ece-580-doe : statistical process control and design of...

1/18/03 ECE-580-DOE Terms Steve Brainerd

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ECE-580-DOE : Statistical Process Control and Design of ExperimentsSteve Brainerd

Distributions: Frequency Distributions and Histograms

• Purpose: To Graphically and quantitatively represent data such that it may be evaluated descriptively for the data’s location, spread (dispersion), and to used to compare to any other data set.

• Note: A normal distribution does not always ensure a process is running in control! But it is a prerequisite to setup control charts and to run most DOEs!

• Amazing Website:

http://www.xycoon.com/toc.htm


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Distributions: Frequency Distributions and Histograms

• Central Limit Theorem: For large sample sizes, the sampling of the mean can be approximated very closely with a normal distribution, no matter what the real distribution of that population.


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Distributions Book: section 2-3 page 26website reference: http://mathworld.wolfram.com/BinomialDistribution.html

• Statistical Distribution: The distribution of a variable is a description of the relative numbers of times (frequency) each possible outcome will occur in a number of trials. The function describing the frequency distribution is called the probability density function (pdf), and the function describing the cumulative probability that a given value or any value smaller than it will occur is called the distribution function.

• Probability Function EXAMPLE:


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OHSU OGI Class Distributions


• 27 Distributions: We typically only deal with 3-9!!


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• Statistical Distribution: Types: Continuous

• A statistical distribution which the variables may take on a continuous

range of values. There are over 61 continuous distributions!

• We will use only: Normal Distribution, Student's t-Distribution, and F-Distribution

• Will briefly mention : Chi-Squared Distribution and Weibull Distribution

• Practical applications: Descriptive Data analysis, Population comparisons, SPC, and Design of experiments

• EXCEL: NORMDIST(x,µ,σ,TRUE or FALSE);

• FDIST(x,df1,df2)

• TDIST(x,df,1 or 2 tail)


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• Statistical Distribution: Types: Discrete• A statistical distribution whose variables can take on only discrete

values. Used for data that is always positive like defect counts! There are 18 discrete type distributions! Each has a slightly different shape, property and application.

• They include: Bernoulli Distribution, Binomial Distribution, Continuous Distribution, Geometric Distribution, HypergeometricDistribution, Negative Binomial Distribution, Poisson Distribution,

• Practical applications: Defect analysis, gambling, sampling plans, and radiation counts

• EXCEL: BINOMDIST(x, n, p, 0); HYPGEOMDIST(x,n,M,N); POISSION(x,λ,TRUE or FALSE)


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OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd

Normal Distribution• Normal

• Application: A basic distribution of statistics. Many applications arise from central limit theorem (average of values of n observations approaches normal distribution, irrespective of form of original distribution under quite general conditions). Consequently, appropriate model for many, but not all, physical phenomena.

• Example: Distribution of physical measurements on living organisms, intelligence test scores, product dimensions, average temperatures, and so on.

• Comments: Many methods of statistical analysis presume normal distribution.

Need to test for normalcy!


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Basic Statistics Normal Distribution


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Basic Statistics Normal Distribution

This area = probability is the KEY IDEA behind all the various distributions and statistical tests we’ll discuss


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Basic Statistics

• Standard deviation: Area under the curve. Developed by Gauss. Hence the Gaussian curve or distribution.


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Basic Statistics• Standard deviation: Area under the curve. Developed by Gauss. Hence

the Gaussian curve or distribution.

• So famous and important, it’s on the Deutsche Zehn Mark = German 10 Mark bill


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Basic Statistics• Normal Curve: Standard deviation: Area under the curve. Developed by

Gauss. Hence the Gaussian curve or distribution.


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ECE-580-DOE : Statistical Process Control and Design of Experiments

Normal Distribution• EXCEL function:• NORMDIST(x,µ,σ,TRUE or FALSE)• NORMDIST(x,mean,standard_dev,cumulative)• x: X is the value for which you want the distribution.

µ: Mean is the arithmetic mean of the distribution.σ: Standard_dev is the standard deviation of the distribution.

• Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, NORMDIST returns the cumulative distribution function; if FALSE, it returns the probability mass function.

• Examples• Returns the normal cumulative distribution for the specified mean and standard

deviation. This function has a very wide range of applications in statistics, including hypothesis testing.

• NORMDIST(42,40,1.5,TRUE) equals 0.908789• NORMDIST(42,40,1.5,FALSE) equals 0.11


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Basic Statistics

• Normal Curve in standard form: Answer each of the following questions

• What percent of the normal distribution lies between one and two standard deviations above the mean?

