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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 3, Number 4 July 2001 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes Math in Everyday Life - How Much Do I Owe? Math in Industry - Finding Out Your Measurement Error Family Math - Summer math Estimating Software Review - Easy Gage R&R Ask Statman - The Poisson Distribution ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "If you ever assume you know all there is to know about something, or even if you accept that you know enough, you have just doomed yourself to mediocrity." Tom Hopkins ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ August 8-10, 2001 Performing Objective Experiments Anacortes, WA
This class includes a free whale-watching trip! Visit http//www.mathoptions.com/class_registration.htm for details. Sept. 12&13, 2001 Creating Custom Experiment Designs Bellingham, WA Learn to create experiment designs to fit YOUR experimental needs. You will no longer have to change your experiment to fit available designs. You can register to attend at http//www.mathoptions.com/class_registration.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Everyday Life ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ How Much Do I Owe? You and a friend go out for lunch. Your friend has a coupon that will purchase two meals for the price of one. You agree to share the savings, and place your orders. You order a meal for $7 and your friend orders a meal for $10. How do you split the $10 (remember, you get two meals for the price of one -- the more expensive one!) so that you each pay a fair amount? The first step is to figure out how much the bill would have been without the coupon -- $17. Next, determine what fraction of the bill would have been owed by each of you - you would have owed 7/17 of the bill and your friend would have owed 10/17 of the bill. To be fair, each of you should pay the fraction of the bill you would have paid without the coupon. So you owe 7/17 of $10 and your friend owes 10/17 of $10. So you pay 7/17 x $10 = $4.12 and your friend pays 10/17 x $10 = $5.88. You saved $7 - $4.12 = $2.88 and your friend saved $10 - $5.88 = $4.12. Notice that the fraction of the savings you received was $2.88 / $7 = 7/17 and the savings your friend received was $4.12 / $7 = 10/17 - so you split the savings fairly too. Now suppose you and your friend go out for dinner on a two for one coupon. You order a $25 meal and your friend orders a $22 meal. How much do you owe? (Neglect the tax and tip. You can find the answer at the end of this issue.) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Industry ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Finding Out Your Measurement Error by John Raffaldi In the manufacturing world, data is what drives decisions, profitable or not, parts, good or bad. The most common method of determining if a part is good or bad is to measure it and determine if the measurement is within the tolerance. But a single measurement in itself may not tell the truth due to measurement variation caused by measurement technique, gage error, and other related measurement errors. If we measured a part many times and made a pile of measurements, called a histogram, the distribution of observed measurements would be created as shown below. The process can be described by its standard deviation, a measurement of how much variation the measurements have, and mean, an average measurement of all the measurements combined. X X X X X X X X X X X X X X X X X X X X X X X X X X X Observed Measurement Variation When a part is measured, usually there is a specified tolerance. If the part is not manufactured within the tolerance, it does not perform as intended and must be discarded or reworked. Ideally, we can measure the part and know if it is within tolerance. Due to measurement errors, however, sometimes we cannot measure the part accurately because of the variation caused by the measurement system. This is because the measurement error consumes part of the tolerance. Near the Upper Specification Limit (USL) and Lower Specification Limit (LSL), the measurement result is not pass or fail; there is an area where we are unsure if the measurement is above the USL or lower than the LSL. We might be wrong even if the part measures within tolerance. In other words, we have pass, fail, and "not sure" measurement results. Occasionally, we might make what is called a Type I error, saying the part is bad when it is good (a false alarm). Or, we might make a Type II error, saying that the part is good, when it is actually bad. People who worry about these things construct what is called an operating characteristic curve (OC). The curve is a graphical representation of the probability of making a Type I or Type II error based on an up-front allowable error for making the mistake. These up-front errors are called alpha and beta respectively and are expressed as a percentage. You decide the acceptable chance of making a Type I error and the chance of making a Type II error is read from the OC curve. When we measure a part and the measurement error is large compared to the tolerance, we may not know if the part is good or bad if some of the measurement distribution overlaps the USL or LSL. The following figure shows the case where a measurement distribution overlaps the LSL. Each X represents an individual measurement of the same part.
