursement check. Hot forging to shape a billet of titanium that will become part of a medical implant for hip, knee, or shoulder replacement. How processes are like populations Think of a population containing all the outputs that would be produced by the process if it ran forever in its present state. The outputs produced today or this week are a sample from this population. Systematic approach to process improvement A systematic approach to process improvement is captured in the Plan-Do-Check-Act (PDCA). Plan the intended work. Then Do the implementation of the solution or change. Check to see if improvement efforts have been successful. Act by implementing the changes. Process improvement toolkit Statistical tools: Histogram – visualize the process in terms of location, variability, distribution. Pareto Chart – rank data about the frequency of process problems or causes. Scatterplot – investigate whether two variables are related, identify root cause of problems. Control Charts – monitor the process, alert us when the process has changed. Process improvement toolkit Non statistical tools: Flowchart Cause and effect diagram Process improvement toolkit : Flowchart Describing processes graphically: A flowchart is a picture of the stages of a process Process improvement toolkit: cause-and-effect diagram Describes processes graphically Organizes the logical relationships between the inputs and stages of a process and an output. Statistical process control Goal: make a process stable over time and then keep it stable unless planned changes are made. All processes have variation. Statistical stability means the pattern of variation remains stable, not there there is no variation in the variable measured. Statistical process control Statistical Control: A variable that continues to be described by the same distribution when observed over time is said to be in control. Control charts: Statistical tools that monitor a process and alert us when the process has been disturbed so that it is now out of control. This is a signal to find and correct the cause of the disturbance. The idea of statistical process control A process that is in control has only common cause variation. Common cause variation is the inherent variability of the system, due to many small causes that are always present. When the normal functioning of the process is disturbed by some unpredictable event, special cause variation is added to the common cause variation. We hope to be able to discover what lies behind special cause variation and eliminate that cause to restore the stable functioning of the process. Control charts Control charts distinguish between the common cause variation and the special cause variations. A control chart sounds an alarm when it sees too much variation. The point X indicates a data point for sample number 13 that is “out of control.” x charts for process monitoring Two-step procedure for applying control charts to a process: 1. Chart setup stage a) Collect data from the process and assess stability. b) Establish control by uncovering and removing special causes. c) Set up control charts to maintain control. x charts for process monitoring 2. Process monitoring a) Observe the process operating in control for some time. b) Understand usual process behavior. c) Have a long run of data from the process. d) Keep control charts to monitor the process because a special cause could occur at any time. x charts for process monitoring Process monitoring conditions: o Measure a quantitative variable x that has a Normal distribution. o Examples: diameter of a part, time to respond to a customer call, etc. o The process has been operating in control for a long period, so that we know the process mean µ and the process standard deviation σ that describe the distribution of x as long as the process remains in control. Law of Large Numbers If we keep taking larger and larger samples, the statistic X is guaranteed to get closer and closer to the parameter µ Law of Large Numbers: Draw observations at random from any population with finite mean µ. As the number of observations drawn increases, the mean X of the observed values gets closer and closer to the mean µ of the population It is reasonable to use X to estimate µ Central Limit Theorem If a population has ( an approximate) N(µ, σ) distribution, then the sample mean X of n independent observations has the Ν (µ, σ/√n) distribution 68-95-99.7 Rule 68% of the data will be between µ − σ and µ + σ 95% of the data will be between µ − 2σ and µ + 2σ 99.7% of the data will be between µ − 3σ and µ + 3σ 18 x charts for process monitoring 1. Take samples of size n from the process at regular intervals. 2. Plot the means x of these samples against the order in which the samples were taken. 3. We know that the sampling distribution of x under the process- monitoring conditions is Normal with a mean µ and a standard deviation σ / n . 4. Draw a solid center line on the chart at height µ. - Continued - The 99.7 part of the 68-95-99.7 rule for Normal distributions says that as long as the process remains in control, 99.7% of the values of x will fall within three standard deviations of the mean. Draw dashed control limits on the chart at these heights. The control limits mark off the range of variation in sample means that we expect when the process remains in control. x charts for process monitoring Sample mean that is out of control. σ µ + 3 n x µ σ µ − 3 n Stopping Rules One attempt to increase the power of control charts is by adding supplementary stopping rules based on runs. 2 of 3 consecutive points fall outside warning (2-sigma) limits, but within control (3-sigma) limits. 4 of 5 consecutive points fall beyond 1-sigma limits, but within control limits. 8 consecutive points fall on one side of the centerline. Assessing improvement efforts a) shows a case where the control chart demonstrates a successful attempt to decrease the time needed to obtain lab results. b) The control chart indicates no impact from the attempted process improvement. Types of control charts There are two main types of control charts: Variable Control Charts: x , range, standard deviation, and cusum charts. Attribute Control Charts: percent defective, number defective, and nonconformities charts. General procedure for control charts Three-sigma (3σ) control charts for any statistic Q: 1. Take samples from the process at regular intervals and plot the values of the statistic Q against the order in which the samples were taken. 2. Draw a center line on the chart at height µQ, the mean of the statistic when the process is in control. 3. Draw upper and lower control limits on the chart 3 standard deviations of Q (σQ) above and below the mean. 4. The chart produces an out-of-control signal when a plotted point lies outside the control limits. s charts for process monitoring x charts are easy to interpret if the process standard deviation remains fixed. Even the simplest description of a distribution should give both a measure on center and a measure of spread. We must monitor both the process center, using an x chart, and the process spread, using an s chart s charts for process monitoring – continued - Note that the standard deviation s does not have a Normal distribution. The sampling distribution of s is skewed to the right. Nonetheless, control charts for any statistics are based on the “plus or minus three standard deviations” idea, motivated by the 68-95-99.7% rule for Normal distributions. Control charts are intended to be practical tools that are easy to use. s charts for process monitoring – continued - 1. The mean of s is a constant times the process standard deviation σ. µσs = c4 This is the center line of an s chart. 2. The standard deviation of s is also a constant times the process standard deviation. σσs = c5 3. The values of the c4 and c5 depend upon the sample size. For large samples, c4 is close to 1. s charts for process monitoring The control limits for an s chart are UCL = B6σ LCL = B5σ The control chart constants c4, B5, and B6 depend on the sample size n. Control chart constants s charts for process monitoring UCL = B6σ µσ= s c4 LCL = B5σ In control Out of control Comparing x to s control charts Do both types of control charts show the same information? Here are two control charts for mesh tension Comparing x to s control charts Lack of control on an s chart is due to special causes that affect the observations within a sample differently. examples: non-uniform material, new and poorly trained operator, mixing results from several machines or several operators. Look at the s chart first. Lack of control on an x chart responds to s-type causes as well as to longer-range changes in the process, so it is important to eliminate the s-type causes first. examples of longer-range change: new raw material that differs from that used in the past or a gradual drift in the process level caused by wear in a cutting tool. Using control charts x and R charts: an R chart is based on the sample range for spread instead of the sample standard deviation.