Stats Exam 2

Long Term Sample

1) Consists of random and assignable causes 2) Collected across a broad inference space 3) Data from several lots, many shifts, many machines and operators

Short Term Sample

1) Free from assignable or special cause 2) Represents random causes only 3) Group of similar things 4) Collected across a narrow inference space 5) Data from one lot of material, on one shift, one part, one machine, one operator

Run Chart Rules

1. More than seven consecutive plots fall above or below the center line. 2. More than seven plots are in a run-up or run-down. Example: Applying Rule 1, there are two runs with length equal to 6 above the center line, and another run with length equal to 11 below the center line after the 45th observation. This shows that the patient is not able to control the fasting blood sugar. Applying Rule 2, the longest run, a run-down, has a length equal to 5, so there is no signal for trends.

Run Chart Procedure

1. Plotting the data chronologically 2. Counting the length of runs and number of runs 3. Deciding the condition of the process based on certain rules

Different Measures (Statistics) to Control Different Parameters

1. Process average (µ) to be controlled, sample average (xbar) plotted, name of chart is xbar-chart 2. Process variability (σ) to be controlled, sample range (R) or sample or standard deviation (S) plotted, name of chart is R-chart or S-chart 3. Process proportion defectives (p) to be controlled, sample proportion defectives (P) plotted, name of chart is P-chart 4. Process defects per unit (c) to be controlled, sample defects per unit (C) plotted, name of chart is C-chart

Choice of Control Limits Part 2

3-Sigma Control Limits for Normal Distribution Probability of type I error (α) is 0.27% Type I error means process is in control but we think it is out of control (false alarm) This could occur because 0.27% of data are suppose to fall out of 3-sigma control limits for a normally distributed data group This gives us the knowledge that, if an out-of-control point is detected, we will only falsely conclude that it is out of control three out of a thousand times. This minimizes our risk of over controlling the process.

Choice of Control Limits Part 3

3-Sigma Control Limits with α = 0.0027 For a production line running at an 8-hour shift and samples being taken once every hour, this error will occur only about three every 125 days - about six times per year assuming 250 days per year or once every two months. This minimizes our risk of over controlling the process.

Improving the sensitivity of the X chart: Use of warning limits

3-sigma control limit 2-sigma warning limit 1-sigma warning limit Look for values outside these limits Improving sensitivity: Use of Runs values will run-down or run-below

Control Charts Basics

A control chart contains a center line, upper control limit, lower control limit A point that plots within the control limits indicates the process is in control, no action is necessary A point that plots outside the control limits is evidence that the process is out of control, investigation and corrective action are required to find and eliminate assignable cause(s) There is a close connection between control charts and hypothesis testing

Measurement Control Charts

A measurement can go off-spec if either the mean changes or the standard deviation changes Need two control charts: 1. One to control process mean 2. Another to control process variability

Chance and Assignable Causes

A process is operating with only chance causes of variation present is said to be in statistical control A process that is operating in the presence of assignable causes (bad x's) is said to be out of control In general, a production process would generate quality data that follow the normal distribution with a target value of µ and a standard deviation σ representing variability of the process The production data deviates from the target value when a chance cause is present

The Binomial Distribution

A random variable X is said to have the binomial distribution with parameters n and p if its probability distribution is given by: p(x) = (n x) p^x (1-p)^(n-x) X represents the number of "successes" of n independent Bernoulli trials where p is the probability of success and (1 - p) is the probability of "failure" in one trial. E(p) = p Var(p) = (p(1-p))/n Based on trials with only two possible outcomes (a.k.a., Bernoulli trials) The trials are assumed to be independent (for example, sampling with replacement)

Rational Sub-grouping in Control Charts

A very important concept. The basis of subgrouping (means "sample") should be such, when an assignable cause is present, the subgrouping enables its discovery. Examples 1) The process changes over time (σ increases only between 9-10AM) 2) The process changes due to shifts (σ increases only when operator A comes to work) In short, your sampling method should be able to find out what has caused the process to change (time or shift?)

