More Control Charts Module 6

Why? There are many probability distributions in our world

The Binomial Distribution, The distribution of coin tosses.

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 Continuo us (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? Ye s No –R–sXmRc-chart u-chartp-chartnp-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 = |x i -x i-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 PointsT

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 over all subgroups

The c-Chart

The u-Chart

Choosing Charts 1.Continuous A. Only 1 observation per subgroup—use iChart B. More than 1 observation/subgroup i) Less than 10 observations/subgroup—use Xbar-R ii) 10 or more observations/subgroup--use Xbar-Sigma 2.Attributes A. Occurrences (heads) and non-occurrences (tails) can be counted. i) Subgroups of equal size—use np-Chart Ii) Subgroups of unequal size—use p-Chart B. Only occurrences can be counted. i) Equal area of opportunity (denominators)—use c-Chart ii) 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 No –R–sXmRc-chart u-chartp-chartnp-chart See Flow Chart on page 72 of Carey and Lloyd