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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 3, Number 2 March 2001 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes Math in Everyday Life - Investing for Retirement How Much Money Will You Need? Math in Industry - Capability Ratios Family Math - Math activities for ages 1-6 Book Review - Applied Reliability 2nd edition Ask Statman - More Tests for Non-Bell-Shaped Data ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "Especially in a technical business where the rate of progress is rapid, a continuing program of education must be undertaken and maintained." David Packard ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ May 14 & 15, 2001 Statistical Process Control and Practical Measurement System Analysis Anacortes, WA May 16-18, 2001 Performing Objective Experiments Anacortes, WA This class includes a free whale watching trip! August 8-10, 2001 Performing Objective Experiments Anacortes, WA This class includes a free whale watching trip! SAVE $300 BY REGISTERING EARLY! Visit http//www.mathoptions.com/class_registration.htm for details. You can learn more about these classes and register to attend at http//www.mathoptions.com/public.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Everyday Life ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Investing for Retirement How Much Money Will You Need? Planning for retirement requires that you estimate the amount of money you will need to have available and develop a plan for accumulating that money. This article will provide the math you will need to estimate the amount of money to accumulate. The next issue will explain the math you will need to develop your plan for accumulating it. The first step is to estimate your monthly expenses during your years of retirement. This step is actually very difficult because our uncertainty about the future is relatively large. What will the rate of inflation be over the next 30 or 40 years? How much will taxes increase? What will your health be like? What interest will you be able to collect on your money? Although this step is difficult, it is essential. Without a monthly requirement, you have no way to calculate the amount of money to accumulate. You should consult with a good financial planner who can help you to estimate this amount based on the lifestyle you expect to have upon retiring. Once you have an estimate of your monthly expenses, you are ready to calculate the amount of money you will need to accumulate. The first step is to determine how you would like to manage your monthly withdrawals after retirement. Would you like to take out only the interest earned on your money, leaving the principle untouched for distribution to your heirs? Or would you be willing to reduce the principle on a regular basis, leaving little behind when you die? The second step differs depending on your decision from the first step. If you want to leave your principle untouched and withdraw only the interest, the calculation of the amount you will need is very simple. You need to know the monthly withdrawal you intend to make and an estimate of the interest your money will be making after your retirement. This will depend on where you keep your money. Here again you should ask a good financial advisor to help you with this estimate. The calculation is, then, Amount to Accumulate = Monthly Withdrawal / Monthly Interest Rate For example, suppose you want to withdraw $4000 per month and you estimate that you will be able to collect 5% interest on your money annually. Your monthly interest rate is 5% / 12 = 0.42%. This monthly interest rate is 0.0042 when expressed as a decimal. Amount to Accumulate = $4000 / 0.0042 = $952,381 - nearly a million dollars. If you are willing to withdraw part of your principle each month, you will need to accumulate less money -- but your calculation will be a little more involved. You will not only need to know the amount of your monthly withdrawal and an estimate of the interest you will make during your retirement, but you also need an estimate of how long you will live. The calculation is, then, 1. Calculate the number of months you expect to live after retiring. 2. Raise 1 + the Monthly Interest Rate to the power of the number of months you expect to live after retiring. 3. Divide 1 by the number from step 2. 4. Subtract the number from step 3 from 1. 5. Multiply the number from step 4 by the monthly withdrawal. 6. Divide the number from step 5 by the monthly interest rate expected. 7. The result is the amount of money you will need to accumulate. For example, suppose you want to withdraw $4000 per month and your estimated monthly interest rate is 0.0042 (5% annually). You want to retire at age 65 and expect to live to age 85. 1. (85 - 65) x 12 = 240 months in retirement. 2. (1 + 0.0042)^240 = 2.734 3. 1 / 2.734 = 0.3658 4. 1 - 0.3658 = 0.6342 5. $4000 x 0.6342 = $2536.80 6. $2536.80 / 0.0042 = $604,000.00 7. You will need to accumulate about $604,000. Next issue The math used to develop a plan to accumulate this money. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Industry ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Capability Ratios by John Raffaldi In manufacturing (and probably any other time we have a tolerance), sometimes our thoughts are directed to determining how much of this tolerance is due to variation in the measurement process and the manufacturing process. Typically, in these situations, we are talking about capability ratios.
There are at least 18 capability ratios, each claimed to be useful by a published statistician at one time or another. But today, we will only worry about two of the most common ones, Cp, and Cpk.
Capability ratios help us to judge whether a process is capable or not. If a process is not capable, it means that chances are good that the process we have for manufacturing parts is unlikely to make us profitable because of scrap production rates. A capable process is a process in which essentially no bad parts are made and is likely to keep us in business.