Lower Specification Limit (LSL) Upper Specification Limit (USL) | | | | | X | | X | X| X X | X| X X | X| X X X | X X| X X X X | X X X| X X X X X X | Measurement Variation with a High P/T (GR&R) Ratio
If we reduce our measurement variation from the above figure so the gaging variation is small relative to the USL and LSL, the gaging variation does not overlap the USL or LSL. This is the situation we want. Lower Specification Limit (LSL) Upper Specification Limit (USL) | | | | | X | | X | | X X X | | X X X | | X X X X | | X X X X X X | | X X X X X X X X X | Measurement Variation with a Low P/T (GR&R) Ratio A GR&R percentage of tolerance calculation (P/T) quantitatively indicates the percentage of the tolerance lost to gaging variation. This P/T ratio should be as low as is economically possible. A very good P/T ratio is up to 10%. A slightly worse, but still acceptable ratio is from about 11% to 20%. P/T ratios between 21% and 29% are usually considered marginal, and ratios greater than 30% are unacceptable. These are not hard and fast rules. If the manufacturing process is centered on target with very little process variation, a higher P/T ratio is acceptable. A non-centered process with greater variation indicated by a higher P/T ratio may be unacceptable. This is because of the difficulty in discriminating between truly good and bad measurement results. Although in some instances GR&R calculations can be done using a calculator, computer software provides an efficient, cost effective way of performing GR&R study calculations, storing the data, and storing the study results. You can learn about an inexpensive software for GR&R studies in the software review later in this issue. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can learn Gage R&R in a practical, hands-on workshop at your company. Let Bill Kappele show you how to use Gage R&R in your work - not just talk about it. You can learn more about "Practical Measurement System Analysis" at http//www.mathoptions.com/practica.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Summer Math Estimating by Beth Heffernan Parents, tread lightly in summer on the math path. There is no faster way to roll back those little eyes than to suggest a continuation of school studies during vacation. However, we can introduce a fun part of math and tie it in with the flights of imagination children enjoy on long purposeless days. I am referring to estimation. One of the best qualities of summer vacation is the lack of hurry. We rush our children along to school, activities, sports practices, and social engagements. Then comes this lovely lull. It is healthy for a child to rest and wonder. It is very healthy for them to become bored enough to marshal their resources and figure out "what if?". If we are present for these imaginative stretches, we can offer estimation as a logical way to reach good enough answers.
How many grains of sand are on this beach? How wide is the Earth? What is the force of gravity at the center of the Earth? How many days till school starts, or my birthday, or Christmas? How many weeks of allowance till I can afford ...? How many hairs did I just cut off in that section of Brother's head? Beloved child, we can answer that question! Lets get out our tape measure and see how big this bald patch is. It's about two inches by three inches. Now look at a section of Brother's head that still has hair. Here's a section one inch by one inch, and counting now, we see it has about fifty hairs. Let's draw this. If one square inch has about fifty hairs, then six square inches would have about three hundred hairs! Wow, all that with one cut. Get out the atlas. Find the picture of the entire Earth. This section of the map has a legend that shows us that one inch of Earth on the map is worth 616 miles on the actual Earth. With our handy tape measure we see that it is 86 inches from one end of the map to the other. So that means 86 x 616..the world is about 53,000 miles around the middle (or throw in the Equator term and double the lesson). Let's see, today is June 24 and Christmas is December 25. That's six months away, there are about 30 days in a month, so it's about 180 days till Christmas. The key word is always "about". Some answers are good enough even if they are not exact. Children are so immersed in math answers having to be right that they have difficulty in getting an answer that's good enough. It is a shame to restrict imaginative thought for fear of working out the math. Your children will probably still be able to argue that there are three or eight billion stars in the galaxy, but, hey, it's summer.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Beth Heffernan is Vice President of Math Options. You can reach her at mailtoBeth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Software Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Easy GR&R by John Raffaldi Computer software provides an efficient, cost effective way of performing GR&R study calculations, storing the data, and storing the study results. Some people hesitate before purchasing software because they aren't sure of the benefit, don't want to chance spending money on a product that won't fit their needs, or feel that the software is over-priced. To help those individuals that are hesitant in purchasing GR&R software for any reason, Math Options has a fully featured GR&R software package that anyone can try. Simply download it from the Math Options web site and see if it helps with your work. The software performs multiple study method calculations, including the GM Long, Short, Ford, Within Part Variation, AIAG MSA, and ANOVA methods, produces graphs for visual analysis, including a Gage Performance Curve, prints reports, and stores the data and results.
