Math Options Be Bold! Inc. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 4, Number 4 July 2002 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This newsletter is best viewed with a Courier font, size 10. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note - It's Time to Be Bold! Schedule of Public Classes Math in Industry - Which Transformation Should I Make? Software Review - The I-Optimal Design Assistant Family Math - Family Math-Playing Cards and Learning Math Guest Author - Anita Foley: The Power of Perhaps Ask Statman - Why Not Just Plot the Averages? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." Thomas Jefferson ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Reader, The Founding Fathers of the United States were bold men. They were willing to do whatever was necessary to secure the rights of Life, Liberty, and the pursuit of Happiness. They succeeded, and we inherited the gift of their boldness in the United States. Fearfulness is the opposite of boldness. Al Qaeda would like us to be fearful, and they have already killed thousands of American citizens in pursuit of their goal. They do not believe in the rights to Life, Liberty, and the Pursuit of Happiness and they would like to scare us into relinquishing them. Now, as much as at any time in history, we Americans need to be bold. We need to be bold at work. We need to be bold in our daily lives. We need to be bold in our political lives. We cannot trade security and safety for Life, Liberty, and the pursuit of Happiness. Let us boldly pursue the goal to preserve the rights to Life, Liberty, and the pursuit of Happiness. If we become timid, Al Qaeda wins. Bill Kappele. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Sept. 17-18 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. Nov. 4 - 6, 2002 Performing Objective Experiments Bellingham, WA Learn to use the power of DOE in your work. When you leave this workshop you will know how to identify the problem to be solved how interactions among factors affect your results how to optimize your product or process how to make contour plots to show to customers and your management how to measure and report precision in your results how to find the most important factors when experimenting on a tight budget how and when to use different types of designs for efficient, cost-effective, yet sufficiently thorough experiments Nov. 7, 8 2002 Objective Experiments for Mixtures Bellingham, WA and Discrete Factors When you leave this workshop you will know how to design and perform an experiment for mixtures how to design and perform an experiment for discrete factors how to design and perform an experiment for mixtures and process factors how to design and perform an experiment for discrete and process factors how to optimize your product or process how to find the most important mixture components when experimenting on a tight budget how and when to use different types of designs for efficient, cost-effective, yet sufficiently thorough experiments 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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Which Transformation Should I Make? In the last two issues we looked at how to make a transformation and how to know if you need to make a transformation. In this issue we will take a look at how you can tell what transformation to make. Remember from the last issue that you can perform three tests to see if you need to make a transformation. None of these tests is perfect, so if at least 2 out of three indicate the need for a transformation, you will make one. One of these tests not only tells you that you may need a transformation, but also which transformation is likely to work well -- the Box-Cox plot. A Box-Cox Plot plots the log of the average for replicate trials vs. the log of the standard deviation for the same replicate trials. You will need at least 5 such points -- at least 5 of your trials must be duplicated. If you see a straight line with no slope, your standard deviations are not likely to be different for different trials. If you see a rising or falling straight line, your standard deviations may be different. The slope of the rising or falling straight line indicates the transformation you should try. The Box-Cox transformation is your response raised to the power of 1 - the slope of the line. Here's an example. Suppose you are studying plating thickness for printed circuit boards. You make a Box-Cox plot of the log of the mean vs. the log of the standard deviation for 5 sets of 2 replicates each and find that the slope is approximately 1/2. The transformation to try is Thickness ^ (1 - 1/2) = Thickness ^ (1/2) = Sqrt (Thickness) So you will try taking the square root of each response and using this as your transformed response. How will you know if the transformation worked? Repeat the three tests from the last article on your transformed data. If you now pass at least 2 out of 3, your transformation worked. If not, you may need to collect some more replicate data to get a better estimate of the slope. Here is a table of common transformations: Slope Transformation -1 Y^2 = Y*Y 0 none 1/2 square root (Y) 1 log (Y) 2 1/Y It is usually a good idea to round your slope to make your transformation a common function. There is no mathematical reason to do this, but most people you explain your work to will be more comfortable with, say, the log of Y than Y raised to the 0.00123645 power! As a final note, a transformation is a property of a response. Once you determine the correct transformation, this transformation will always be required for this response when it is measured in the same way. So if you measure a particular response frequently in your work, you can determine the required transformation once and use it from then on. Good experimenting! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The I-Optimal Design Assistant I-Optimal designs provide many benefits to industrial experimenters. They are very flexible, allowing you to perform more complex experiments than you can perform with classical designs and they only require a few more trials to fit a quadratic model than an interaction model. Unfortunately, most DOE software manufacturers do not provide I-Optimal designs in their software. Currently only Process Builder offers I-Optimal designs in the software, STRATEGY. These are called Hardin-Sloane designs, and are only available for process factors. The I-Optimal Design Assistant provides you with easy access to a thorough library of I-Optimal designs. With it you can design experiments for combinations of factor types, such as process, mixture, and discrete factors in the same experiment. Designs are also available that are protected against the loss of any trial -- so you can lose an experimental trial entirely and still rest assured that you will be able to analyze your results! Generating a design is straightforward -- select the numbers and types of factors to find a design, then enter your factor names and high and low levels to generate your design. You can also add replicates and save your design for import into a spreadsheet, STATISTICA, or STRATEGY. Best of all, this software costs less than one custom design! And it comes in versions for Windows and Solaris operating systems. The I-Optimal Design Assistant does not analyze data for you. You collect data using the designs it provides, then you analyze your data using a commercial DOE software such as STATISTICA or STRATEGY. You can learn more about this tool at http://www.mathoptions.com/i-optima2.