There are primarily three things necessary for determining a capability ratio for a process with only random variation present An Upper Specification Limit (USL) - a number which a measurement should not exceed to be called good. A Lower Specification Limit (LSL) - a number which a measurement should not be less than to be called good. A process standard deviation - A measurement of how the process varies over time. ("Sigma" is another term for standard deviation.) A small standard deviation indicates a process where there is little variation -- a good characteristic to have in any process. It turns out that 3 times the standard deviation less than the average is about the lowest measurement you will normally see, and 3 times the standard deviation above the average is about the largest measurement you will normally see. This range covers 6 times the standard deviation and is often referred to as a "6 sigma range." For "6 sigma range" 99.73% of all parts produced, or about 27 out of 10000 parts, would be outside of the range. This range is the "natural tolerance" for a manufacturing process. Most math books have reference for calculating standard deviations, and almost any calculator will perform the math for you. (Before trusting your calculator to calculate standard deviation you can test it at http//www.mathoptions.com/calculatortest.htm) The capability ratio (Cp) is the required tolerance range (Upper Specification Limit (USL) minus the Lower Specification Limit (LSL)) divided by the natural tolerance range (6 sigma range). In equation form, Cp=(USL-LSL) / 6*Standard deviation As stated above, this is a measurement of how much of the tolerance is consumed by the process and measurement variation. A Cp of 1.0 indicates that the required tolerance range is equal to the natural tolerance range. Usually a Cp of 1.0 is considered marginally adequate because 99.73% of the parts are acceptable if the process average is centered between the USL and LSL. If the process average shifts or the variation increases, many more out-of-specification parts would exist. A process with its average centered that has a Cp index of 1.0 would have 27 out of 10,000 parts manufactured out of tolerance. Most processes should have a Cp of at least 1.33. With a Cp of 1.33 the required tolerance range is 8 standard deviations wide (8 Sigma). A process with its average centered will only produce 64 parts out of a million that are likely to be bad due to random variation. The Cpk is another type of capability ratio, but it is better than the Cp because it takes into account the process average location relative to the USL and LSL. Cp can be divided into two values the Cpu and Cpl (where the "u" and "l" indicate the Cp for upper and lower specifications). Cpu =(USL-X2bar) / 3sigma Cpl =(X2bar-LSL) / 3sigma (X2bar is the overall average measurement of parts measured for prior samples.) The Cpk is the smaller of the Cpu and Cpl. Just like the Cp, the lower ratio should not be less than 1.33. If the process is exactly centered, the Cp will equal the Cpk. From a gauge error standpoint, if we decrease our measurement error, we are likely to increase our capability ratio. This is because the gauging error is included in the process variation. If you are interested in more detailed information on capability ratios, a good place to look is at the SEMATECH/NIST on-line statistics manual at http//www.itl.nist.gov/div898/handbook/, then do a search on process capability. Similarly the StatSoft web site www.statsoft.com has an on-line statistics book at http//www.statsoft.com/textbook/stathome.html ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can learn about Statistical Process Control and Gage R&R Studies May 14 & 15, 2001, in Anacortes, WA, in the workshops, Statistical Process Control and Practical Measurement System Analysis You can learn more about these classes and register to attend at http//www.mathoptions.com/public.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math Activities for Ages One Through Six by Beth Heffernan Caution! You may have fun while stretching your child's mind. Up to age Four or so, numbers are mainly fun sounds with meaning only in relation to your child's immediate present. Children enjoy counting in many languages for this reason-more fun sounds to roll off their tongues. Beginning at age One you should count everything, even before they can talk. It is fun, it is real and it establishes a base for precise language that you will always use to describe their unfolding world. No baby talk (at least with your children)! Using counting rhythms and rhymes is even more fun, and is useful in introducing music to your child. Geometry begins with exact language. Point out the shapes of surrounding objects, the triangle of their cracker, the sphere of the cotton ball, the octagon of the stop sign. Count the sides. How many sides in a circle? Introduce two and three dimensions using their bodies, then items you point out on neighborhood walks. Children understand positioning, high vs. low, under, over and in between, and definitely more or less. At age Two, when children begin talking (and never stop), it is especially important to count like objects, i.e. three marsh- mallow cylinders or four cup cylinders. When the cookies are "all gone" everyone understands zero. Food is a wonderful tool for displaying fractions and sets. Let your child help you cut his or her sandwich into thirds or fourths or tenths. Let him or her divide the snack into equal sets for friends. Great games