Variables control charts

𝑥 -chart Purpose: Watch for shifts in the mean Each point represents average of n observations Control limits based on distribution of sample mean Center line = µ Standard deviation = 𝜎 𝑥 = 𝜎 𝑛 Problem: µ, σ often unknown Note: Not necessarily sensitive to changes in variability

R-chart Purpose: Monitor variability in the process Each point represents the range of n observations Control limits based on sampling distribution of the range We need to explore this Notes: Often included with x-bar chart Range is simpler to calculate than standard deviation

Question What would you expect the sampling distribution of the range to look like?

Sampling distribution of the range Shape is often not symmetric Control limits will not be symmetric Center is not at the true value of the range How to define “true range” for normal? ∞? 6σ? Variability decreases as sample size increases So n will play an important role

What can happen? Change in mean Increase in the variability Range will not necessarily change Increase in the variability Mean will not necessarily change Range should increase * So we need to look at x-bar chart in conjunction with R chart to make sure process is stable

Relationship b/w range and std. deviation   Where 𝑅 = 𝑅 1 + 𝑅 2 +…+ 𝑅 𝑚 𝑚 = the average range Values for d2 are provided in Appendix VI Why do this? The range is easier to calculate than the standard deviation (especially in “real time,” as parts of coming off an assembly line), but we need to know σ for the distribution of the mean.

Recall: control limits for x-bar charts

What if µ and σ are not known?

What if µ and σ are not known?

What if µ and σ are not known? So 𝐴 2 = 3 𝑑 2 𝑛 . Values are provided in Appendix VI.

Estimated control limits for x-bar charts Center line = 𝑥 = 𝑥 1 + 𝑥 2 +…+ 𝑥 𝑚 𝑚 = historic process mean

Estimated control limits for R charts Center line = 𝑅 - Values for D3 and D4 are provided in Appendix VI

Example A small company supplies keys and locks to the automotive industry. For one of their products a “ward groove” is cut into the key. The depth of this groove is a key characteristic of the key.

Example Data represents 20 subgroups of measures from this process (note: m=20, n=5) Based on this data Verify the provided UCL and LCL for both charts

Example (x-bar chart)

Example (x-bar chart)

Example (x-bar chart)

Example (R chart)

Example (R chart)

Example (R chart)

s chart Purpose: Monitor variability in the process Alternative to the X-bar and R chart, preferred if sample size is moderately large (n > 12) or is variable across subgroups Easy if computer based Each point represents the sample standard deviation Control limits based on the sampling distribution of the standard deviation

Sampling distribution of standard deviation Shape is skewed for small sample sizes Control limits will not be symmetric Center is not at the true value of σ s2 is an unbiased estimator of σ2, but s is not an unbiased estimator of σ s estimates c4σ (c4 values provided in Appendix VI) Variability decreases as sample size increases So n will play an important role (affects c4)

Estimated control limits for s charts Center line = 𝑠 = 𝑠 1 + 𝑠 2 +…+ 𝑠 𝑚 𝑚 = average value of s - Values for B3 and B4 are provided in Table VI

Example-Verify UCL and LCL

Example (s chart)

Example (s chart)

Control chart for individuals (X Chart) Purpose: Plot individual measurements rather than means Each point represents an individual (i.e. average of one value) Control limits Centered around overall process mean Problem: Don’t have a range or standard deviation Notes: Often used when it is too difficult or costly to sample several measurements

Moving range Take the range of 2 consecutive measures: Necessary for calculating control limits for X-chart Can also create a Moving Range Chart to monitor process variance

Estimate of standard deviation Recall: For R charts Use same idea for this situation (since n=2) Where 𝑀𝑅 = average of the moving ranges over all subgroups of 2 consecutive observations.

