Six Sigma TrainingSix Sigma Training

Dr. Robert O. NeidighDr. Robert O. Neidigh

Dr. Robert SetaputraDr. Robert Setaputra

Variable Types – Page 235Variable Types – Page 235

Attribute Data – a variable is either Attribute Data – a variable is either classified into categories or used to classified into categories or used to count occurrences of a phenomenon, count occurrences of a phenomenon, also referred to as classification or also referred to as classification or categorical data. Examples: gender, categorical data. Examples: gender, reasons for defects, and votes for reasons for defects, and votes for candidatescandidates

Measurement Data – results from a Measurement Data – results from a measurement taken on an item or measurement taken on an item or person of interest, also called person of interest, also called continuous or variables data. continuous or variables data. Examples: height, weight, Examples: height, weight, temperature, and cycle timetemperature, and cycle time

Measures of Central Measures of Central TendencyTendency

Measures that try to describe or Measures that try to describe or quantify the middle of a data set.quantify the middle of a data set.

Measures of Central Measures of Central TendencyTendency

Mean – average of all data pointsMean – average of all data points Median – value such that at least half the data Median – value such that at least half the data

points are less than or equal to the value and points are less than or equal to the value and at least half the data points are greater than at least half the data points are greater than or equal to the value or equal to the value

Mode – value in the data set that occurs most Mode – value in the data set that occurs most frequentlyfrequently

First Quartile – value such that at least 25% of First Quartile – value such that at least 25% of the data points are less than or equal to the the data points are less than or equal to the value and at least 75% of the data points are value and at least 75% of the data points are greater than or equal to the value greater than or equal to the value

Third Quartile – value such that at least 75% of Third Quartile – value such that at least 75% of the data points are less than or equal to the the data points are less than or equal to the value and at least 25% of the data points are value and at least 25% of the data points are greater than or equal to the value greater than or equal to the value

Minitab example on Page 249Minitab example on Page 249

Measures of VariationMeasures of Variation

Measures that try to describe or Measures that try to describe or quantify the amount of spread or quantify the amount of spread or variation in a data setvariation in a data set

Measures of VariationMeasures of Variation

Range – distance from the smallest data point Range – distance from the smallest data point to the largest data pointto the largest data point

Variance and Standard Deviation – measure of Variance and Standard Deviation – measure of how much the data points fluctuate around the how much the data points fluctuate around the meanmean

Minitab example on Page 249Minitab example on Page 249

What is standard deviation?What is standard deviation?

Standard deviation is a measure of Standard deviation is a measure of variation within a data set. variation within a data set.

The larger the standard deviation, the The larger the standard deviation, the more variation in the data set and vice more variation in the data set and vice versa. versa.

Technically, standard deviation is a Technically, standard deviation is a measure of variation about the mean.measure of variation about the mean.

Roughly speaking, standard deviation Roughly speaking, standard deviation is the average distance between each is the average distance between each data point and the mean.data point and the mean.

Motivate measure of variation through Motivate measure of variation through examples of small data sets.examples of small data sets.

Population – divide by Population – divide by nn, sample – , sample – divide by divide by nn - 1 - 1

Show students Normal.xls file.Show students Normal.xls file.

Continuous Probability Continuous Probability DistributionsDistributions

Can assume an infinite number of Can assume an infinite number of values within a given rangevalues within a given range

Probability of any one point is zeroProbability of any one point is zero Probabilities are measured over Probabilities are measured over

intervalsintervals Area under curve defines probabilityArea under curve defines probability Use calculus to calculate probabilitiesUse calculus to calculate probabilities Ugh!!!Ugh!!! Normal probability distribution is one Normal probability distribution is one

typetype Fortunately, probabilities already Fortunately, probabilities already

calculated and contained in a table for calculated and contained in a table for normal distributionnormal distribution

Characteristics of Normal Characteristics of Normal ProbabilityProbabilityDistributionDistribution

1)1) Bell-shapedBell-shaped

2)2) SymmetricalSymmetrical

3)3) Mean, median, and mode are the sameMean, median, and mode are the same

4)4) Asymptotic – tails never touch X-axisAsymptotic – tails never touch X-axis

5)5) Completely described by its two Completely described by its two parameters – mean(parameters – mean(µ)µ) and standard and standard deviation(deviation(σσ))

6)6) There are an infinite number of There are an infinite number of possible normal probability possible normal probability distributionsdistributions

How do we calculate How do we calculate probabilities?probabilities?