Specification Limits

Also known as tolerance limits. Are the limits of variability that a customer or a product designer has imposed. Given as upper specification limits (USL) and lower specification limits (LSL). Often set as the target value +/- 3 to 6σ (assuming σ is known)

Natural Tolerance Limits

Are tolerance limits a process is capable of meeting in its current condition (UNTL and LNTL) Are chosen as 3σ of the process and often added to the target value of the part (estimated by xbar being produced as xbar +/- 3s) In general, do not equal specification limits. Use s to estimate σ because it's time consuming to count population

Disadvantages of Attribute Charts

Attribute information indicates whether a certain quality characteristic is within specification limits. It does not state the degree to which specifications are met or not met Variable control charts can forewarn us when the process is about to go out of control (by monitoring the trend of the data points) Attribute control charts won't give warning signals until the process has already gone out of control Attribute control charts require larger sample sizes then variable charts to ensure adequate protection against a certain level of process changes Larger sample sizes can be problematic if the measurements are expensive to obtain or the testing is destructive

P bar

Average proportion defectives in about 25 samples, and n is the size of each sample When true Po is known then replace P bar with Po

Benefits from Use of Control Charts

Avoidance of defectives by producing products right the first time Reduction of waste and increase in throughput Satisfied customers and improved customer relationships Better knowledge of the processes and their capabilities Improved worker morale because of the satisfaction they get from fruitful efforts Improved image for the producer and better market share Improved profitability

Data Missing for Control Chart

Calculate different control limits for the different n for those points since A2, D3, and D4 are different.

Types of Control Charts

Charts may be used to estimate process parameters, which are used to determine capability Two general types of control charts: Variables - continuous scale of measurement, quality characteristic described by central tendency and a measure of variability Attributes - conforming/nonconforming, counts Control chart design encompasses selection of sample size, control limits, and sampling frequency

What are Control Limits based on?

Control limits are based on an estimate of the variance that exists within the subgroup. There is also variation that exists between subgroups over time.

Xbar and R Charts

Control limits for R-Chart UCL(R) = D4Rbar CL(R) = Rbar LCL(R) = D3Rbar Control limits for Xbar-Chart UCL(Xbar) = Xbarbar + A2Rbar CL(Xbar) = Xbarbar LCL(Xbar) = Xbarbar - A2Rbar Xbarbar and Rbar are averages of 25 sample averages and sample ranges respectively. A2, D3 and D4 are factors obtained from standard tables. (To establish control charts, we normally start with 25 samples)

The Control Charts

Dr. Walter Shewhart proposed a set of procedures in early 1920's called this. According to him, variability in a process parameter (or product characteristic) can be from any of the two sources: 1. "stable system of chance causes" (common causes) 2. "assignable causes" (special causes) - this is what we need to discover and remove The control chart is the means of differentiating between the two sources of variability

Type I Error

False Alarm, null hypothesis true but rejected, want this one

C&E Diagram

Fishbone diagram - combined with machines, materials, measurement, methods, manpower, environment to bond failure

How are control limits determined?

For X bar charts: UCL = µxbar + Lσxbar Center Line = µxbar LCL = µxbar - Lσxbar We know σxbar = σ/√n so if we choose L = 3 then UCL = µxbar + 3σ/√n Center Line = µxbar LCL = µxbar - 3σ/√n µxbar can be estimated by xbarbar and 3σ/√n can be estimated by A2Rbar

Poisson Distribution Example #1

For a particular textile manufacturer, the number of flaws in bolts of cloth is assumed to be Poisson distributed with a mean of 0.1 flaw per square meter. a) What is the probability that there are two flaws in 1 square meter of cloth? b) What is the probability that there are no flaws in 20 square meters of cloth? λ = 0.1 flaw per square meter P(X = 2) = (e^(-0.1)*0.1^2)/2! =0.0045 λ = 2 flaws per twenty square meters = 0.1 flaw per square meter P(X = 0) = (e^(-2)*2^0)/0! = 0.1353 Note: You must enter λ = 2 here because the unit is in 20 square meters.