Example A metal plating process makes use of a nickel plating bath. The concentration of electroless nickel is important in the process. Each day at the start of a shift the concentration (in ounces/gallon) in the bath is analyzed. Data for 25 measurements is given

Example MR2 = |4.33 - 4.30| = 0.03 MR3 = |4.30 - 4.31| = 0.01 MR4 = |4.31 - 4.11| = 0.20 MR5 = |4.11 - 4.26| = 0.15 … 𝑀𝑅 =0.11

Control limits for X-chart For X-bar chart use 3 standard deviations of the mean ( 𝜎 𝑥 ) For X chart we are working with a single value Use 3 standard deviations of the population (σ) Estimate with 3 𝜎 =3 𝑀𝑅 1.128 =2.66 𝑀𝑅

Control limits for X-chart Center line = 𝑥 = process average 𝑈𝐶𝐿= 𝑥 +2.66 𝑀𝑅 𝐿𝐶𝐿= 𝑥 −2.66 𝑀𝑅

Control limits for moving range chart Recall-for R chart: Since n = 2: D4 = 3.267 and D3 = 0 Thus: Center line = 𝑀𝑅 𝑈𝐶𝐿=3.267 𝑀𝑅 𝐿𝐶𝐿=0

Example: 𝑀𝑅 =0.11, 𝑥 =4.287 X chart: Moving range chart: Centerline: 4.287 UCL = 4.287 + (2.66)(0.11) = 4.5796 LCL = 4.287 - (2.66)(0.11) = 3.9944 Moving range chart: Centerline: 0.11 UCL = (3.267)(0.11) = 0.3594 LCL = 0

Example

Example

Use when Items not in subgroups Items are continuously measured Items come from the process on an individual basis Subgrouping would be very artificial  Items are continuously measured Automated systems inspect every item Have data on all items use it in the charting process  

Use when Time between items is large Processes that take long periods of time Product is produced in batches that are relatively homogenous Chemicals, foods, liquids, plastics Very little variability within batches

Cautions Sensitivity to non-normal Individuals chart assumes population is normally distributed Does not average any values If population is not normal control limits will not be appropriate. 

Cautions Out of control signal will be seen much more often than it should be Because average run length is less than for the x-bar chart Average run length = average number of points plotted before process deemed out of control: where p = chance process is out of control

Cautions Not good for detecting small process shifts (less than 1.5σ) Need other charts (e.g. EWMA, CUSUM) for this

Questions for you

Exponentially Weighted Moving Average (EWMA) Basic idea Take point and average it with previous value Then average the next point with the previous average Points contribute to a running average, but amount of contribution decreases over time

EWMA

EWMA  is a weighting constant Typically between 0.1 and 0.25 Larger values tend to discount older observations faster Process value or subgroup mean

Questions for you How to start zi calculations? Set z1 = x1 Calculate z1 per formula, using target value as z0

Control limits for EWMA chart Based on L and  (more on these later) Depends on the value of i More narrow near start of process

Control limits for EWMA chart If =1.0 and L=3 the chart becomes a typical x-bar or individuals control chart

Example Calculate the zi values of the nickel concentration data Use λ = 0.2 Process has a target value of 4.2 ounces/gallon and a historical standard deviation of 0.1 ounces/gallon

Example i Concentration 1 4.33 4.226 14 4.18 4.282 2 4.30 4.241 15   1 4.33 4.226 14 4.18 4.282 2 4.30 4.241 15 4.25 4.276 3 4.31 4.255 16 4.39 4.298 4 4.11 17 4.299 5 4.26 4.233 18 4.42 4.323 6 4.17 4.220 19 4.41 4.340 7 4.44 4.264 20 4.16 4.304 8 4.19 4.249 21 4.29 4.301 9 4.27 4.253 22 4.14 4.269 10 4.32 4.267 23 4.21 4.257 11 4.273 24 4.256 12 4.34 4.287 25 4.293 13 4.307

Example Find the UCL and LCL at i=4 if λ=0.2 and L=3 Also find the UCL and LCL at i=20

Example Find the UCL and LCL at i=4 if λ=0.2 and L=3

Example Also find the UCL and LCL at i=20

Example

Example Note: 𝑥 is 1σ higher than µ0; x-chart did not pick up this shift but the EWMA chart did.

Choice of Parameters Chosen based on ARL

Choice of Parameters Chosen based on ARL QC Manager specifies desired ARL & anticipated shift in mean

Choice of Parameters Chosen based on ARL QC Manager specifies desired ARL & anticipated shift in mean

Choice of Parameters Note: If the process does not shift the ARL is the same for all parameter choices Note: If the  is small chart does not react quickly to shifts in the opposite direction.

Notes: The points of the EWMA chart are not independent. Can not apply action rules EWMA charts are generally more robust to the normality assumption than standard individuals charts Better to use if data is skewed