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can purchase the I-Optimal Design Assistant at https://secure1.selfnet.com/SSLx01/ You can purchase STATISTICA 6 and receive a free copy of the I-Optimal Design Assistant at the same address. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math-Playing Cards and Learning Math by Beth Heffernan Do your children like to play cards? With the long days of summer in front of us, card games can provide an unobtrusive look at several principles of math. At the simplest level, cards teach children to sort and categorize. First red is sorted from black, then numbers from face cards. Later they learn the order and value of the number and face cards. With a small child who shows interest, say, "Find all the threes (or Jacks or whatever)", or "Put all the reds in one pile." Later a game called Battle provides hours of entertainment. Two players split a deck, each playing one card at a time against the other's card, and the higher value takes the match. If the cards tie, there is a rematch. The winner takes all the cards, ensuring that the game lasts a very long time. My children loved to play Concentration. Gauging your child's ability, choose from five to all the pairs in the deck and lay them out in orderly rows, upside down. Each player turns over two cards, and, if they match, takes the pair. A match gives that player another turn. If they don't match, turn both cards over and it is the other person's turn. Play till all cards are matched. The player with the most pairs wins. This is easy for an adult, so make a few mistakes (with loud and silly exclamations of grief) and stretch out the game. It is highly challenging for a child's memory; it can also offer an opportunity for introducing a little strategy into their playing as well. Parents offer this game with pictures first, but the cards add the quality of making numbers familiar and fun, not scary. Your older child is ready to play 21, with a little betting if it's OK with you. The dealer gives each player two cards. An ace with a face card or ten is an automatic win (=21). Otherwise each player adds the value of the cards in their hand, and tries to get as close to or equal to 21 without going over 21. All face cards have a value of ten. If the player wants another card(s), he or she tells the dealer and receives one or more cards. The obvious math lesson here is adding the values of the cards in the players' heads. The more subtle value for older players is calculating the odds or probability of hitting 21 without exceeding it. Playing 21 also gives you a great chance to teach your child about the dangers of gambling, namely that the odds are stacked in favor of the house. Play long enough, and the child will see that winning is fleeting, and losing is the ultimate fate! If you or your family love another card game, look at it from the Family Math point of view and see what your child(ren) can learn about sorting, categorizing, greater or less than comparisons, probability and strategy. Please let us know what you come up with (mailto:Beth@mathoptions.com) and we will share this with other families in a future issue. Children love to play games with their parents, and will have so much fun they won't even realize you've sneaked in the M word again. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Beth Heffernan is Vice President of Math Options. You can reach her at mailto:Beth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Guest Author: Anita Foley ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Power of Perhaps by Anita Foley http://www.wealth-happens.com Do you find that every time you think about making a change in your life you sabotage yourself by thinking it is unlikely that you'll be able to do it? Do all the negative, self-limiting thoughts come to the forefront and stop you in your tracks? You must learn to change these self-imposed limitations that are preventing you from reaching your potential. An unlikelihood can become a possibility, which can lead to a probability, if you use the power of PERHAPS. You can use this simple word to change your negative beliefs into possibilities; and that is the first step toward changing them into probabilities. Remember when you were a kid and you asked your parents if you could have something? If their answer was "No, you can't," you knew there wasn't much chance you'd get it. But, if their answer was, "Perhaps, we'll see," it usually meant you'd be able to convince them. The word PERHAPS left the door open to negotiation; it meant there was a possibility that you'd get what you'd asked for. All you had to do was be persistent from that point on and you would probably get what you wanted. There's POWER in the word PERHAPS. You can use the power of PERHAPS today, also, just like when you were a kid. How? By using the word PERHAPS in place of the words I CAN'T and I WON'T. Let's say, for example, you have a belief that you can't do math. Either you did poorly in the subject when you were in school (internal belief from past experience) or someone told you that you were not good in math (external belief). What would happen if you simply changed "I can't do math" to "PERHAPS I can do math"? The word PERHAPS opens up the belief to other possibilities. It allows for some action to be taken that could result in a change in the belief. "PERHAPS I can do math if I . . ." - get a tutor - take a class - study harder - concentrate more - buy some math software - work with numbers more often - read a math textbook Instead of just giving in to a negative belief, you are open to taking some ACTION that will help you change the belief. "I CAN'T do math" can become "PERHAPS I can do math if I take some action" and then become "I CAN probably do math". Likewise, "I WON'T make money with an online business" becomes "PERHAPS I can make money with an online business if I start one and work at it" to "I CAN probably make money with an online business." Do you see how changing that one word will change the whole feeling of the belief? Use the word PERHAPS to change your negative beliefs to possibilities that invite ACTION and, ultimately, to positive beliefs and probabilities. Think about the potential outcome for your changed beliefs. What are your possibilities? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Anita Foley publishes Wealth Happens E-zine. Inspiration, motivation, and information for the serious netpreneur. Visit online at: http://www.wealth-happens.com or subscribe: mailto:WH-Ezine@InfoGeneratorPro.com title: The Power of Perhaps website: http://www.wealth-happens.com This article provided by the Family Content Archives at: http://www.Family-Content.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/averagevsind.doc with the full text and graphics. Dear Statman: Is there any problem with plotting just the averages instead of all the data when performing a regression analysis? Signed, Your Average Joe Dear Joe- Good question. The answer is that it does matter if you plot all the data or just the averages when you perform a regression. It turns out that you are usually better off doing your analysis with all the data. Let me tell you why. First, some of you out there might not know what a regression is. A regression is a type of analysis that fits a straight line to your data. It uses a technique called "least squares". The method is called least squares because it finds the equation of the line that minimizes the squared distance between the fitted line and the data. You don't really need to know that because your software does that for you. Let's consider a fictitious data set for %transmission of a particular wavelength of light thru a liquid solution containing various weight percentages of some component we'll call "X". Table I gives the data. Table I. Fictitious data set Observation No. Weight %X %Transmission Avg. %transmission 1 6 25.98 28.28 2 6 29.16 3 6 29.10 4 6 28.74 5 6 28.37 6 8 36.36 36.16 7 8 33.10 8 8 34.11 9 8 36.83 10 8 35.41 11 10 41.09 41.45 12 10 43.11 13 10 39.97 14 10 42.27 15 10 40.83 How can we plot this data? Well, one way would be to plot %transmission vs. weight %X for all the data. This plot is shown in Figure 1. Another way would be to plot the average %transmission vs. weight %X. This is shown in Figure 2. It turns out that when the number of observations for each weight %X is the same, the slope and intercept you calculate for the lines in both plots is the same. (If the number of observations for each level of %X is different, then the least squares coefficients calculated by using the averages will be different than those calculated by the individual data points. The reason why could be discussed in a future Statman article.) If the equations for the lines are the same, why would plotting all the data be preferred to plotting the averages? The answer is evident when you want to make predictions. Suppose we are interested in predicting the %transmission when the weight %X is 24. Figures 1 and 2 also show the "true" value of %transmission for weight% X = 24. I say "true" value because the data are simulated, i.e. I get to peak behind the curtain and know exactly what the true value is supposed to be. When you run an experiment in the lab, you don't get to do that. You can see from the plots that both methods predict the "true" value pretty well. Again, what distinguishes the two methods? Suppose we ran the same experiment a second time. Would we get identical values of %transmission? Probably not. If we did, we would likely think something was wrong. Suppose we run the same experiment 25 times and plot the regression lines for each experiment (see Figure 3). Clearly, the lines are not all the same. There is some variation. What we want to do is take that variation into account when predicting the "true" value of %transmission at weight %X = 24. We do that by constructing a confidence interval. (The confidence intervals discussed here are for the mean value of %transmission at %X = 24 with a confidence level of 95%.) What is a confidence interval? It is a range that encompasses the "true" value of the response (%transmission) most of the time. When you actually do the experiment, either your confidence interval does or does not bracket the true value. You don't know for sure. However, you do know that if you were to do the experiment again and again and form the confidence interval (CI) the same way, then some high percentage of the time (like 95%) you would capture the true mean. A narrow confidence interval means the quality of our predictions is high. The formula for the confidence interval is a little complicated so I won't show it to you here. However, your software should give you the estimates you need. Figures 4 and 5 show similar plots to Figures 1 and 2, but this time 95% confidence intervals are added (shown as dashed lines). Do you see that the confidence interval for the averaged data is actually narrower than that for the individual data? What is going on here? I have been saying that it is better to use individual data but the CI for the averages is smaller. Isn't this the opposite trend of what I have been saying? Yes, but?. It turns out that in this case we were unlucky. The estimated confidence interval for the averages is less stable than that of the individual data. That is, the CIs bounce around a lot from experiment to experiment. Look at Figures 6 and 7. These graphs are like Figures 5 and 6 for a similar set of simulated data. Now the confidence intervals for the averages are much wider than that for the individual data. I ran this simulation many times and found that on the average, the confidence intervals for the averaged data are wider than that of the individual data. Further, it can be shown theoretically that using the averaged data will usually produce wider intervals. For a particular set of data, is it OK to use the method that produces the narrower interval? No, you ought to pick a method before you do the analysis and stick to it. That is only fair. Since using all the data usually produces more narrow limits, that is the method that should be used. Both methods use all the data, just in a different way. Why is it that the confidence intervals for the average are not as stable as those for the individual data? Well, while both methods do use all the data, the "average" method has only three data points, as opposed to 15 for the individual data. It turns out that the number of data points is important in determining the width of the confidence interval. The more data you have, the tighter (better) the confidence interval. So, the moral of the story is: always use all the data when running the regression. It will give your predictions more confidence! 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 2002 by William D. Kappele, Beth Heffernan, Anita Foley, 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, 98229 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.