can be played with M&Ms, right down to subtracting them to zero into the lucky players' mouths. Measurement is an integral part of math and science. Water is a perfect starting material. Dig a hole in the sand and see how many buckets of water fill it. Better toy stores sell sets of telescoping cups that are volumetrically correct, and make the best bath toys for years at different levels of under- standing. Skills achieved with pouring water translate easily to later cooking projects with lots of measuring and fractions involved. A Five or so can plant a garden and learn linear measurement of rows and columns. Measure depth while planting seeds to the proper depth. At this age money becomes important and so real. Many children love to play store. Provide them with all your canned goods and a basket of small change or toy cash register. Let them wheel and deal with each other. If they receive an allowance or gift money, point out to your Six that a pack of Pokemon cards is $3.99 at this store and $2.99 elsewhere. Let them pay for their purchases themselves and receive the change. Learn successive counting in card games which require dealing equal numbers of cards to players. An inexpensive pad of puzzles is a handy tool when waiting with your child. Compare pictures with six differences. Match like shapes. The truly desperate parent, during a long car trip, can offer as many cents as the child can count, in any language. Infinity is the amount of love we have for each of our children. Next issue Math activities for ages 6-12. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Beth Heffernan is Vice President of Math Options. You can reach her at mailtoBeth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Book Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Applied Reliability 2nd edition, by Paul A. Tobias and David C. Trindade by Dr. Charles Whitman This is an excellent book for the beginner to the world of reliability. The authors are light on mathematical proofs and heavy on practical examples. While you should be familiar with basic calculus to understand some of the concepts, you can still benefit with a good knowledge of algebra. The first chapter covers the basics of probability and descriptive statistics. No prior knowledge is assumed. It then proceeds to describe the more common life distributions like the exponential, Weibull, extreme value, normal and lognormal. The authors provide guidance on how to tell which distribution to use for your data. For cases when you don't know the distribution, they demonstrate the utility of the Kaplan-Meier technique. Different approaches of calculating a distribution's parameters are discussed, including graphical methods. The concept of censoring is introduced along with ways of dealing with censored observations. Special emphasis is placed on plotting data. This helps the reader get a "feel" for what is going on and how to think about the concepts presented. The book also has an entire chapter on accelerated testing. The Arrhenius model is discussed, along with variations of it. In a later chapter, acceptance sampling is explained. Throughout the book, the authors use plain language and practical examples to help the reader understand key concepts. For those who need to know the basic techniques of reliability, this book is a must. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dr. Whitman writes the "Ask Statman Column in E-Math News." He is a Statistician for Agere Systems. You can purchase a copy of "Applied Reliability 2nd edition," by Paul A. Tobias and David C. Trindade from Amazon.com at the link below http//www.amazon.com/exec/obidos/ASIN/0442004699/mathoptionsinc ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ask Statman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Written by Dr. Charles Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ As promised last issue, Statman will explain another test for data that doesn't fall in Bell-Shaped piles - non-Normal data. The test is the Mann- Whitney test and is described below Let's consider one other test called the rank sum test (or Mann-Whitney test). This is used for comparing two independent samples when the observations are quantitative. The rank sum test is different than the sign test (discussed in the last issue)in that the data need not be paired. In fact, the data can even come from samples of different sizes. Let's suppose we want to know if, on the average, treatment 1 produces observations which are different (larger or smaller) than treatment 2. Note that I say "on the average". Suppose treatment 1 really is different than treatment 2. Even though this is true, the data may indicate no difference. Since we are dealing with data, and data have variation, there is some chance that we will observe no difference when in fact there is one. Suppose we have two different processes for drawing wire we want to compare (Drawing wire means pulling it through a small hole that will deform it uniformly, decreasing its diameter to that of the hole.) Sometimes after drawing, small cracks appear in the wire. The cracks are undesirable and we want to find out if one drawing method is preferred. Several pieces of wire, each of the same length, are randomly selected from the parent wire spool produced by each process. The size of the sample for drawing process 1 is N1= 10, and the size of the second sample for drawing process 2 is N2 = 12. Let N = N1 + N2 = 22. The number of cracks in each sample is counted for each method. First, we tabulate the results.
