Math Options Be Bold! Inc. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 4, Number 1 January 2002 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This newsletter is best viewed with a Courier font, size 10. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note - Changes Schedule of Public Classes Math in Industry - New Years Resolutions Software Review - An RPN Calculator on the Web Family Math - Real life Math for Teenagers Guest Author - Richard Clark: Numbers - From the Sands of Ancient Egypt to the Las Vegas Strip Ask Statman - How to Determine Your "Burn-In" Time ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Whatever you can do, or dream you can, begin it. Boldness has genius, power and magic in it. Goethe ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Reader, You will notice a few changes in E-Math News starting with this issue. First, Math In Everyday Life is gone. While the other columns receive a lot of feedback and are clearly worthwhile, Math In Everyday Life does not seem to have been widely read. In its place you will find an article by a guest author each month. These articles can be on virtually any subject involving math, so you will get a variety of reading. If you'd like to be a guest author, please send me a note at mailto:Bill@MathOptions.com. Also notice that the Book Review may now be a review of books, software, hardware, or anything that can help you use math more effectively in your work. Thank you for reading E-Math News. Happy New Year! Bill Kappele. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Feb. 25-27, 2002 Performing Objective Experiments Bellingham, WA Learn to use designed experiments to make you a top performer in your company. Feb. 28 - Mar. 1 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. May 14 - 16, 2002 Performing Objective Experiments Bellingham, WA Can't make the February class? Here's another chance. You can learn more about these classes and register to attend at http://www.mathoptions.com/public.htm You can learn about hosting these classes at your company at http://www.mathoptions.com/training1.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Industry ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ New Years Resolutions The beginning of each new year is a time to analyze our lives and look for new behaviors that will make our futures brighter. This article lists a few resolutions you may not have considered. 1.Sharpen the Saw -- In a recession it is easy to spend so much time working that you forget to improve your skills. Improving your skills is the most important thing you can do to make yourself more valuable to your company. So how do you convince your company to spend money on training? This may be difficult, but a study by the American Society for Training and Development (ASTD) may help you. The ASTD found that a company's investment in training can improve future financial performance, including total stock holder return. You can learn more at http://www.astd.org/CMS/templates/index.html?template_id=1&articleid=26040 2.Beware of Folklore -- Folklore is a body of common beliefs about how the world works. For instance, "Only inks with high surface tension dry quickly." The problem with folklore is that it is never proven and is often wrong. Yet people treat it as if it were a law of nature. Before accepting your company's folklore, check carefully to see if it has really been proven -- or if it is just one possible explanation for past observations. Inventions hide behind folklore. Challenging folklore can lead to wonderful inventions. 3.Be Bold! -- Your greatest chances for success come from bold actions. When you experiment, make big changes. When you choose factors, choose as many as you can afford to study. Choose factors nobody has studied before. But don't be reckless! Gambling is for vacations -- not your work. Use Designed Experiments to help you manage your bold experimentation. Happy New Year! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can learn Design of Experiments in a practical, hands-on workshop at your company. Let Bill Kappele show you how to USE DOE in your work - not just talk about it. Please visit http://www.mathoptions.com/training1.htm for details. Have you taken "Performing Objective Experiments" but are feeling pretty rusty? You can repeat the workshop for $495. Please call Bill Kappele for details - (888) 764-3958. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Software Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ An RPN Calculator on the Web I am a confirmed RPN fan. I have difficulty using a calculator with an equals key. Many of you have the same trouble. Most of the time this is no problem -- I just pull out my trusty HP calculator. From time to time, though, I am at a computer away from my office and I don't have my calculator with me. Usually I end up trying to use the built-in calculator, but it is very frustrating. But no more -- I have found an RPN calculator on the web. For those of you who prefer an RPN calculator but like to use one on your computer, please take a look at the Java Calculator at http://www.fpsmith.com/calculator.htm This RPN calculator runs in your browser and offers several useful functions, including trigonometric functions. You can also see all four levels of the stack. While it won't replace a good HP calculator, it is much better than anything that comes with Windows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Real life Math for Teenagers by Beth Heffernan If you have a child in high school, and you have not begun an introduction to the world of budgeting, checking accounts and negative balances, now is the time to do so. Our office is in a real estate building in a college town, and we hear some highly imaginative stories from renters about why they haven't got this month's rent. If a young person is released to the world with a checking account and budget, but no practice in using them, they are walking into trouble. We parents teach our children to use a car, but