• What percent of the normal distribution lies above three standard deviations above the mean?

• If there were 100,000 persons arrayed in a normal distribution of heights, how many would be expected to lie more than three standard deviations above the mean?

• Note: Z requires population µ and σ to be known!! Which is typically not the case


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Basic Statistics Z-Score

• Standard deviation: Z SCORE: Normalize the curve: mean = 0.0. Easier to work with. Z is called a transformed statistic. Also called Standardized normal deviate or unit normal deviate and Z-Score: Now can use to compare populations in terms of “standard deviation units”. These of course have probabilities associated with them! I.e. A is different than B by 5 standard deviations!

• KEY IDEA: We will use this quite a bit in the future to compare populations and to make inferences.


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Basic Statistics

• Standard deviation: Area under the curve Z=Score: Normalized statistic.

• Note the use of Z is for comparisons of the sample data mean to the true populations mean and standard deviation.


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OHSU OGI Class


• Normal Curve in standard form and density functions for various sigma values :


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Basic Statistics Normal Distribution Example


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So what does this say? Probability of getting a 159.4 or greater value due to random chance is 11.9%

Or about 12% of the time you would normally expect a value of 159.4 or higher!


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Basic Statistics Normal Distribution Example confidence intervals

Where did this 1.96 come from? It is the +/- Z scores representing the upper and lower limits containing 95% of the sample population data! Can calculate in Excel as:

Z NORMSDIST(Z) 1 -NORMSDIST(Z)

1.96 0.975 0.025

Z 1-2*(1-((NORMSDIST(Z)))) 2*(1-((NORMSDIST(Z))))

1.96 0.950 0.050

P NORMSINV(P)0.01 -2.33

0.025 -1.96

In the days before computers we had to look up in tables!


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Basic Statistics Normal Distribution Example confidence intervals

For our example here:

µ = 152.1 σ = 19.64

z for 95% confidence interval = 1.96

Thus: we are 95% confident that the true mean is contained in the interval

= (152.1 – 1.96*19.64) < µ < (152.1 – 1.96*19.64) =

113.6 < µ <190.6

Value

mean 152.1

std 19.64

LOWER UPPER

z 95% 1.96 113.61 190.59

z 99% 2.326 106.42 197.78


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Basic Statistics Normal Distribution Example 2

• Example 2: Excel NORMDIST(x,µ,σ,0)

Normal Distributions for data example: Excel

Generated

0.000%

5.000%

10.000%

15.000%

20.000%

25.000%

30.000%

35.000%

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20

data

Prob

abili

ty %

Normal P(X) (Avg =0 ;SD =5)Normal P(X) (Avg =0 ;SD =2.5)Normal P(X) (Avg =0 ;SD =1.2)


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Student's t-Distribution

•t distributions for mean when σ unknown• The t distributions were discovered in 1908 by WilliamGosset who was a chemist and a statistician employed by the Guinness brewing company.• William Sealey Gosset (1876-1937) needed a distribution that could be used with small samples. Since the Irish brewery did not allow publication of research results, he published under the pseudonym of Student. We know that large samples approach a normal distribution. What Gossetshowed was that small samples taken from an essentially normal population have a distribution characterized by the sample size.


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•t distributions for mean when σ unknown• When we speak of a specific t distribution, we have to specify the degrees of freedom. The t density curves are symmetric and bell-shaped like the normal distribution and have their peak at 0. However, the spread is more than that of the standard normal distribution. The larger the degrees of freedom, the closer the t-density is to the normal density. •Normal and t Distributions–Theoretically, the normal distribution and the t-distribution are identical only for the infinite number of the degrees of freedom.


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Student's t-Distribution •Properties of the Student t distribution.

•The Student t distribution is different for different sample sizes. •. As the sample size increases, the distribution approaches a normal distribution. For n > 30, the differences are negligible. The mean is zero (much like the standard normal distribution). •The variance is greater than one, but approaches one from above as the sample size increases (=1 for the standard normal distribution). •The distribution is symmetrical about the mean. •The population standard deviation is unknown.


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• A term called Degrees of freedom:• Degrees of freedom is a fairly technical term

which permeates all of inferential statistics. In simple cases, it is n-1. n = sample size

• In general, the degrees of freedom is the number of values that can vary after certain restrictions have been imposed on all values.