Since there are an infinite number of normal Since there are an infinite number of normal distributions, how can we possibly calculate probabilities distributions, how can we possibly calculate probabilities for all of them? Fortunately, there is a unique for all of them? Fortunately, there is a unique characteristic of all normal distributions that allows us to characteristic of all normal distributions that allows us to do so. The probability of having a value above/below a do so. The probability of having a value above/below a point that is X standard deviations above/below the mean point that is X standard deviations above/below the mean is the same for every possible normal distribution. The is the same for every possible normal distribution. The probabilities for the standard normal distribution (probabilities for the standard normal distribution (µ = 0 µ = 0 and and σσ = 1) can be used for every other normal = 1) can be used for every other normal distribution. These probabilities can be found in the distribution. These probabilities can be found in the standard normal probability table. Our task is to convert standard normal probability table. Our task is to convert every normal distribution to the standard normal, this is every normal distribution to the standard normal, this is called standardizing. called standardizing.

How do we standardize?How do we standardize?

The distance between any point on our normal The distance between any point on our normal distribution of interest and the mean is found. We now distribution of interest and the mean is found. We now want to put this distance in units of standard deviation, to want to put this distance in units of standard deviation, to do so we divide the distance between the point and the do so we divide the distance between the point and the mean by our standard deviation. This value is called a Z-mean by our standard deviation. This value is called a Z-value and tells us how many standard deviations above or value and tells us how many standard deviations above or below the mean a point is. If the z-value is positive, the below the mean a point is. If the z-value is positive, the point is above the mean and if the z-value is negative the point is above the mean and if the z-value is negative the point is below the mean.point is below the mean.

Z-value = (point minus the mean)/standard deviationZ-value = (point minus the mean)/standard deviation

The standard normal table always gives the probability of The standard normal table always gives the probability of having a value less than the Z-value.having a value less than the Z-value.

Finding the probability of Finding the probability of having a value less than a having a value less than a

given pointgiven point Find the Z-value for the given point Find the Z-value for the given point The Z-value lets us know how many The Z-value lets us know how many

standard deviations above/below the standard deviations above/below the mean the point is mean the point is

Look up the probability in the Look up the probability in the standard normal table standard normal table

This is the probability of having a This is the probability of having a value less than the given pointvalue less than the given point

μμ = 70 and = 70 and σσ = 10, find probability of = 10, find probability of having a value less than 66having a value less than 66

40 50 60 70 80 90 100

-3 -2 -1 0 1 2 3

Finding the probability of Finding the probability of having a value greater than having a value greater than

a given pointa given point Find the Z-value for the given point Find the Z-value for the given point The Z-value lets us know how many The Z-value lets us know how many

standard deviations above/below the standard deviations above/below the mean the point is mean the point is

Look up the probability in the Look up the probability in the standard normal table standard normal table

This is the probability of having a This is the probability of having a value less than the given pointvalue less than the given point

Subtract this probability from one to Subtract this probability from one to find the probability of having a point find the probability of having a point greater than the given pointgreater than the given point

μμ = 70 and = 70 and σσ = 10, find probability of = 10, find probability of having a value greater than 56having a value greater than 56

40 50 60 70 80 90 100

-3 -2 -1 0 1 2 3

Finding the probability of Finding the probability of having a value between two having a value between two

pointspoints Find the Z-values for the given pointsFind the Z-values for the given points The Z-values let us know how many The Z-values let us know how many

standard deviations above/below the standard deviations above/below the mean the points are mean the points are

Look up the probabilities in the Look up the probabilities in the standard normal table for the two Z-standard normal table for the two Z-valuesvalues

These are the probabilities of having a These are the probabilities of having a value less than the given point value less than the given point associated with each Z-valueassociated with each Z-value

Subtract the probability associated Subtract the probability associated with the smallest Z-value from the with the smallest Z-value from the probability associated with the largest probability associated with the largest Z-valueZ-value

This is the probability of having a This is the probability of having a value between the two pointsvalue between the two points

μμ = 70 and = 70 and σσ = 10, find probability of = 10, find probability of having a value between 57 and 76having a value between 57 and 76

40 50 60 70 80 90 100

-3 -2 -1 0 1 2 3

Finding the point on a Finding the point on a normal distribution normal distribution

associated with a given associated with a given probabilityprobability

Find the probability in the standard Find the probability in the standard normal tablenormal table

Find the Z-value associated with the Find the Z-value associated with the probabilityprobability

Convert the Z-value to a point on the Convert the Z-value to a point on the normal distributionnormal distribution

Mean plus (Z-value times standard Mean plus (Z-value times standard deviation)deviation)

μμ = 70 and = 70 and σσ = 10, find the value such = 10, find the value such that 70% of the charge amounts will be that 70% of the charge amounts will be greater than that amountgreater than that amount

