lecture4 control charts

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STATISTICAL QUALITY CONTROL

AND IMPROVEMENT

Lecture 4 – Control Charts

Shashikant Sathaye

Polytechnic Institute of NYU



Variable Control ChartsHow to Use the Process Control Charts

• A process is said to be in control when the performance of the process

falls within the statistically calculated control limits and exhibits only

random variation due to chance causes

• Process under control does not mean that the process meets designspecifications

• Identifying Patterns:

– Trends are changes in levels; for e.g the center line might be driftingdownwards

– Change or jump in level can be detected if a consecutive set of observationsare consistently above(below) the center line.

Statistical Quality Control and

ImprovementPolytechnic Institute of NYU Lecture 4 – MN611 2



Variable Control ChartsExample: Printer Assembly –

• An assembly area has been experiencing serious delays in the

manufacture of computer printers. As a quality assurance manager you

are asked to investigate and determine the cause of delays and fix theproblems. Given the limited time, you get together with the

representatives from all the departments and develop an cause andeffect diagram

• Set the qualitycharacteristicto measured –

length of a shaft





Variable Control ChartsExample: Printer Assembly –

• Choose a rational sub-group to sample –

needs to be homogeneous

– Sub-group should be at least 4.

– Normally use range R for estimating std dev.If sub-group size is more than 10, then use‘s’.

• Collect the data

• Determine the trial central line for the X

chart.

X =

where X = average of the subgroup

averagesXi = average of the ith groupm = number of subgroups



ΣXi

m



Variable Control Charts• Determine the Trial Control Limits for the Xbar Chart

UCLx = X + 3 σx

LCLx = X - 3 σx

Since the std deviation, σ is not known and we have a small sample sizefor each subgroup (4), we use the range R as a proxy for the std deviationand a factor A2 as the multiplier for the 99% confidence limit. A2 dependson the sample size.

Thus

UCLx = X + A2RLCLx = X – A2R

So for our example: X = (11.99+12.00,…+12.00)/21= 11.99

and R = (.08+.07_.04+….+.04)/21 = 0.05

UCLx = X + A2R = 11.99 + .577(.05) = 12.02LCLx = X – A2R = 11.99 - .577(.05) = 11.96





Variable Control Charts• Determine the Trial Control Limits for the R Chart

R = = .05

The upper and lower control limits are determined using another set ofmultiplier D3 and D4; for a sample of n=5, D3 and D4 are 0 and 2.114.

Thus

UCLR = D4R = 2.114(.05) = 0.11LCLR = D3R = 0(.05) = 0



ΣRi

m



Variable Control Charts



11.88 11.90 11.92 11.94 11.96 11.98 12.00 12.02 12.04

‐

0.02

0.04

0.06

0.08

0.10

0.12

LCLr

UCLr

CL

X

bar

CH

ART

R

C

HART

UCLr

LCLr

CLine



Variable Control Charts• Examine the process: Control Chart Interpretation

– Avoid misinterpretations

Blaming people for problems they cannot control

Spending time and money on process adjustments that are not necessary Asking for workers to improve when it the equipment that needs fixing

– State of Process Control – Process is under control when the performancefalls within the statistically calculated control and exhibits only chance orrandom causes.

– +/-3σ limits 99.73% of the measurements will be within the limits. – Based upon the normal curve



Region A = 68.3%Region B = 27.2%

Region C = 4.2%



Variable Control Charts• Examine the process: Control Chart Interpretation

Thus the basic guidelines for a process under control are: – Two thirds of the points are close to center

– A few points on or near the center line

– Points equally spread around the center

– No points outside the control limits

– No patterns or trends on the chart

• The range chart reveals the amount of variation in the process; so lookfor amount of variation in addition to the rules listed above.

• Types of patterns (non random variations)

– Trends (upward or downward)

– Change or shift in level

– Runs – Recurring cycles

– Non homogenous data





Variable Control Charts• Revising the charts

– R chart should be under control (it appear so)

– Isolate the cause of out of control and fix the problem



11.88 11.90 11.92 11.94 11.96 11.98 12.00 12.02 12.04

‐

0.02

0.04

0.06

0.08

0.10

0.12

LCLr

UCLr

CL

Xba

r

CHART

R

CHART

UCLr

LCLr

CLine



Variable Control Charts• Revise the charts

– Eliminate the out of control points from both the charts and re-calculate thecenter line and control limits

Eliminate the three points –

Xnew = = (251.77-11.94-11.95-11.95)/(21-3) = 12 = Xo

and

Rnew = = (1.06-.04-.05-.6)(21-3) = .05

New Control Limits

σo = Rnew /d2 = .05/2.326 = 0.02

UCLx = Xo + A σo = 12 + 1.342(.02) = 12.03

LCLx = Xo - A σo = 12 - 1.342(.02) = 11.97



ΣXi - Xd

m - md

ΣRi - Rd

m - md





ImprovementPolytechnic Institute of NYU Lecture 4 – MN6113 - 12

Revising the control charts

11.88

11.90

11.92

11.94

11.96

11.98

12.00

12.02

12.04

UCLo

LCLo

Center Lineo

LCLx

UCLx




• X and s charts

– s chart is more precise than R charts but require larger sample subgroup size (> 10)

– The control limits are calculated in a similar manner as X and R charts except that the

control limits are based upon s.

s = = (.03+.029+…..+.011)/21 = 0.414/21 = .02

UCLx = X +A3s

= 11.99

+ 1.427(.02)

= 12.02

LCLx = X ‐A3s = 11.99 – 1.427(.02) = 11.96

UCLs = B4s = 2.089(.02) = .04

LCLs = B3s = 0(.02) = 0


ImprovementPolytechnic Institute of NYU Lecture 4 – MN6113 - 13

Σsi

m



Variable Control Charts (Xbar and s charts)



11.88 11.90 11.92 11.94 11.96 11.98 12.00 12.02 12.04

X

bar

CH

ART

S

C

HART

UCLr

LCLr

CLine

‐

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0.04

0.05

LCLs

UCLs

CL

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