How are Control Limits Determined?

For xbar charts UCL = µxbar + Lσxbar Center Line = µxbar LCL = µxbar - Lσxbar We know that σx = σ/√x so if we choose L = 3 then... UCL = µxbar + 3(σ/√n) Center Line = µxbar LCL = µxbar - 3(σ/√n) µxbar can be estimated by xbarbar and 3(σ/√n) can be estimated by A2Rbar

Poisson Distribution Rules

Given an interval of real numbers, assume events occur at random throughout the interval. If the interval can be partitioned into subintervals of small enough length such that: 1) the probability of more than one event in a subinterval is zero 2) the probability of one event in a subinterval is the same for all subintervals and proportional to the length of a subinterval 3) the event in each subinterval is independent of other subintervals the random experiment is called a Poisson process.

Additional Notes on Attribute Charts

If no historical info is available, attribute charts are generally used first. When problem areas are identified, attribute charts may be replaced by variable charts to collect more info.

Recalculated Limits

If points fall outside UCL or LCL, remove them and recalculate UCLs, CLs, and LCLs.

Recalculating Limits with "Remaining Data"

If the process was not in control, the assignable causes will be searched and eliminated. Then those values of Xbar and/or R that are outside the limits can be removed from the data, after eliminating the assignable cause(s) New limits can be recalculated for future use from the "remaining" data to save time and money While making these recalculations, start with the R-chart first because calculation of limits for the Xbar chart requires the value for a good Rbar.

Reconstruct a P chart

If values fall outside control limits, remove offending samples and calculate new limits using remaining subgroups

More about variation

If we haven't identified the variability in our process and then minimize the variation within each subgroup, our control limits will be too wide The result will be that our ability to make decisions with regard to controlling the process will be degraded. Therefore, we often use the R chart to check variation within group first

Poisson Distribution

Important to use consistent units λ = Average number of flaws equals 2 per inch = Average number of flaws equals 20 per ten inches = Average number of flaws equals 200 per one hundred inches

Control Charts Practice

Individual measurement xi ~ (10, 1) Samples (n=5) are taken hourly. What would be the control limits for X chart? σ/√n = 1/√5 = 0.447 CL = 10 UCL = 10 + 3(0.447) = 11.341 LCL = 10 - 3(0.447) = 8.695

What is a Process?

Insert: Input (material), controllable variables (Xi), environment (uncontrollable variables, Zi) Inside: Machinery, manpower, methods, measurements Output: Product

C Chart Limits

Let C = X (as in the previous slides), then λ = X bar = C bar CLc means our production line is expected to produce this many defects per product E(C) = c V(C) = c σ(C) = √c UCLc = c bar + 3√c bar CLc = c bar LCLc = c bar - 3√c bar

Shewhart Control Chart Model

Let w be a sample statistic that measures some quality characteristic of interest, and suppose that the mean of w is μw and the standard deviation of w is σw. Then the center line, the upper control limits and the lower control limits become UCL = μw + L σw Center Line = μw LCL = μw - L σw Where L is the "distance" of the control limit from the center line, in standard deviation units

Type II Error

Miss, null hypothesis is false and fail to reject

np Chart and 100P Chart

Most people prefer dealing with whole numbers rather than decimal fractions. The advantage in using the nP-chart or 100p-chart can easily be seen. The nP-chart avoids calculation of P for each sample. The 100P-chart gives people a good sense about the expected number of defective units in a sample containing 100 units and the control limits associate with such sample. (e.g., more than 11 units fall above the UCL out of 100 units would be considered out of control.)

C Chart Distributioon

Number of defects per unit of product is C (proportion non-conforming which is a number) and does not equal defective ratio or P (%) Follows Poisson Distribution

Preparing Instruments in Control Charts

Often lack of adequate instruments is cause for poor quality

Selecting the Variable for (Control) Charting

Only important variables should be tracked using the charts

CL(p bar)

P bar represents the defective ratio of our production line

P Chart notation

P is proportion defective in the population being controlled P is the statistic that represents the proportion defective in a sample Pi is the proportion defective in any one sample and p bar is the average of the pi's

Some Special Attribute Control Charts, P-Chart with Varying Sample Size

P-Chart with Varying Sample Size, based on normal p bar formula. UCLp becomes pi and n becomes ni, where ni is the size of the ith sample Another method is to use n-bar, an average n, in the formula and obtain constant control lines When doing this method, replace n with n bar

What is Process Control?