If you like the software and continue to use it beyond 30 days, you are only required to spend $75.00 for the software, much less than other software which has fewer features and is more difficult to use. This software is easily within the budget of most any organization or company. Once you pay for the software, you will receive a printed manual that will provide full documentation on using the software, performing GR&R studies, and interpreting the results. The downloaded software is fully functional and is not programmed to stop working within a specified time period. If you download the software, but decide not to use it now, keep it on your computer for future use. When you start using it, please send the money and receive the manual to help you with the studies. No salesman will call trying to talk you into buying the software at any time. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can download your evaluation copy of "Easy GR&R" from http//www.mathoptions.com/easygr&r.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ask Statman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Written by Dr. Charles Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Statman- I would like to know more about the Poisson distribution. What is it, what are the assumptions behind it, and how is it used? Signed, Poised to learn.
Dear Poised- The Poisson distribution is a very useful tool for predicting every day things. It can be used to predict the number of traffic accidents per year in a city, the number of errors in a computer program, the number of phone calls you receive per day, or the number of goals scored in a hockey game. This distribution is used to predict "discrete" random variables. A discrete random variable is one that takes only integer values (0, 1, 5, 17, etc.). For example, the total number of a certain type of flaw in a semiconductor chip is a random variable (since we can't predict it exactly) but it can only be an integer. In other words, you can't have 12.6 defects. The assumptions behind the Poisson are not too restrictive. That’s why it can be used in so many situations. The first assumption is that the probability of exactly one "event" in a very short "interval" is proportional to the length of that interval. An example of an "event" might be a defect, an accident, or a goal scored. An "interval" can be the surface area where you are looking for defects, the time you wait for your team to score, etc. This assumption means that, for example, the larger the area you search, the greater the chance you will find a defect. The second assumption is that the probability of two events occurring in a short interval is approximately zero. So, the chance of the two particles of dust occupying the same square millimeter is very low. The third and last assumption is that these intervals are independent of each other. So, the rate at which the Knicks score doesn’t depend on how many points they have already scored or how far behind they are in the last five minutes of the game. (This last assumption may not work that well in a real sports game. Like most applied mathematics, we use it where it works well as an approximation to reality).
The formula for finding the probability that the random variable, X, will take on a particular value, x, is given by P(X=x) = m^x e^(-m)/x! where m is the average number of events, e is the exponential function and x! is read "x factorial". The factorial function just takes a number and multiplies it by every number less than it all the way down to one. For example, you may remember that 5! = 5 x 4 x 3 x 2 x 1 = 120. Also, 0! is defined to be 1. Let’s use the Poisson formula in an example. Suppose we make metal castings and want to predict the probability of a casting having exactly three defects. The number of defects in the casting might very well follow a Poisson distribution. Considering the first assumption, the probability of finding a defect will likely increase if we look over a larger area of the casting. Second, if we imagine dividing the casting into very tiny pieces, the probability of two defects occurring in one piece is very small. Lastly, we are assuming that the existence of a defect in one part of the casting does not depend on where the other defects are. Now, in order to use the formula, we need to know m. From past experience, we know that on the average, there are m = 5 defects per casting. So, using the above formula, we would have P(X=3) = 5^3 e^(-5) /3! = 0.14 Thus, there is a 14% chance of exactly three defects occurring in the entire casting. If we wanted to know the probability of up to three defects, we would just sum the probabilities