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• Degrees of freedom:• Where does the term degrees of freedom come from? Suppose, for

example, that you have a phone bill from Ameritech that says your household owes $100. Your mother and father state that $70 of it is theirs and that your younger sibling owes only $5. How much doesthat leave you? Here, n=3 (parents, sibling, you), but once you have the total (or mean) and two more pieces of information, the last data element is constrained.

• The same is true with the degrees of freedom, you can arbitrarily use any n-1 data points, but the last one will be determined for a given mean. Another example is with 10 tests that averaged 55, if you assign nine people random grades, the last test score is not random, but constrained by the overall mean. Thus for 10 tests and a mean, there are nine degrees of freedom.


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• Degrees of freedom:


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ECE-580-DOE : Statistical Process Control and Design of ExperimentsStudent's t-Distribution

• EXCEL function:• TDIST(x,degrees_freedom,tails)• X is the numeric value at which to evaluate the distribution.• Degrees_freedom is an integer indicating the number of degrees of freedom.• Tails specifies the number of distribution tails to return. If tails = 1, TDIST

returns the one-tailed distribution. If tails = 2, TDIST returns the two-tailed distribution..

• TDIST is calculated as TDIST = p( x<X ), where X is a random variable that follows the t-distribution.

• Examples• Returns the Student's t-distribution. The t-distribution is used in the hypothesis

testing of small sample data sets. Use this function in place of a table of critical values for the t-distribution.

• TDIST(1.96,60,2) equals 0.054645• TINV(significance level, degrees of freedom)• TINV(0.05,9) = 2.26 ( two tail at 0.05 or 0.025 one tail)


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Basic Statistics Student t Distribution Example

• Student’s t distribution• . Distribution of the mean of n randomly

picked numbers divided by the sample standard deviation follows the Student’s t distribution with n-1 degrees of freedom.

• t α,υ = t value with υ degrees of freedom that’s gives a probability α.

• t = x-µ/(s/sqrt n)


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Student's t-Distribution Example


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Student's t-Distribution Example 1


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Ho: µ1 = µ2 Hi: µ1 = µ2 KEY NOTE: In these examples we are using the t statistic to compare our sample mean to the population mean. Later we will use the t statistic to compare two sample means.

First calculate the test statistic t as:


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We calculated T = 1.41

Rule is If T cal > t critical, we have a significant difference.

DF = 30 - 1 = 29

Alpha = 0.05 2 tail (0.025% per tail)

T critical from t distribution = 2.04

Since tcal < tcrit we accept the Null Ho : µ1 = µ2

So what the heck does this mean????


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Student's t-Distribution Example 1We calculated T = 1.41.DF = 30- 1 = 29

Probability of this mean value being at this location in the distribution is 17 % due to random chance ( normal distribution variation. (area under this part of curve)

This is not good enough for us to reject the Null!

i.e. We are not going to risk being wrong at least 17% of the time! We are willing to take a 5% risk in wrongly rejecting thenull!!

Alpha = 0.05 2 tail (0.025% per tail) (risk of rejection )

T critical from t distribution = 2.04Since tcal < tcrit or P-value > 0.05 we accept the Null Ho : µ1 = µ2


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Student's t-Distribution Example 1atwo tail test just looking for difference

Population MeanAverage 20.8 23.7 T-test t t t t

SD 11.3 two tail 0.1 0.05 0.02 0.01n 30 one tail 0.05 0.025 0.01 0.005

df degrees of freedom 29 29 29 29 29

T calculated (TINV(%P,df) 1.406 1.699 2.045 2.462 2.756P value (2tail)

TDIST(t,df,2tail) 0.170 0.100 0.050 0.020 0.010The probability or chance of being wrong when I reject the null. Desire small value 17.04% 10.00% 5.00% 2.00% 1.00%This is also called the risk of rejection or alpha risk I.e. The probability of being wrong when I state that mean 1 not equal to mean 2!degrees of freedom 29


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Student's t-Distribution Example 1bone tail test looking to see if sample is smaller that population

Ho: µ1 = µ2 Hi: µ1 < µ2 Population Mean

Average 20.8 23.7 T-test t t t tSD 11.3 two tail 0.1 0.05 0.02 0.01

n 30 one tail 0.05 0.025 0.01 0.005df degrees of freedom 29 29 29 29 29

T calculated (TINV(%P,df) 1.406 1.699 2.045 2.462 2.756P value (1tail)

TDIST(t,df,1tail) 0.085 0.050 0.025 0.010 0.005The probability or chance of being wrong when I reject the null. Desire small value 8.52% 5.00% 2.50% 1.00% 0.50%This is also called the risk of rejection or alpha risk I.e. The probability of being wrong when I state that mean 1 not equal to mean 2!degrees of freedom 29


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Basic Statistics Student t Distribution T statistic use

• This t statistic will be used in future tests to compare differences between 2 means: We need to evaluate that difference relative to their spread

(Standard deviation). 3 cases below.