not money, and they are both powerful and potentially dangerous tools in the real world. Here is one approach. Wait for that rare moment when your teen is feeling reasonable. Make an agreement about which items you will buy, and which your teen is to cover. Their list should grow with each year. Some or all of their necessary funds will come from you as an allowance. Some families tie the allowance to chores and some do not. Further necessary funding may come from part-time jobs, anything from baby sitting, help with the family business or work in the community. Now you must, on paper, help your child balance the outflow with their earnings. This is the critical introduction to budgeting all of us receive, some painfully from creditors and some gracefully in the arms of their family. A famous comedienne handed his teenage son $1000 of Monopoly money. His son's eyes glowed. The father removed $500 for rent, $100 for utilities, $100 for a car payment and gas and $100 for food. Still, the son's eyes shone with the pleasure of future pur- chases. His father said "Got a girlfriend?" and the remaining $200 vanished. As we all know on payday, when the bills are paid, without budgeting we have a much less likelihood of pleasurable outings and purchases than with a plan in place. The easiest way to show a child how to manage a checking account and/or ATM account is to have your child balance your accounts for several months (maybe you can pay him or her for doing so). Here they will see the importance of writing down each trans- action as it occurs. They may see that credit card bills do not always match receipts. They will see how cash with- drawals erode the bottom line. The math of adding deposits and subtracting withdrawals is not hard, but it can be compli- cated. Trust them with a growing responsibility for daily, monthly and annual necessities. Arm them with adequate funding, a written budget and lots of practice in record keeping. Let them fill out the 1040EZ tax form when their wages become public. Co-sign a credit card with them in their senior year and go over each month's statements, receipts and payment. This child will go out into the world with confidence and your priceless gift, good credit. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Beth Heffernan is Vice President of Math Options. You can reach her at mailto:Beth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Guest Author: Richard Clark ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Numbers - From the Sands of Ancient Egypt to the Las Vegas Strip Several years ago I had the opportunity to visit Egypt and tour the Great Pyramids at the Giza plateau. Since that time I have read material and watched television programs that focus on how the Egyptians could construct these structures. They would have needed a crane that could lift x amount of weight or they would need a mill that created y amount of downward force in order to get the stones to fit as precisely as they did. To sit and wonder from a pure construction approach may be overlooking the most phenomenal fact. The ancient people had to either A) Solve the most difficult geometry and trigonometry problems to perfect accuracy by literally drawing them in the sand, or B) They possessed some fundamental understanding of numbers that today, in our world of cellular Internet, we have Overlooked. Let me give you an example of the magic of numbers. This can be found in the first 50 pages of about any number theory book. If we add consecutive odd numbers, the sum will always equal the next perfect square. 1 + 3 = 4 square root of 4 = 2 1 + 3 + 5 = 9 square root of 9 = 3 1 + 3 + 5 + 7 = 16 square root of 16 = 4 1 + 3 + 5 + 7 + 9 = 25 square root of 15 = 5 1 + 3 + 5 + 7 + 9 + 11 = 36 square root of 36 = 6 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 square root of 49 = 7 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 square root of 64 = 8 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81 square root of 81 = 9 And it never ends !!!!!!!!!!!!!! Here's another -- if we sum odd numbers in increasing intervals (first 1, then the next 2, then the next 3, then the next 4, etc.) the cubed root of the sums are in perfect order. 1 = 1 cubed root of 1 = 1 3 + 5 = 8 cubed root of 8 = 2 7 + 9 + 11 = 27 cubed root of 27 = 3 13 + 15 + 17 + 19 = 64 cubed root of 64 = 4 21 + 23 + 25 + 27 + 29 = 125 cubed root of 125 = 5 And this one never ends either !!!!!!!!!!!!!!! Now, let's go to Vegas. There is no better example of a Billion-dollar gamble that is a "sure thing" than the Blackjack tables at any casino on the strip. You can bet (no pun intended) the last penny you will ever have that the executives of these casinos know the probability of anything and everything that can happen, and the game will be set up so the house gets the odds. You may remember from the movie "Rain Man" the scene where Dustin Hoffman can predict the cards that will be dealt by counting the cards that have been played. Counting cards at Blackjack is not illegal in Vegas but if you are caught you will be asked to leave. You do not have to pay back any of your winnings but the house won't tolerate anything the tips the odds back towards the even side. The process of counting cards in a standard deck is very simple. Keeping track of the probabilities can be very difficult. Here's how it's done. The four #8 cards in the deck are given a value of zero. Cards #9, #10, Jack, Queen, King, and Ace are given a value of plus 1. Cards #7, #6, #5, #4, #3, and Deuce are given a value of minus 1. This means of the 52 cards in the deck, 4 are zero, 24 are plus 1, and 24 are minus 1. If you shuffle the deck and lay the cards out one at a time on top of one another, you can keep a running total of the current "count", and the 52 cards will add up to zero. For example, a King is shown (your count is +1), a 9 is next (now your count is up to +2), an 8 is shown next (your count remains +2), a 3 is shown (your count drops back to +1). If this count continues, after all 52 cards