P(X<=3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 5^0 e^(-5) /0! + 5^1 e^(-5) /1! + 5^2 e^(-5) /2! + 0.14 = 0.0067 + 0.034 + 0.084 + 0.14 = 0.265 If X follows a Poisson distribution, then on the average, X = m. Said another way, the mean of a Poisson random variable is m. Interestingly, the variance of X (a measure of how spread out X is around the mean) is also m. This is in contrast to another well known distribution, the normal. The normal distribution (or bell curve) is described by a mean (m) and variance (s^2). For the normal distribution, the mean doesn't have anything to do with the variance. You can have any combination of mean and variance you like. In the Poisson distribution, the mean is the variance. Suppose we are observing events from two Poisson random variables with different means. The one with the higher mean will have observations more spread out than the other one. Let’s consider a different example. Suppose we have 100 pages of text to check for typos. We pick 10 pages at random and find a total of four typos on the ten pages. So, we estimate the true value of m will be 4/10 = 0.4 typos per page. Now we pick 20 more pages. What is the probability that there will be up to and including two typos in these 20 pages? First, we need to find the appropriate estimate of m. In this problem, we expect the average number of typos to be 0.4 typos/page x 20 pages = 8 typos. Now all we need to do is "plug and chug". P(X<=2) = P(X=0) + P(X=1) + P(X=2) = 8^0 e^(-8) /0! + 8^1 e^(-8) /1! + 8^2 e^(-8) /2! = 0.00034 + 0.0027 + 0.011 = 0.014 It looks like there’s a pretty slim chance (1.4%) of observing up to 2 typos. In fact, we can say that there is 98.6% chance of observing more than two typos (100% - 1.4% = 98.6%). By the way, many statistics books have Poisson tables so you can look the probability up rather than calculating it yourself. Lastly, I'd like to talk about a Poisson "process". A Poisson process is a kind of "stochastic" process, or a collection of random variables. One example of this is the amount of time people wait to be seated in a restaurant. Suppose you arrive at the local eatery and there are several groups ahead of you. After you get there, you note the time it takes for the next party to be seated. Suppose that occurs 10 minutes after you arrive. Then the second party is seated after you have been there 17 minutes, the third after 25 minutes, etc. These "waiting times" (10, 17 and 25 minutes) are a stochastic process. Let’s look at the "interarrival" times. That is, the length of time each party waited after the last one was seated. In the above example, the interarrival times would have been 10, 7 and 8 minutes respectively (10 - 0 =10 and 17 - 10 = 7, 25 - 17 = 8). Imagine measuring many of these interarrival times and plotting them in a histogram -- a pile of data. It turns out that if the histogram bars match a particular type of smooth curve, then we have a Poisson process. That type of curve is the exponential distribution. Its formula is f(t) = re^(-rt) where r is called the rate parameter, t is time and f(t) is the height of the curve at time t. Also, the Poisson parameter and r are related by m = rt. One of the properties of the exponential distribution is that it has no "memory". That is, the amount of time party #7 has to wait after party #6 is seated has nothing to do with the amount of time party #13 has to wait after party #12. This is related to the assumption stated earlier that the intervals (in this case time increments) are independent. What all this means is that if we are dealing with a Poisson process, then the interarrival times have an exponential distribution. It turns out that the mean of the exponential distribution is 1/r. The value of r in the graph was chosen to be 0.2, so the average waiting time is 1/0.2 = 5 minutes. If you are 10th in line at the restaurant, on the average you will have to wait 10*5 = 50 minutes. So now you know how a restaurant can figure out how long you will have to wait. Of course, they'll probably tell you it will only be "20 minutes or so". To summarize, the Poisson distribution is a very powerful tool in many different applications. Before you use it, however, you have to make sure the assumptions behind it are reasonable. If so, then I think this article should be of help. If not, then you might have to ask a statistician for help. Of course, you can always Ask Statman. Thanks, Statman. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you have a question for Statman, please send it to mailtoStatman@MathOptions.com. Statman will answer questions about basic statistics that are of general interest to people working in industry. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Answer to the price of dinner question You owe $13.30 - 25/47 x $25. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Copyright 2001 by William D. Kappele, John Raffaldi, Beth Heffernan and Charles S. Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you like E-Math News, please forward it to a friend. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A free newsletter published every other month by Math Options Inc. http//www.MathOptions.com 814 Lakeway Drive #179 FAX (503) 218-6587 Bellingham, WA, 98221 Toll Free (888) 764-3958 William D. Kappele, Editor Bill@MathOptions.com To subscribe to or unsubscribe from E-Math News please visit http//www.mathoptions.com/e-math.htm. 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Vol. 3, No. 4