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• Statistical Process Control (SPC) applies statistical techniques to ensure that processes meet standards.
• SPC provides a statistical signal when assignable (special) causes of variation are present.
• SPC eliminates assignable causes of variation.
• Variability exists in every process.
• Natural or common causes refers to the inherent variability in a process.
• Special or assignable causes refers to unusual or unexpected variations in a process.

Variations Explained

• Natural variations affect all processes and are considered common causes.
• Expected variations are when output measures follow a probability distribution within acceptable limits, the process is "in control".
• Assignable causes indicate changes in a process that can be traced to a specific reason.
• Misadjusted equipment, machine wear, fatigue, untrained workers, and new raw materials are examples of specific reasons.
• Bad causes are eliminated, and new causes are incorporated.

Key Definitions

• Population: The entire group or set of items being studied.
• Sample: A subset of the population selected for study.
• Sampling: Selecting a subset of data points from a larger population for analysis.
• Sample Size: The number of observations included in a sample.
• Distribution: A systematic way to describe the likelihood of different outcomes in a random process.
• Normal Distribution: A symmetric, bell-shaped probability distribution characterized by mean and standard deviation.
• Sampling Distribution: The distribution of a statistic (e.g., mean) calculated from multiple samples of the same size.
• Dispersion: The extent to which a dataset is spread out from its central tendency.
• Central Tendency: Center of a distribution – commonly measured by mean, median, or mode.
• Mean: The average of a set of values.
• Standard Deviation: A measure of the typical deviation of values from the mean.
• Variation: Natural and special differences in a process.
• Control Limits: Upper and lower bounds on a control chart define the acceptable range of variation.
• Centerline: Represents the average/mean of the process; a baseline for accessing the stability.
• Out of Control: Points on a control chart fall outside the control limits due to special cause variation. Corrective action may be needed.
• In Control: All points on a control chart fall within the control limits. The process is stable and operating as expected.

Samples and Distributions

• Taking and analyzing samples is how you measure the process.
• Samples will vary from each other in weight.
• After taking enough samples from a stable process, they form a pattern called the distribution.
• Distributions can differ in terms of central tendency (mean), standard deviation/variance, and shape.
• The output of a process with only natural causes of variation forms a distribution that is stable over time and predictable.
• With assignable causes present, process output is not stable over time or predictable.

Control Charts

• Constructed from historical data.
• These distinguish between natural variations and variations due to assignable causes.

Process control

• A process in statistical control is capable of producing within control limits.
• A process can be in statistical control yet not capable of producing within control limits.
• A process that is "out of control".

Central Limit Theorem

• Regardless of the population’s distribution, the distribution of sample means drawn from the population will tend to follow a normal curve.
• Mean of the sampling distribution will be the same as the population mean (μ).
• The standard deviation of the sampling distribution (σ_x) will equal the population standard deviation (σ) divided by the square root of the sample size, n.
• Distributions of sample means always has a normal distribution
• 95.45 represents the fall within in +/-2ox
• 99.73 represents the fall within in +/-3ox
• As the sample size increases, the sampling distribution narrows.

Types of control charts

• Continuous variables: x-chart which measures changes in mean & R-chart which measures changes in dispersion
• Categorical variables: p-chart measures, fraction, proportion or percentage defects & c-chart which measures count defects per unit output

Control Charts for Variables

• Continuous random variables with real values appear in whole or fractional numbers.
• The x̄-chart tracks changes in central tendency (mean), due to tool wear, gradual temperature increase or new materials.
• The R-chart indicates a gain/loss of dispersion, due to changes and loose tools or operators.

Setting Chart Limits (when σ is known)

• Lower Control Limit (LCL) = x̄ - zσ̄x
• Upper Control Limit (UCL) = x̄ + zσ̄x
• x̄ = mean of the sample means/target value
• z = number of normal standard deviations
• σ̄x = standard deviation of the sample means = σ/√n
• σ = population (process) standard deviation
• n = sample size

Example Chart Limits

• To set control limits at 99.73% of the sample means(z=3), select and weigh (n=9) boxes of cereal at each hour.
• Population standard deviation known at 1 oz when s = 1
• Steps: find average weight in the first sample, average mean of 12 samples, calculate upper and lower control limits.
• Using an chart can identify Out of control samples.

Setting Chart Limits (when σ is unknown)

• Lower control limit (LCL)= ㄡ - A₂R
• Upper control limit (UCL)= x + A2R
• x = mean of the sample means
• A2 = control chart factor found in Table S6.1
• A2 = average range of the samples

Control Chart Facors (Table S6.1) Table values for factors for computing Control Chart Limits (3 sigma)/

• Sample size.
• Mean Factors.
• Upper Range.
• Lower Range.

Setting Control Limits using Table value example

• Given Process Average = 12 Ounces
• Average Range = .25 Ounces
• Sample Size 5
• We have a UCL of 12.44, UCL of 12 and LCL of 11.856

The R-Chart

• Shows sample ranges over time/
• Finds the Difference between smallest and largest values in the sample/
• Monitors process variability'/
• is Independent from the process mean.

Setting Chart Limits for R-charts

• Lower control limit (LCL_R) = D3R
• Upper control limit (UCL_R) = D4R
• UCL_R = upper control limit for the range
• LCL_R = lower control limit for the range
• D3 and D4 = values from Table S6.1

Mean and Range Charts

• The mean chart is sensitive to shifts in the process mean.
• The R-chart is sensitive to shifts in the process standard deviation.
• The mean chart detects shift in central tendency where as the R-Chart does not detect change in mean.
• The mean Chart indicates no change in central tendency and the R-Chart detects increase in dispersion

Steps for Creating Control Charts

• Step 1: Collect samples
• Step 2: Compute overall means.
• Step 3: Set appropriate control limits
• Step 4: Calculate UCL and LCL
• Step 5: Graph x and R charts
• Step 6: Investigate patterns.
• Step 7: Identify and address assignable causes
• Step 8: Revalidate with new data

Control charts for variables

• Catergorical variables are defective / non defecitve good or bad, yes or no,
• Measurment is typically counting defects
• Control charts can measure percent defective/ and number of defects
• P-Chart requieres a sample size
• C-Chart does not require a sample size

Comtroll Limits for p-charts

• Using the central limit theirm to compute the normal distributionfor the samples.

Calculating P-Charts using data entry

• Need numbers of errors, fraction defective totals.

Control Limits for C-Charts

• The population will be a poisson distribution.
• Based no a central limet theirim we use the average defects and the standard deviation of the average defects LCL= c= 3/c UCL= c + 3Vc

Managerial Issues and control Charts.

• Select points in the process in need of SPC, determine the use of which charting technique, and setting clear polices.
• Variable charts monitos weights and dimensions, when attribute charts are more of a yes, go or no- go system.

Patterns in Control Charts

• By examining the patterns in a control chart, abnormalities in a prcess can be identeified.
• A run test is used to examin nonradom variation.

Acceptance Sampling

• This is a form of qaulity used for incoming materials.
• Sampling is done to take a random batch items.
• Inspections are used to inspect each item in the sanple, the inspection will determine accept or rejet in the sample

Acceptance Sample Cons

• Only screens lots and does not drive quality.
• Rejections can lead to returning ti the suppier
• May need to re-grade a lower specification

Operating charteristics (OC)

• The poor level is quality we want to accept is caleld AQL
• The good lecel is called Lot torlerenece percent defecitv LTPD

Types of Risks in OC Curves

• producers can have a risk of good lots being rejected
• Consumers have the rish of accepted bag lots.