are used the count will end at zero. The 8s are zero, and the 9s and up balance out with the 7s and below. How you increase your odds is by being able to calculate the probability of the next card by knowing which cards are left. If 45 cards have been played and your count is - 3, you know of the 7 cards remaining there are 3 plus cards (this gets the count back to zero). The only combination of the other 4 cards that equal zero is 2 cards of plus 1 and 2 cards of minus 1, so the seven cards remaining are 5 plus cards and 2 minus cards. The probability of a plus card being dealt next is 5 in 7, or 71.4%. In the above scenario, if you've got 13, there is a 71% chance that taking a hit will place you over 21. Let the dealer take it, he'll bust and you'll win the hand by holding on 13. So much for 13 being unlucky. In order to do this successfully, you need to keep the running "count" and keep track of how many cards are left in the deck. Amazing isn't it, plus 1, minus 1, and zero - and it adds up to over $1,000,000,000! Go figure ????????????? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Richard Clark works in Measurement Science and Measurement Systems Analysis as a consultant for Qualtech Tool and Engineering. E-Mail feedback to mailto:qtmetrology@qualtechtool.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ask Statman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Written by Dr. Charles Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note: This article includes graphics that are not easily represented in text. You can find a Microsoft Word document at http://www.MathOptions.com/statman.burn.in.web.doc with the full text and graphics. Dear Statman: I would like to know how long I should "burn-in" my devices. I know I should be able to use the Weibull distribution for this. How can I do it? Signed, Burned-out Dear Burnie: So you want to do a burn-in? I can tell you how to do that. The Weibull distribution can be used for this, but it's a little more complicated than that. Let's start off with a little background for those who aren't familiar with this stuff. First, failure times for parts (capacitors, car tires, ball bearings, etc.) usually pile up according to a particular equation. One particular equation is called the Weibull distribution. It is called a distribution because the parts don't all fail at the same time but are spread out, i.e. they are "distributed". Suppose we had many different failure times for our parts and graphed them in the form of a histogram. A histogram is just a pile of data. You may have heard that data can form a pile in the shape of a bell (the bell curve), but sometimes they don't. If the failures times come from a Weibull distribution, the shape of the data pile (histogram) looks like a lop-sided bell. Figure 1 is an example of a histogram for some simulated Weibull data with a Weibull curve superimposed over it. If the assumption of a Weibull distribution for your parts is correct, then the probability of a part failing at or before a particular time, t, is given by F(t) = 1 - exp{-(t/a)^b} where F(t) is the probability of the part failing by time t, a is a constant called the "scale parameter" and b is a different constant called the "shape parameter". Roughly speaking, the scale parameter is related to how long the parts last on the average, and the shape parameter determines the shape of the histogram. The next concept we need to talk about is something called the "bathtub curve". That is a term used in reliability circles to describe how the failure rate changes with time. Suppose we are testing capacitors that come off the production line. Some of them (hopefully a small fraction) will be defective. If we have a test set that can apply a voltage to the capacitor, we can monitor how long it takes for it to fail (either become an open or short circuit). Imagine a thousand capacitors on test. Initially, the failure rate (number of failures per unit time) will be large. That is because all the defective capacitors fail early. This is called "infant mortality" because relatively young devices are dying (failing). After all the defective parts are gone, the failure rate will level off to a relatively constant value. This is called the "intrinsic region" of the bathtub curve because it represents what the failure rate is, free of any defects. This is the best the production process can do. After a while, however, the capacitors start to wear out. When that happens, the failure rate goes back up. For obvious reasons, this region of the curve is called "wear out". It is called a bathtub curve because it looks kind of like a bathtub. Figure 2 is an example of a bathtub curve for some fictitious data. For Figure 2, the "infant " region is from 0 to ~25 hours, the intrinsic region is from around ~25 to ~50 hours and the wear out region is beyond ~50 hours. A burn-in is a type of test where the part is run under high stress (high temperature, high voltage, etc.) for a given amount of time. The idea is to kill any defective parts, leaving behind only those that will last a long time. Of course, you don't want to do a burn-in for too long because then you will have gone beyond the infant region and started to take away from the useful life of the part. The question is, how long at how high a stress can you go before the failures are from either the intrinsic or wear out regions? If you knew the bathtub curve for your capacitors, you could pick off the time where all the defects were gone and the intrinsic region is just starting. From Figure 2, this would be at ~25 hours. This long a burn-in time may not be possible in practice. Figure 2 implies that we need to look at burning in at higher temperature (or voltage, etc.). Another possibility would be to see if we can get away with a burn-in of less than 25 hours. More on this later. The Weibull distribution is particularly flexible in that it can be used to approximate any one part of the bathtub curve: infant mortality, intrinsic or wear out. The values for the