Calculating Probabilities Using the Normal Distribution

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来自 University System of Georgia 的课程

Six Sigma Tools for Analyze

128 个评分

University System of Georgia

Six Sigma Tools for Analyze

128 个评分

课程 3（共 4 门，Specialization Six Sigma Yellow Belt）

This course will cover the Measure phase and portions of the Analyze phase of the Six Sigma DMAIC (Define, Measure, Analyze, Improve, and Control) process. You will learn about lean tools for process analysis, failure mode and effects analysis (FMEA), measurement system analysis (MSA) and gauge repeatability and reproducibility (GR&R), and you will be introduced to basic statistics. This course will outline useful measure and analysis phase tools and will give you an overview of statistics as they are related to the Six Sigma process. The statistics module will provide you with an overview of the concepts and you will be given multiple example problems to see how to apply these concepts. Every module will include readings, discussions, lecture videos, and quizzes to help make sure you understand the material and concepts that are studied. Our applied curriculum is built around the latest handbook The Certified Six Sigma Handbook (2nd edition) and students will develop /learn the fundamentals of Six Sigma. Registration includes online access to course content, projects, and resources but does not include the companion text The Certified Six Sigma Handbook (2nd edition). The companion text is not required to complete the assignments. However, the text is a recognized handbook used by professionals in the field. Also, it is a highly recommended text for those wishing to move forward in Six Sigma and eventually gain certification from professional agencies such as American Society for Quality (ASQ).

从本节课中

Data Analysis

In this module you will be diving into the statistical side of Six Sigma. You will begin with learning about the basic distribution types which include normal and binomial. You will then proceed to variation and will learn the difference between common and special cause variation.

Characteristics of Probability Distributions2:05

Calculating Probabilities Using the Normal Distribution4:56

Calculating Probabilities in the Normal Distribution: Example Problems3:50

The Binomial Distribution2:44

Binomial Distribution Calculations3:31

Binomial Distribution Calculations: More Examples3:12

与讲师见面

Christina Scherrer, PhD
Professor of Industrial Engineering
Department of Systems and Industrial Engineering
David Cook, PhD
Assistant Professor of Mechanical Engineering Technology
Department of Mechanical Engineering Technology
Gregory Wiles, PhD
Interim Chair and Assistant Professor
Department of Systems and Industrial Engineering
Bill Bailey, PhD
Assistant Professor of Industrial Engineering
Department of Systems and Industrial Engineering

Calculating probabilities in the normal distribution.

The normal distribution is probably the most commonly used distribution for

Six Sigma.

You will learn later that due to something called the central limit theorem,

we can often do statistical analysis of samples

under the assumption that the sample mean follows a normal distribution.

Mathematically, the formula for the distribution si the one on the slide.

But we always use tables or technology, rather than applying the formula directly.

The area under the normal curve between two x values, represents the probability

that the random variability would fall between those two values.

To calculate that value, we first determine the z scores of the points.

The z score is the number of standard deviations away from the mean

the measurement is.

Therefore the formula is z equals the quantity x minus mu divided by sigma.

Where mu represents the mean of the population and

sigma represents the standard deviation.

Once z is calculated you can then use the table or

technology to determine the area to the side of that z.

The table in the certified Six Sigma Yellow Belt book

provides the area to the right of the z.

However, most tables and technology provide the area to the left, so

that's the way I'll explain it for the rest of the module.

There is also a PDF table provided in the module for you.

Since the total area of the curve is equal to exactly 1,

you can count the probability of being to the right of z by subtracting

the area to the left from 1.

Suppose after you plug in the x mu and sigma, you obtain a z of -1.21.

In the table, go down the first column until you get to -1.2.

And then go across the columns at the top until you get to 0.01 for

the hundredths place.

The value there is 0.1131 which means that the fraction of

the area under the curve that is to the left of the x is 0.1131.

For example, suppose you have a process that is normally distributed with

a mean of 12 and a standard deviation of 3, and

you want to know the probability of being less than 15.

First, we calculate that the z score is 15-12 divided by 3 which is equal to 1.

Then we look up 1.0 in the table provided in the module.

In first column, go down to 1.0 and then across the top look for

0.00 since the hundreds place is 0.

We obtain 0.8413,

which is the area to the left of z = 1.00.

This means at the probability of being less than 15 on a normal curve

with mean 12 and standard deviation 3 is 0.8413.

If you want to calculate the probability of being greater than 16.5,

you would calculate z equals 16.5 minus 12 divided by 3 equals 1.5.

Look up 1.5 in the table by going down to 1.5 and across to 0.00.

This will be the area to the left of 16.5.

And since you want the probability of being greater than 16.5,

you would subtract that value from 1.

So from the table the area to the left of 16.5 is 0.9332 and

your answer is 1-0.9332 which is equal to 0.0668.

If you want to calculate the probability of being between 16.5 and 12.

You would need to subtract the area to the left of 12 from

the area to the left of 16.5.

So this value would be

0.9332-0.5000 = 0.4332.

This is even easier to do in Excel.

If you want the area to the left of x using Excel you don't even

need to calculate your z.

Simply enter, =norm.dist(x,mu,sigma, the word "true") and

the area to the left of x will automatically be calculated.

If you want the area to the right of x instead,

then enter =1- =norm.dist(x,mu,sigma,"true") instead.

In either case, you put in the correct numbers for your x, mu and sigma and

type that full word, true.

For example, to find the area to the left of 15 in the previous example,

it would be =norm.dist 15, 12, 3, true.

And for the area to the right of 16.5,

1-norm.dist(16.5, 12, 3, true).

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