Parameter Selection - Designed Experiments Parameter Control - Control Charts SPC - A way to find bad Xi and correct them so that Y can be good

Example of Unusual Patterns in Control Charts

Pattern is very nonrandom in appearance 19 of 25 points plot below the center line, while only 6 plot above Following 4th point, 5 points in a row increase in magnitude, a run up There is also an unusually long run down beginning with 18th point Variability with cyclic pattern can exist

Reasons for Poor Process Capability

Poor process centering (LSL is higher than LNTL) Excess process variability (Defects outside LNTL and UNTL)

Some Special Attribute Control Charts, np Control Chart

Possible to base a control chart on the number of nonconforming rather than the fraction nonconforming. Called a number nonconforming (np) control chart. UCL = np + 3√np (1-p) CL = np LCL = np - 3√np (1-p) If a standard value for p is unavailable, then p bar can be used to estimate p. Many nonstatisically trained personnel find np chart easier to interpret than the usual fraction nonconforming control chart. To get CL, UCL, LCL of np chart, multiple CL, UCL, LCL of a p chart by n To get CL, UCL, LCL of 100p chart, multiple CL, UCL, LCL of a p chart by 100

Normal Distribution to Poisson Distribution

Probability involving Poisson random variable X can be approximated by using a standard normal distribution The approximation is good when λ > 5

Control vs. Capability

Process in control simply means process is consistent. Capability means the process is producing products within customer's specifications. A process in control does not automatically mean that the process is capable (example: low consistent GPA) A capability study is needed to verify if the process incontrol is also in-specification (or capable)

The S-Chart

S-chart is used instead of R-chart when the sample size has to be large, because the R-chart loses in efficiency when the sample size becomes large, n > 10. When shift in the mean is < 1.5σ, R-chart cannot detect such small shift effectively, so S chart has to be used Large sample size leads to higher sampling cost Sometimes we need large sample size because of the dictates of the circumstances or because of the extra sensitivity from larger sample size is desired Note that the control limits width of Xbar-S chart is smaller than those of Xbar-R chart, so a small shift in the process mean can be detected earlier by the Xbar-S chart

How do Control Charts work?

Sample averages are plotted on this control chart - not individual data points. Distribution of individual measurements x and xbar have same mean and different sigmas (σ and σx)

More P Chart Notes

Sample size determination Let p bar represent the process average nonconforming rate. Then based on the number of nonconforming items that you wish to be represented in a sample - say 5, on average - the bound is expressed as npbar >= 5 or n > 5/p bar E(D) = np V(D) = np(1-p) E(P) = p V(P) = √(p(1-p)/n) In this case P follows the normal distribution with σp = √p(1-p)/n Since true σp is unknown, σphat = √Pbar(1-Pbar)/n to estimate σp

Attribute Charts Sample Size Selection

Sample size should be large enough to allow nonconformities to be observed in the sample For example, if a process has a nonconformance rate of 2.5%, a sample size of 25 is not sufficient because average number of nonconforming items per sample is only 0.625. Thus, misleading inferences might be made, since no nonconforming items would be observed for many samples A sample size of 100 is sufficient for this case, as the average number of nonconforming items per sample would thus be 2.5

A Typical Control Chart

Samples are taken at regular intervals, suitable measure of quality is plotted. Process in control/not in control Assignable cause (special cause) The control limits represent limits of natural variability (inherent in the process) of the measure being plotted

Xbar and R-Charts Constructions

Samples or sub-groups of 4 or 5 observations are taken at regular time intervals Sample average, Xbar , and sample range, R, are computed for each sample from the sample observations At least 25 samples or subgroups are needed to calculate the limits (subgroup = sample)

