~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 3, Number 6 November 2001 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes Math in Everyday Life - Social Security Math in Industry - Finding Out Your Measurement Error Family Math - Math Head Games Book Review - Measurement System Analysis 2nd edition Ask Statman - Reliability, Confidence, and Sample Size ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "The principle of science, the definition, almost, is the following: The Test of All Knowledge is Experiment. Experiment is the sole judge of scientific 'truth'." Richard Feynman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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. September 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 learn more about these classes and register to attend at http://www.mathoptions.com/public.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 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. The Type I error is decided upon and the 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. This is because we don't know which measurement point is making the measurement distribution we just measured. The following figure shows the case where a measurement distribution overlaps the LSL. 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 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, store the data, and the study results. Some people hesitate purchasing the 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 by downloading it from the Math Options web site and seeing if it helps them with their 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, 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 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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math Head Games by Beth Heffernan Do you play with numbers in your head? Some people enjoy looking for the symmetry in numbers or the patterns in numbers. Others work with the way numbers can be rearranged and manipulated. For example, when our daughter has trouble with 9 x 9, we ask her "What is 9x10?" That's an easy one. Now subtract 9 from that number and you have the answer to 9 x 9. Repeatedly we look for ways to show her that multiplication is a fast shortcut to addition. "What is 2+2+2+2+2?" It is the same as 2 X 5. No-one ever tells children this in school. They delight in the idea of a shortcut. There are lots of tricks to make multiplication easier that are not taught in school. Anything multiplied by two will be an even number. Actually, anything multiplied by an even number will be an even number. Numbers multiplied by ten must end in zero. What seems elementary to us is not obvious to our children, and these "tricks" add to their personal math power. Division is a breeze with certain tricks. Similarly, finding the least common denominator is a lot easier this way. If it's an even number, divide by two. If the digits of a number add up to a number divisible by three, then the original number is divisible by three (i.e. 564? 5+6+4=15, 1+5 =6, this is divisible by three). All numbers with digits adding to nine are divisible by nine. If a number ends in 5 or zero, it is divisible by five. Cut to the dining room table with the child wailing loudly "I can't factor this number, Mom (Dad)!!" "Beloved child, if you always follow this sequence you can factor anything. Begin with two. If your problem number is even, divide by two. Continue till the number is odd. Add the digits. Is it divisible by three? Yes, divide by three. No, try five. Does the problem number end in five or zero? Divide by five. After this, most numbers are factored, but try the next four or five prime number just to be sure (7,11,13,17 and 19). If this doesn't find the least common denominator, your child has a very mean teacher. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Beth Heffernan is Vice President of Math Options. You can reach her at mailto:Beth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Book Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Measurement System Analysis "Measurement System Analysis" is one of the most valuable references you can have for Gage Repeatability and Reproducibility studies. This manual, developed by Chrysler, Ford, and General Motors, details every step of the Gage R&R process, from developing a study through collecting data to various methods for analyzing your data. The presentation is clear and accompanied by worked example. You won't want to miss Section 5, "Gage Performance Curve." This curve is a billiant way to communicate the results of a study in in a visual way that everyone can understand, and this book shows you exactly how to create it. If you make measurements in your work, you can't afford to be without this manual. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can purchase a copy of "Measurement System Analysis," from the AIAG (Automotive Industry Action Group) by calling (810) 358-3570. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ask Statman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Written by Dr. Charles Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Statman- What is the relationship between reliability, confidence and sample size? Signed Dawn Dear Dawn- That is a very good question. It turns out that all these terms are related. Let’s take the first: reliability. The reliability of a part (ball bearing, capacitor, toaster oven, etc.) is defined as the probability of it surviving to a given time or later. So, if you have 20 capacitors on test and after 100 hours 6 of them have failed, then the reliability, R, at 100 hours can be approximated as 14/20 = 70%. Why do I say approximated? That’s because if we did the experiment again with another 20 capacitors, we might get a different number (maybe only 4 would fail, etc.). So, any experiment to find the reliability gives only an estimate of the reliability as opposed to an absolute number. From an engineering perspective, we might say that if we tested another capacitor, we are 70% confident that it will survive to 100 hours. However, in statistics, confidence means something else. Statisticians use confidence intervals to form a bracket around the estimate. An underlying assumption in calculating a confidence interval (CI) is that there is a “true” reliability. The true reliability would be estimated by testing thousands or millions of capacitors. In real life, finding the exact reliability might be difficult to do. Instead, we compromise by estimating the reliability and finding a confidence interval that will likely enclose the true value. In order to find the CI, we need to know the distribution of our estimate of R. Said another way, if we performed the same experiment again and again and got several values of R, these different values would pile up around the true value. This pile of data is a distribution. It turns out that for large samples (20 or more), the distribution of the estimated R is approximately bell shaped (it has a normal distribution). All we have to do is find the spread (standard deviation or s) in the estimated R. The derivation for the expression for s is somewhat complicated. Therefore, I will simply state it. Assuming none of the data are censored (removed before the failure time of interest), the expression for s is given by: s = sqrt{(R(100) x [1 - R(100)])/n} where R(100) is the reliability at 100 hours and n is the sample size. Of course, if we were interested in finding s at a different time, say 50 hours, we would substitute R(50) for R(100). From the formula, we see that the value of s depends on the number of hours and on the sample size. The larger the sample size, n, the smaller s is. Since the estimate of R has a normal distribution, we expect that 95% of the time, the true R will be within about 2 standard deviations. Further, we expect that 99.7% of the time, the true R will be within 3 standard deviations. Taking the estimated R and subtracting 2s forms the lower 95% confidence limit. The upper 95% confidence limit is found by adding 2s to the estimated R. Taken together, the lower and upper limits form the confidence interval. Let’s do an example. Suppose I test 20 capacitors and 5 of them fail at the following times (in hours): 9, 58, 61, 72, 92 The other 15 survived beyond 100 hours. So, my estimated R(100) is 15/20 = 75%. The estimated s is given by: s = sqrt{(0.75 x [1 - 0.75])/20} = sqrt{(0.75 x 0.25)/20} = 0.097 So, the 95% confidence interval is just 0.75 +/- 2*0.097 = [0.56, 0.94]. Similarly, the 99.7% CI is 0.75 +/- 0.29 = [0.46, 1.04]. Note that increased confidence comes at the cost of an increase in the width of the CI. It turns out that the actual value of R was 0.78. Thus, the CI’s were able to bracket the true R. If the sample size were larger (say 80), the value of s would have been smaller by a factor of 2. Thus, the confidence intervals would have been narrower. We were able to bracket the true value of R this time. However, if we did this experiment over and over and constructed confidence intervals the same way, we would find that on the average we bracket the actual R 95% or 99.7% of the time, depending on the confidence level chosen. Since this was simulated data, we could check and see if the interval contained the true R or not. In real life we never know the true R. The best we can do is to calculate an interval in which we have some confidence. Of course, there are fancier ways of forming confidence intervals for life data. The technique I have described here is a little crude, but good enough in most situations. Thus, we see that reliability, confidence and sample size are related. The estimated reliability and its associated confidence are calculated from the data. The sample size influences the size of the confidence interval. The larger the sample (i.e. the more time and money you are willing to spend) the tighter the interval created. Keep those questions coming! 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 2000 by William D. Kappele, 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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