No. cracks No. cracks observed observed for for process 1 process 2 20 100 0 55 1 0 15 77 3 86 55 61 37 30 3 15 3 59 0 0 5 10
The next step is to rank all the observations from lowest to highest, keeping track of the treatment. If the number of cracks from each sample were different, then a sample with no cracks would have rank 1, another with 1 crack would be rank 2, etc. Two or more samples with the same number of cracks are "tied". Unfortunately, ties make life more difficult. For example, the four samples with 0 cracks are tied. Should they all get rank 1? Should they all get rank 4? When there are ties, an "average" rank is used. To find the average rank, we start by assigning all the observations an intermediate rank. In this intermediate step, we don't need to pay attention to the sample's treatment. In our example, the four samples with 0 cracks are assigned intermediate ranks 1, 2, 3, and 4 since they are the four smallest observations. Next, we average the intermediate ranks. The average of 1 through 4 is 2.5, so all observations with 0 cracks have a rank of 2.5. The other ties receive a rank that is the average of their intermediate ranks. The following table should help make this clear. No. cracks Process Intermediate Rank Rank 0 1 1 2.5 0 1 2 2.5 0 2 3 2.5 0 2 4 2.5 1 1 5 5 3 1 6 7 3 1 7 7 3 1 8 7 5 2 9 9 10 2 10 10 15 1 11 11.5 15 2 12 11.5 20 1 13 13 30 2 14 14 37 1 15 15 55 1 16 16.5 55 2 17 16.5 59 2 18 18 61 2 19 19 77 2 20 20 86 2 21 21 100 2 22 22 I think you can see why we ignored the drawing method when assigning intermediate ranks. All ties received the same rank, regardless of the treatment. The next step is to sum the ranks from the different treatments (hence the name "rank sum test"). We'll call the sum of the ranks for process 1 R1, and the sum of the ranks for process 2 R2. Their values are R1 = 87 and R2 = 166. As a check, R1 + R2 should equal N*(N + 1)/2 = 22*23/2 = 253. After finding R1 and R2 we calculate U and U' using the equations below. The smaller of U and U' is then compared to a tabulated critical value to get the p-value. The formulas for U and U' are U = N1*N2+N1(N1+1)/2-R1 U' = N1*N2-U If there are no ties, then the value of U is the number of times a sample from method 1 is ranked before a sample from method 2. Similarly, the value of U' is the number of times a sample from method 2 is ranked before a sample in method 1. In the presence of ties, this is only approximately true. Using the above formulas, we find that U = 88 and U' = 32. To make the interpretation of U and U' clearer, consider the extreme case where every sample from method 1 is ranked before method 2. In that case, method 1 clearly produces fewer cracks and U' would be zero. If the reverse were true, then U would be zero. In our case, U'=32 is smaller than U=88 indicating that observations from process 1 are usually ranked before process 2. Thus, process 1 appears superior. We must use statistical tests to determine if the difference between the two methods is significant. Now we compare the smaller of U and U' to a critical value. From the Mann-Whitney table (http//www.mathoptions.com/utable.htm) the critical value of 24 (N1=10, N2=12) corresponds to a p-value of 0.02 and the critical value of 34 corresponds to a p-value of 0.1. (If there are ties, then the p-values corresponding to the critical values will only be approximate.) Since 32 is between 24 and 34, the p-value for this data must be between 0.02 and 0.1. Said another way, we are more than 90% confident that the two methods of drawing wire produce a different number of cracks. Since we want a process that produces fewer cracks, we should pick process 1. As in the sign test, we can use an approximation when the sample size is large (N1 and N2 > 9). (As in the case for U and U', this formula for z assumes no ties. The presence of ties requires a small correction. This correction is not important in most situations.) In this case, the z-score is calculated as z = [ |N1(N+1)-2R1| -1]/sqrt(N1*N2(N+1)/3)
where all the terms have been previously defined. In this example, z = 1.81. This corresponds to a probability of 0.035, or a p-value of 2 x 0.035 = 0.07 (93% confidence) (http//www.mathoptions.com/normal.htm). (As noted in the previous Statman article, we must double the probability since this is a two-tailed test.) This is comparable to the prior analysis that had a p-value < 0.1. Again, it appears that process 1 produces fewer cracks than process 2. In summary, we have discussed two different methods for comparing treatments when we don't have bell-shaped data. At this point you may be asking, "If we can use a non-parametric test instead of a parametric one (which must assume a distribution), why not just use the non-parametric test all the time?" It turns out that non-parametric tests are generally less powerful at determining differences between samples. That is, it can be more difficult to tell if the two treatments are different (get a significant p-value) using a non-parametric test. In the t-test, the assumption of bell-shaped data (normality) adds power. So, the moral of the story is that you should use a parametric test like the t-test when you can. But if you can't, use a non-parametric test instead. Many other non-parametric tests exist to help you. If you'd like to hear more just let me know.
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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Copyright 2000 by William D. Kappele, Beth Heffernan, John Raffaldi 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 mailtoBill@MathOptions.com with either "Subscribe E-Math News" or "Unsubscribe E-Math News" as the subject. |
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