40 50 60 70 80 90 100

-3 -2 -1 0 1 2 3

Sampling MethodsSampling Methods

Reasons for sampling:Reasons for sampling:

Too time consuming to check entire Too time consuming to check entire populationpopulation

Too expensive to check entire Too expensive to check entire populationpopulation

Sample results are adequateSample results are adequate Destructive testingDestructive testing Impossible to check entire populationImpossible to check entire population

Sampling DefinitionsSampling Definitions

Simple random sample – each item in the Simple random sample – each item in the population has the same probability of population has the same probability of being selectedbeing selected

Sampling error – difference between a Sampling error – difference between a sample mean and the population meansample mean and the population mean

Sampling distribution of the sample mean Sampling distribution of the sample mean – probability distribution of all – probability distribution of all possible sample means of a given possible sample means of a given sample sizesample size

Standard error of the mean – standard Standard error of the mean – standard deviation of the sampling distribution deviation of the sampling distribution of sample means (average sampling of sample means (average sampling error)error)

When is sampling When is sampling distribution normal?distribution normal?

If population distribution is normal, then If population distribution is normal, then sampling distribution is normal for sampling distribution is normal for any sample sizeany sample size

If sample size is greater than or equal to If sample size is greater than or equal to thirty, then sampling distribution is thirty, then sampling distribution is always normalalways normal

Properties of normal Properties of normal sampling distribution?sampling distribution?

Sampling distribution mean (Sampling distribution mean (µµx-barx-bar) equals ) equals population mean (µ)population mean (µ)

Standard error (Standard error (σσx-barx-bar)) equals population equals population standard deviation (standard deviation (σσ) divided by the ) divided by the square root of the sample size (square root of the sample size (nn))

Once we know the mean and standard Once we know the mean and standard error of the sampling distribution and error of the sampling distribution and we know it is normally distributed we we know it is normally distributed we are set to compute probabilitiesare set to compute probabilities

NotationNotation

nX

X

/

ExampleExample

Captain D’s tuna is sold in cans that have a net Captain D’s tuna is sold in cans that have a net weight of 8 ounces.weight of 8 ounces.

The weights are normally distributed with a mean of The weights are normally distributed with a mean of 8.025 ounces and a standard deviation of 0.125 8.025 ounces and a standard deviation of 0.125 ounces.ounces.

You take a sample of 36 cans.You take a sample of 36 cans.

Example – Cont.Example – Cont.

020833.036/125.0

025.8

X

X

Example – Cont.Example – Cont.

What is the probability of having a sample mean What is the probability of having a sample mean greater than 8.03 ounces?greater than 8.03 ounces?

-3 -2 -1 0 1 2 3

7.962 7.983 8.004 8.025 8.046 8.067 8.088

Example – Cont.Example – Cont.

What is the probability of having a sample mean less What is the probability of having a sample mean less than 7.995 ounces?than 7.995 ounces?

-3 -2 -1 0 1 2 3

7.962 7.983 8.004 8.025 8.046 8.067 8.088

Example – Cont.Example – Cont.

What is the probability of having a sample mean What is the probability of having a sample mean between 7.995 ounces and 8.03 ounces?between 7.995 ounces and 8.03 ounces?

-3 -2 -1 0 1 2 3

7.962 7.983 8.004 8.025 8.046 8.067 8.088

Hypothesis TestingHypothesis Testing

Hypothesis – a statement about a Hypothesis – a statement about a population developed for the purpose of population developed for the purpose of testingtesting

Hypothesis test – a procedure based on Hypothesis test – a procedure based on sample evidence and probability theory to sample evidence and probability theory to determine whether the hypothesis is a determine whether the hypothesis is a reasonable statementreasonable statement

Key Point – Anytime a decision is made Key Point – Anytime a decision is made about a population based upon sample about a population based upon sample data an incorrect decision may be madedata an incorrect decision may be made

Type I and Type II ErrorsType I and Type II Errors

Type I Error – rejecting a true null Type I Error – rejecting a true null hypothesishypothesis

Type II Error – accepting a false null Type II Error – accepting a false null hypothesishypothesis

Unfortunately, in hypothesis testing the Unfortunately, in hypothesis testing the probability of a Type I Error (probability of a Type I Error (αα)) is inversely is inversely related to the probability of a Type II Error related to the probability of a Type II Error ((ββ)). If we decrease the probability of a Type . If we decrease the probability of a Type I Error, then the probability of a Type II I Error, then the probability of a Type II Error increases and vice versa.Error increases and vice versa.

What are Type I and Type II errors in the What are Type I and Type II errors in the U.S. Legal System?U.S. Legal System?