Advantages of Attribute Charts

Save sampling time and cost: go/no go Attribute charts are often used at high levels to identify areas which have high proportions of nonconforming items. For example, which plant, which department, or which line? Attribute charts assist in going from the general to a more focused level Once the lowest problem level has been identified, a variable chart is then used to determine specific causes for an out-of-control situation

Short-Term and Long-Term Variations

Source of variations include: lot-to-lot, time-to-time, physical or positional, measurement error, stream-to-stream, and piece-to-piece variation Variation within a subgroup is called *short-term variation* Total variation is the sum of variation from all sources and is called *long-term variation*

Transmission Example

Specifications were given by car designer. It was +/- 3σ in US and +/- 6σ in Japan. Quality characteristic follows normal distribution with µ representing the target value and σ representing the variability of the process

Statistical Process Control (SPC)

Standard SPC is a collection of of very basic tools that help us understand systems and processes 7 Basic Quality Control Tools 1. Check sheet 2. Pareto charts and analysis 3. Cause and effect (fishbone) diagrams 4. Frequency histograms 5. Defect concentration diagram 6. Scatter diagrams 7. Control charts (most important tool, 6 sigma green belt tools)

Statistical Methods for Quality Control and Improvement

Statistical Process Control (SPC) - Control charts, plus other problem-solving tools Useful in monitoring processes, reducing variability through elimination of assignable causes On-line technique Designed Experiments (DOE) - Discovering the key factors that influence process performance Process optimization (Find best combinations of key factors such as temp, pressure) Off-line technique Acceptance Sampling - Determining the number of samples needed to be inspected, sample size and sampling frequency

C Chart

The C-chart is used where the quality is measured by counting the number of blemishes, defects or non-conformities on units of a product. Note: a product that contains a few defects might be acceptable by the customer

P Chart

The P-chart is an attribute control chart, used with count data arising from inspection by attributes. This is also known as fraction defective or fraction non-conforming chart. The P-chart will be typically used where there is large continuous production. The P-chart requires usually a large sample size, larger than 20. Take a sample at regular time interval, inspect the sample and count the defective units. Then the proportion defective in each sample (Pi ) is computed and plotted on the chart with control limits drawn on it The P-chart is base on a binomial distribution. P = D/n, D = number of defective parts, n = inspected parts in a sample Probability involving D can be approximated by using a standard normal distribution to save calculation time The approximation is good when n is large relative to p, in particular, when np > 5 and n(1-p) < 5

Poisson Distribution

The Poisson distribution emerges as the number of trials in a binomial experiment increase to infinity while the mean of the distribution remains constant

False alarm in Xbar chart:

The Type I error in control chart is called False alarm. When a control chart declares a process not-in-control when in fact it is in-control, it is a false alarm. The Shewhart charts with 3-sigma limits have a false alarm probability of 0.0027 in any one sample since X bar follows normal distribution. So 3/1000 samples could cause a false alarm.

How the Control Chart Works?

The distribution of product quality characteristics follows normal distribution with mean=μ and standard dev. = σ. Distribution of sample mean also follows normal distribution with the same mean but a smaller standard deviation. σx = σ/√x The idea here is to set up control limits so that if a sample mean is observed to fall out the control limit, we know the process is out of control.

Poisson Distribution Example #2

The number of telephone calls that arrive at an operator's station is often modeled as a Poisson random variable. Assume that on average there are 5 calls per hour. Draw the probability distribution. Suppose X has a Poisson distribution with a mean of 4. Determine the following probabilities: (a) P(X=0) = 0.0183 (b) P(X≤2) = 0.2381 (c) P(X=4) = 0.1954 (d) P(X=8) = 0.0298

Blood Glucose Level Example

The patient's target level is not close to xbar and therefore not in control

Purpose of a Control Chart

The purpose of a control chart is to estimate the inherent variability of the process (variation within a subgroup), and use it to determine how much variability we can allow in the process over time (variation between subgroups).