constants a and b are estimated from the data (most software packages do that automatically for you). It turns out that if the value of the constant b from the Weibull formula is less than one, then the failure rate is decreasing and so the failures are likely due to defects (infant mortality region). If the value of b is exactly one, then the failure rate is constant (intrinsic region). Lastly, if b is greater than one, the failure rate is increasing and so the failures are likely due to wear out (wear out region). Unfortunately, the Weibull distribution cannot be used for all three regions of the bathtub curve at the same time. That's because b can only have one value. In fact, no distribution can generate the entire bathtub curve. So, how can we use the Weibull distribution for burn-in? There is a highly sophisticated technique that can generate a bathtub curve. In this case, a "mixture" of Weibull distributions is used to predict the failure rate for all three parts of the curve. However, this is a little too advanced so I will demonstrate a much simpler method. It turns out that if we plot the life data assuming a Weibull distribution, there will be a "knee" in the curve at the boundary between two regions of the bathtub curve. Table I gives some fictitious failure times for capacitors. Also included is the probability of failure, F on or before a particular time, t. These probabilities were generated empirically, that is, they don't assume the data are from a Weibull distribution, or any other distribution. Most software packages that deal with reliability will generate these numbers for you. Now, if the data do come from a Weibull distribution, it turns out that a plot of log[-log(1-F)] vs. log(time) should be a straight line. Figure 3 is such a plot for the Table I data. On the right hand side of the graph is the Weibull probability, F, derived from the corresponding value of log[-log(1-F)]. Clearly, the data do not lie on a straight line. There appears to be two linear parts to the curve. The first from 0 to ~50 hours, the second is from ~50 to ~200 hours (see Figure 4). By eye, there is a change in the slope of the line ("knee"), indicating a transition from one failure mode to another. Let's segregate the data into two groups: early and late failures. Early failures are those before 50 hours, and late failures are those after 50 hours. Suppose we let the software plot the two groups individually (see Figure 5). For early failures, the software gives a = 15.3 hours and b = 0.71. For later failures, a = 120 hours and b = 5.1. Thus, the early failures have a value of b that is less than one, indicating a defect driven failure mechanism. The later failures have a b that is greater than one, indicating wear out. Technically speaking, there is no intrinsic region, since we don't find a b = 1. Thus, ~50 hours is the transition from infant mortality to wear out and would make a reasonable choice for a burn-in time. Now, there are a couple of more issues to deal with. First, what do we do if 50 hours is unacceptably long? Well, we can try burning in at a higher temperature. Suppose our original data were acquired at 150 C. What happens if we run a similar test at 200 C? Figure 6 is a Weibull plot of the two sets of data. Do you see that the 200 C data also has a knee? If we use a similar trick and break up the data (before and after 10 hours), then we get b = 0.88 for early failures and b = 3.1 for later failures. Thus, 10 hours is the transition time from infant mortality to wear out and could be used for a burn-in time. Of course, you can't use an arbitrarily high temperature. It is possible that if we set the burn-in temperature too high, the parts could fail due to a completely different mechanism than they would under standard operating conditions. For example, a metal part might melt or sag, which would not happen if it were run under use conditions. How do we make sure the mechanism hasn't changed? Just look at the probability plot. If both sets of data are roughly parallel, then the mechanism is probably the same. If they are not parallel, the temperature (or whatever stress you use) is too high and you need to lower it. In Figure 6, the probability plots for both temperatures are roughly parallel, so we don't have to worry about a change in mechanism. Another issue we need to deal with is: what if b > 1 for the early failures? It turns out that really doesn't matter. All you want to do is get beyond that front end of the bathtub curve. It turns out that you can still get roughly a bathtub shape for the failure rate if both early and late failures have a b > 1. The technique discussed here should still work. Lastly, it is possible that you may try this and the burn-in time you come up with is still unacceptably long. In that case, you won't be able to get rid of all the defects. Even so, it still could be that after the burn-in, the number of defects has been reduced to an acceptable level. That is, your customer might be willing to tolerate one or two early failures, so your burn-in doesn't have to screen out everything. After doing the burn-in, check the reliability by whatever methods you usually use. If the reliability after your "incomplete" burn-in is good enough, then you're done. If not, you need a more aggressive burn-in. I hope I have helped. If you have any more questions on this or any other statistical subject, I'd love to hear them. Thanks, Statman. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you have a question for Statman, please send it to mailto:Statman@MathOptions.com. Statman will answer questions about basic statistics that are of general interest to people working in industry. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Copyright 2001 by William D. Kappele, Beth Heffernan, Richard Clark, 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. If you don't have access to the World Wide Web, please send E-Mail to mailto:EMathNews-request@listdelivery.com with either "subscribe" or "unsubscribe" in the message body.