More Control Charts

Module 6

Why?

There are many probability distributions in our world

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Two types of data

• Variables----Continuous

• Attributes—Discrete, Countable– Two types of attributes data

• You can count occurrences and non-occurrences.• You can only count occurences.

Examples?????

Some Variables Shewart Charts

• X-mR aka i-Chart, Individuals Chart

• X-bar-range

• X-bar-sigma

Some Attributes Shewart Charts

• p-Chart

• np-Chart

• u-Chart

• c-Chart

Decide on type of data

Continuous

(Variables)

Data

Discrete (Attributes)

Data

More than one observation per

subgroup?

< 10 observations

per subgroup?

Can both occurrences &

non-occurrences be

counted?

Are there equal area

of opportunity

?

Are the subgroup

sizes equal?

Yes

Yes

Yes

Yes

Yes

No

NoNo

No

No

–R –s XmR c-chart u-chart p-chart np-chart

Example Individuals Chart

Example X-Range Chart

Example X-Sigma Chart

How did they do that?

The basic pattern….

• Plot observed measurements over time.– Measurements, counts, rates

• Plot Centerline– Average measurement or count, pooled rate.

• Plot Control Limits– Centerline +/- Multiplier X “Standard Deviation”

Multiplier does 3 Things

• Determines the number of sigmas– usually 3

• Converts standard deviations to standard errors (variables data).

• Can include factor to adjust for unusually small or large number of subgroups or time intervals.

Note: How multiplier is constructed and used varies by author.

“Standard Deviation”

• Based on sample estimate of population standard deviation.

• Based on moving ranges.

• Based on ranges.

The i-Chart or XmR Chart

•Calculate average of all individual values = x•Calculate all the moving ranges (MRi)

•MRi = |xi-xi-1|•Calculate the average MR = Rbar•Calculate control limits = xbar +/- 2.66Rbar•Plot xbar•Plot control limits•Plot individual values, points

The Xbar-Range Chart

The Xbar-Sigma Chart

The Xbar-Sigma Chart (Part II)

Is “3” always OK?

• Notice 3 is multiplied by the SD.

• This gives +/- 3 Sigma Control Limits.

• Designed for 25 observations.

• When you have only 7 observations– β risk is too high

• When you have 200 observations– α risk is too high

• Can use T-Sigma Limits

T-Sigma Limits

No. of Plotted Points T

2 1.5

3-4 2.0

5-9 2.5

10-34 3.0

35-199 3.5

200-1500 4.0

How to use T-Sigma Limits

• Substitute the T-Sigma limit from the table for the “3” in A3, B3, and B4 above.

• For attributes charts, simply substitute the T-Sigma Limits for the multiplier in front of the standard error.

The attributes Shewart Charts

• p-Chart

• np-Chart

• u-Chart

• c-Chart

Example p-Chart

Example np-Chart

Example u-Chart

Example c-Chart

How did they do that?

The p-Chart

The np-Chart

Pooled overall subgroups

The c-Chart

The u-Chart

Choosing Charts1. Continuous

A. Only 1 observation per subgroup—use iChart B. More than 1 observation/subgroup

i) Less than 10 observations/subgroup—use Xbar-Rii) 10 or more observations/subgroup--use Xbar-Sigma

2. AttributesA. Occurrences (heads) and non-occurrences (tails) can

be counted.i) Subgroups of equal size—use np-ChartIi) Subgroups of unequal size—use p-Chart

B. Only occurrences can be counted.i) Equal area of opportunity (denominators)—use c-Chartii) Unequal area of opportunity– use u-Chart

Decide on type of data

Continuous(Variables)

Data

Discrete (Attributes)

Data

More than one observation per

subgroup?

< 10 observations

per subgroup?

Can both occurrences &

non-occurrences be counted?

Are there equal area of opportunity?

Are the subgroup

sizes equal?

Yes

Yes Yes

Yes

Yes

No

NoNo

No

No

–R –s XmR c-chart u-chart p-chart np-chart

See Flow Chart onpage 72 of Carey and Lloyd