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• This t statistic will be used in future tests to compare 2 population samples

Difference between means

This is called the standard error (SE) of the difference

=sqrt(vt/nt + vc/nc)

v – variance

n = sample size


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• t distribution EXCEL function:

• TDIST(x,degrees_freedom,tails)Student t

Distributions Vs Normal: Excel Generated =TDIST(t,degrees of freedom,1 tail)

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

-2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

t value SD units

Prob

abili

ty %

Student t P(X) (df =5, 1 tail)

Student t P(X) (df = 1000, 1 tail)


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• t distribution table; of values 1 tail/2tail use to check calculation etc in EXCEL

Note Z score for infinite df

•TDIST(1.96,100,2) equals 0.05•TINV(significance level, degrees of freedom)•TINV(0.05,10) = 2.23 ( two tail at 0.05 or 0.025 one tail)


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OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve BrainerdBasic Statistics Student t Distribution Confidence Intervals

Estimating Population mean from sample


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Excel functions

P NORMSINV(P) TINV(P,df) df =1000

0.01 -2.33 2.580.025 -1.96 2.240.05 -1.64 1.960.1 -1.28 1.650.2 -0.84 1.28


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In Excel:

P TINV(P,df) df =29

0.01 2.760.025 2.360.05 2.050.1 1.70


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So we are 95% confident the true mean is contained in this interval!


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F-DistributionFisher F distributionDistribution of the ratio of the standard deviation of n1 randomly

picked numbers by the standard deviation of n2 randomly picked numbers follows an F distribution with n1 –1 and n2-1 degrees of freedom . Note larger standard deviation in numerator!


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Basic Statistics Fisher F Distribution

Fisher F distribution used to compare variances from two populations and used in ANOVA and Factorial experiments


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F-Distribution• EXCEL function:• FDIST(x,degrees_freedom1,degrees_freedom2)• X is the value at which to evaluate the function.• Degrees_freedom1 is the numerator degrees of freedom.• Degrees_freedom2 is the denominator degrees of freedom.• FDIST is calculated as FDIST=P( F<x ), where F is a random

variable that has an F distribution.• Examples• Returns the F probability distribution. You can use this function to determine

whether two data sets have different degrees of diversity. For example, you can examine test scores given to men and women entering high school and determine if the variability in the females is different from that found in the males.

• FDIST(15.20675,6,4) equals 0.01


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F-Distribution• EXCEL functions


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Fisher F distribution


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Fisher F distribution example# Sample A Sample B1 7.5 7.92 8.2 8.4 F-Test Two-Sample for Variances3 8.1 84 8.4 7.8 Sample A Sample B5 7.1 7.5 Mean 7.7 7.6666666676 7.3 7.2 Variance 0.176 0.1677 7.1 7.4 Observations 15 158 7.8 7.3 df 14 14

9 8 8.1 F 1.0510 7.3 7.6 P(F<=f) one-tail 0.4611 7.3 7.7 F Critical one-tail 2.4812 7.9 8.113 7.8 7.714 8.1 6.815 7.6 7.5 you set

MANUAL P crits 0.4192 0.4082 F cal 1.05

Variance 0.1757 0.1667 F Critical FINV 2.48 0.05df 14 14 If Fcal > FINV signifcant difference

P value FDIST 0.46

P-value : Statement I’ll be correct 46% of the time if I say the variances are equal. To reject p <0.05 criteria!


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Basic Statistics Chi Square Distribution

• Chi-square: to test a sample variance against a hypothetical variance

• The distribution of the standard deviation of n randomly picked numbers follows a Chi squared distribution with n-1 degrees of freedom

• X2 = (n-1)s2/σ2

• The probability density curve of a chi-square distribution is asymmetric curve stretching over the positive side of the line and having a long right tail. The form of the curve depends on the value of the degrees of freedom.


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Fisher F distribution


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Basic Statistics

• Distributions Relationships :

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