Poisson Distribution Explanation

The random variable X that equals the number of events in the interval is a Poisson random variable Probability Mass Function: f(x) = ((e^-λ*λ^x)/x!) where x = 0,1,2,... λ = mean number of events per unit of measure e = 2.71828 Mean: µ = E(x) = λ Variance: σ^2 = V(X) = λ Standard Deviation = σ = √σ² = √λ

The Run Chart

The run chart can be used with both measurements and attributes. In this chart, there are no limit lines. The condition of the process, whether it is in control or not, is determined by merely counting the length of runs and the number of runs A run is a sequence of plots having a common characteristic like all plots in the sequence falling above/below the center line

More Control Charts

They can be used both to watch sporadic deviation as well as to solve chronic deviations of processes. Event logs are used and related to signals on charts.

More Run Chart Stuff

This chart can be used with single observations (X values) or averages (X bar values), ranges (R values), standard deviations (S values), proportion defectives in a sample (p values), etc The main advantage of the run-chart is its simplicity and universal application This chart is widely used in healthcare industry

Trial control limits

This is the name for the limits calculated from the first batch of data.

Probability Limits

Type I error probability can be chosen by the user directly. For example, type I error = 0.001 (0.1%) on each side of a normal distribution curve gives 3.09-sigma control limits (typically used in Europe). As a result, total type I error = 0.002. (Note: Is US, type I error=0.0027)

Warning Limits

Typically selected as 2-sigma limits Note: nearly 5% of your data will fall out 2- sigma limits.

Frequency of Sampling in Control Charts

Use more frequent sampling in beginning and taper off when better control is reached

Sensitizing Rules for Control Charts

Western Electric Rule (Top 4) *A single point beyond Zone A (outside a control limit) *Two out of three points in a row in Zone A (μ+ 3σ) or beyond; the odd point may be anywhere. *Four out of five points in a row in Zone B (μ+ 2σ) or beyond; the odd point may be anywhere. *Eight points or more in a row all in Zone C (μ+ 1σ) , either above or below the centerline. Eight point in a row on both sides of the center line but none falling in Zone C. Systematic or cyclic patterns. A long series of points all going in the same direction. Even more rules in the Statistical Quality Control Handbook. (AT&T, 1958

Choice of Control Limits

Why W. Shewhart chose µ+/-3σx as control limits for the X bar chart? UCLxbar = µ + 3σxbar = µ + 3σ/√n CL = µ LCLxbar = µ - 3σxbar = µ - 3σ/√n Why not 1, 2, or 4 sigma? Because this would give an acceptable Type 1 Error.

Unusual Patterns in Control Charts

Your process could be out of control even all data points fall within the control limits.

Patterns in Control Charts

a. Natural pattern: The pattern expected from process in control b. Sudden change in level c. Gradual change in level d. Grouping and bunching: process goes out of control temporarily e. Instability: many causes in action: possibly overcontrol by operator f. Interaction: the plot in one period is related to plots in the previous period g. Mixtures: sampling done from two different populations mixed together h. Cycles: affected by cyclic behavior of some process parameters i. Stratification: one population neutralizing other j. Trend: a continuous change occurring in a process parameter

Many Uses of (Control) Charts

a. To control a process at a given target or nominal value b. To maintain a process at its current level c. As a trouble shooting tool d. As an acceptance tool

C Chart Notation

c is the parameter, the number of defects per unit in the population, which is to be controlled C is the statistic representing the number of defects in any sample unit ci is an observed value of the statistic C in any one sample unit, and c bar is the average of the observed values

Determining Sample Size in Control Charts

n = 4 or 5 is the norm (because calculators weren't available when created)

Variation Equation

σ²Total = σ²Between + σ²Within Total Variation = Between Subgroup (monitored by X bar chart) + Within Subgroup (monitored by R chart) Long-term capability = Accuracy + Precision (Short-term capability) Control limits are based on an estimate of the variance that exists within the subgroup. There is also variation that exists between subgroups over time. The purpose of a control chart is to estimate the inherent variability of the process (variation within a subgroup), and use it to determine how much variability we can allow in the process over time (variation between subgroups).