04/21/23IENG 486 Statistical Quality &

Process Control 1

IENG 486 - Lecture 05

Interpreting Variation Using Distributions

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Assignment:

Reading: Chapter 1: (1.1, 1.3 – 1.4.5)

Cursory – get Fig. 1.12., p.34; Deming Management,1.4.4 Liability Chapter 2: (2.2 – 2.7)

Cursory – Define, Measure, Analyze, Improve, Control Chapter 3: (3.1, 3.3.1, 3.4.1)

HW 1: Chapter 3 Exercises: 1, 3, 4 – using exam calculator 10 (use Normal Plots spreadsheet from Materials page) 43, 46, 47 (use Exam Tables from Materials page – Normal Dist.)

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Distributions

Distributions quantify the probability of an eventEvents near the mean are most likely to occur, events

further away are less likely to be observed

35.0 2.5

37()

41.4(+2)

32.6(-2)

43.6(+3)

30.4(-3)

39.2 (+)

34.8 (-)

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Normal Distribution

Notation: r.v. This is read: “x is normally distributed with mean and

standard deviation .”

Standard Normal Distributionr.v.

(z represents a Standard Normal r.v.)

~ ,x N

~ 0, 1z N

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Simple Interpretation of Standard Deviation of Normal Distribution

( ) .6827P x

( 2 2 ) .9546P x

( 3 3 ) .9973P x

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Standard Normal Distribution

The Standard Normal Distribution has a mean () of 0 and a variance (2) of 1 (thus, standard deviation is also 1)

Total area under the curve, (z), from z = – to z = is exactly 1

The curve is symmetric about the meanHalf of the total area lays on either side, so:

(– z) = 1 – (z)

z

(z)

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Standard Normal Distribution

How likely is it that we would observe a data point more than 2.57 standard deviations beyond the mean?

Area under the curve from – to z = 2.57 is found by using the table on pp. 693-694, looking up the cumulative area for z = 2.57, and then subtracting the cumulative area from 1.

z

(z)

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Standard Normal Distribution

How likely is it that we would observe a data point more than 2.57 standard deviations beyond the mean?

Area under the curve from – to z = 2.57 is found by using the table on pp. 693-694, looking up the cumulative area for z = 2.57, and then subtracting the cumulative area from 1.

Answer: 1 – .99492 = .00508, or about 5 times in 1000

z

(z)

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What if the distribution isn’t a Standard Normal Distribution?

If it is from any Normal Distribution, we can express the difference from a sample mean to the population mean in units of the standard deviation, and this converts it to a Standard Normal Distribution.

Conversion formula is:

where:x is the sample location point, is the population mean, and is the population standard deviation.

x

z

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What if the distribution isn’t even a Normal Distribution?

The Central Limit Theorem allows us to take the sum of several means, regardless of their distribution, and approximate this sum using the Normal Distribution if the number of observations is large enough.

Most assemblies are the result of adding together components, so if we take the sum of the means for each component as an estimate for the entire assembly, we meet the CLT criteria.

If we take the mean of a sample from a distribution, we meet the CLT criteria (think of how the mean is computed).

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Example: Process Yield

Specifications are often set irrespective of process distribution, but if we understand our process we can estimate yield / defects.

Assume a specification calls for a value of 35.0 2.5.

Assume the process has a distribution that is Normally distributed, with a mean of 37.0 and a standard deviation of 2.20.

Estimate the proportion of the process output that will meet specifications.

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Continuous & Discrete Distributions

Continuous Probability of a range

of outcomes is the area under the PDF (integration)

Discrete Probability of a range of

outcomes is the area under the PDF (sum discrete outcomes)

35.0 2.5

37()

41.4(+2)

32.6(-2)

43.6(+3)

30.4(-3)

39.2 (+)

34.8 (-)

35.0 2.5

36 40324230 3834 ()

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Discrete Distribution Example

Sum of two six-sided dice: Outcomes range from 2 to 12.

Count the possible ways to obtain each individual sum - forms a histogram

What is the most frequently occurring sum that you could roll?Most likely outcome is a sum of 7 (there are 6 ways to

obtain it) What is the probability of obtaining the most likely sum in a

single roll of the dice?6 36 = .167

What is the probability of obtaining a sum greater than 2 and less than 11?

32 36 = .889

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How do we know what the distribution is when all we have is a sample?

Theory – “CLT applies to measurements taken consisting of many assemblies…”

Experience – “past use of a distribution has generated very good results…”

“Testing” – combination of the above … in this case, anyway! If we know the generating function for a distribution, we can

construct a grid (probability paper) that will allow us to observe a straight line when sufficient data from that distribution are plotted on the grid

Easiest grid to create is the Standard Normal Distribution … because it is an easy transformation to “standard“ parameters

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Normal Probability Plots

Take raw data and count observations (n) Set up a column of j values (1 to j) Compute (zj) for each j value

(zj) = (j - 0.5)/n

Get zj value for each (zj) in Standard Normal Table Find table entry((zj)), then read index value (zj)

Set up a column of sorted, observed data Sorted in increasing value

Plot zj values versus sorted data values Approximate with sketched line at 25% and 75% points

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Interpreting Normal Plots

Assess Equal-Variance and Normality assumptions Data from a Normal sample should tend to fall along the line, so

if a “fat pencil” covers almost all of the points, then a normality assumption is supported

The slope of the line reflects the variance of the sample, so equal slopes support the equal variance assumption

Theoretically: Sketched line should intercept the zj = 0 axis at the mean value

Practically: Close is good enough for comparing means Closer is better for comparing variances If the slopes differ much for two samples, use a test that

assumes the variances are not the same

Normal Probability Plots

Tools for constructing Normal Probability Plots: (Normal) Probability Paper

In-class handout

Normal Plots Template Materials Page on course website

Interpretation Fat Pencil Test

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