Math Options Be Bold! Inc. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 4, Number 2 March 2002 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This newsletter is best viewed with a Courier font, size 10. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes Math in Industry - Transformations Web Site Review - WebDOE Family Math - Another Way to Look at the "Right Answer" Guest Author - Bob McElwain: What's it Cost to Start an Online Business? Ask Statman - Where Does the Word "Regression" Come From? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "I don't have time to sharpen the saw ... I'm too busy sawing!" Stephen Covey ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Mar. 21, 22 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 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 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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Transformations Design Of Experiments (DOE) assumes that the standard deviation is the same for all possible trials of interest. So you should measure the same standard deviation at high pressure with high temperature as you do at low pressure with low temperature. When you discover that this assumption is wrong, what can you do? You can make a transformation. A transformation is a mathematical operation, like square root or log, that makes the standard deviation the same for all of your experimental trials. So while hardness may have different standard deviations at high pressure-high temperature and low pressure-low temperature, log hardness will not. Let's take a look at an example. Suppose you are studying the hardness of a plastic. You collect the data below for two of the experimental trials: Binder Level Plasticizer Level Hardness High High 7.29 High High 7.89 High High 6.62 Average 7.27 Std. Dev. 0.64 Low Low 3.18 Low Low 3.84 Low Low 3.67 Average 3.56 Std. Dev. 0.34 The standard deviation is actually lower at the low-low trial than at the high-high trial. Now let's make a transformation. We'll take the log (base 10) of each response. Binder Level Plasticizer Level log(Hardness) High High 0.863 High High 0.897 High High 0.821 Average 0.860 Std. Dev. 0.038 Low Low 0.503 Low Low 0.584 Low Low 0.565 Average 0.551 Std. Dev. 0.042 The standard deviation for the transformed response is now actually the same for both trials. (The numbers look slightly different due to measurement noise.) Finding the right transformation will be the subject of a future article, but it is important to note that a transformation is always the same for the same response measured in the same way. So now that we know the log transformation works well for hardness, as we have measured it, we will know to make this transformation in future experiments. In the next issue you will learn how to know when a transformation is necessary. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Web Site Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ WebDOE Would you like to try DOE without buying any software? A new tool has appeared on the web to help you meet your goal. WebDOE is a unique web site that lets you design your own 1, 2, or 3-factor experiment and analyze it over the Internet. You can can choose from classic designs, or create your own I-Optimal design. You will receive many measures of the quality of your newly created design and the opportunity to download it to your computer. After you have collected your data, you can return to WebDOE to analyze your data. How much will this cost you? Not a cent! The current functions are free. In the future this site should allow you to create your own I-Optimal designs for more than three factors for a modest fee. WebDOE is a service of the Crary Group and can be found at http://www.WebDOE.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Another Way to Look at the "Right Answer" by Beth Heffernan Children are accustomed to looking to an outside source for the right answer to their mathematical problems, usually the teacher or parents or an answer key. This effectively dilutes their confidence in their own independent reasoning. In her book "A Collection of Math Lessons from Grades 3 through 6," Marilyn Burns suggests a class lesson where the answer is not as important as is the students' logic in arriving at that answer. You can try this with your child or children at home or give it to a responsive teacher as a suggestion for a stimulating mathematical discussion. Here is the problem: A man buys a horse for $50. He sells it for $60. Then he buys the horse for $70. He sold it again for $80. What is the financial outcome of these transactions (not including food, boarding, etc.)? Do not tell the children the answer. Have each participant offer their conclusion, till no different ones emerge. Divide the children into groups of those who agree on any given answer. Have each individual defend their conclusion, and make sure all children know that, if they change their mind due to the persuasiveness of an individual or group, they can join that new group. Burns' experience is that children will become quite involved in the discussion when they perceive that their reasoning processes are valued, and that there is no stigma attached to changing their mind. An additional value in a class setting is the experience of how difficult it is to listen to another's conclusions when the child is vested in their own reasoning. Burns recommends acting out the horse scenario to finally reveal the "right answer". She gives "the man" $100 in play money and has this child buy and sell "the horse". What is the Right Answer?? Why do you think so?? Marilyn Burns has many books with a fun and different approach to math for children. There are "A Collection of Math Lessons" books for grades 1-3, 3-6, and 6-8. She even has a book about probability for Grades 3 and 4!. Marilyn Burns' books are available at Amazon.com-use our Math Options link for easy browsing and buying. P.S. The answer is Gained $20. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can browse Marilyn Burns' books at Amazon.com using this link: http://www.amazon.com/exec/obidos/tg/stores/static/-/books/search/ref=b_tnbhbo/002-6641335-7956804 Beth Heffernan is Vice President of Math Options. You can reach her at mailto:Beth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Guest Author: Bob McElwain ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ What's it Cost to Start an Online Business? It's all a function of that extremely precious commodity called time. When the alternator in your car quits, you can fix it yourself or turn to a mechanic. Working the Web is no different in this regard. Doing it yourself saves bucks, but may not be cost-effective. And it can be a serious mistake if you lack required skills. If you want your site to become a significant source of income, judicious use of time is mandatory. No one person can do it all. And what you need but don't have time to do, will cost. Going Into Business If you are starting a new business, you must file a DBA (Doing Business As statement) or the equivalent in the county in which you will work. After filing, it needs to be published, then you need to open a bank account. Costs vary, but the minimum is about $50. Also consider any state or local licenses required. If you need an accountant, costs go up. Turn to an attorney, and they may skyrocket. But you may need to consider these options because of the products or services you will market, just as in an offline business. You may need to consider liability insurance. Incorporation may provide even more protection. HTML vs Web Page Editors You must understand the basics of HTML, the language in which web pages are written. There is a time cost here. But at some point, most will find it more effective to turn to a web page editor to save time. Costs range from about $50 to $200. Building Your Site Hiring someone to put a site together can cost thousands of dollars. More important, you may find making changes later brings significant added cost. It is best to build your own pages, for then you have total control. But the template used throughout the site is so critical to success, consider hiring an artist to get it right. Not the site, just the basic page template. Once the site is established, it can be very cost-effective to hire out the creation of new pages and updating. A good page template with original art work can run anywhere from $200 on up, but $500 should cover even special needs. Free vs Paid Hosting Services There is only one option. You must have your own domain name ($35/year from Internet Solutions) and a good hosting service. I use both Pair.Com and JumpLine.Com. While there are other fine services available, these two offer attractive entry level pricing. $5.95/month will buy ample resources at Pair.Com provided you do not need cgi initially. If you do, JumpLine.Com at $14.95/month may be the better choice. In both cases, you can save a bit by paying a year in advance. If you know how to build your own forms, do so. Many, however, will prefer a shopping cart service. Americart is very good, and is available through Pair.Com at $15/month, but anyone can use it at $21/month. JumpLine.Com offers a shopping cart service as part of their package at $24.95/month. However, it is limited. If you can live with the format available, JumpLine may be the best choice. Note forms or shopping carts only take the orders. You will need a merchant account to deal with credit cards. Set up fees run from about $300 on up. If you need online processing, add a similar amount. Opening An Office While getting started, you will likely keep your present job, and it may make sense to work from your home. Even so, you still need an "office," including stationery, invoices, business cards, and possibly brochures to be handed out wherever you happen to be. Costs here are the same as in an offline business, and will be a function of your needs. Don't overlook software. If you want to do some of the graphics for your site, Paint Shop Pro at $99 is a good value. For your accounting, Quicken is good. For mail list handling and personalized mailing, including emailing, Easy Mail Plus at $50 is an excellent choice. Then there are other things, such as supplies. Printers chew up enormous gobs of paper. For competitive prices on consumables, try Office Depot. Call 800-463-3768 for a free catalog. A Phone Is A Must An email address is not enough. You need a phone and someone to answer it. Even if you expect to receive few calls, this is a must. People often call just to see if you're for real. If there's no phone, you've lost a sale. Some argue that voice mail is a reasonable alternative, but it will not help if you can not get back quickly. If you have a spouse who can answer, go for it. If not, find someone in your area who can take calls as your secretary. If you provide up-to-date information about your business, your "secretary" can often save you the need for a later reply. Further, there is simply no less expensive way to appear to be working the business full time. Where We're At The above is not the whole of it. For example, there has been no mention of search engines, yet good positioning can make a big difference. Again, if you know how to optimize pages and can do so easily, it is worth your time to do so. If not, hire it out. Writing skills are important. If yours aren't so hot, factor in some cost for editing, or even creating both page content and the advertising message behind all. Good service is available at $25-$50/hour. You must add up these costs relative to your particular needs. But it is unlikely you can start a serious online business for less than $500 to $1000, even if you do all the work yourself. Time Cost Analysis Starting any business means commitments in time you can not expect to recover except over the long run. So good cost analysis is difficult initially. Even so, put a dollar value on your time, perhaps as low as $5/hour, to help you make good decisions about how you will use your time. Even at $5/hour, it will be clear that some things should be hired out. Building web pages with HTML when you could be generating leads may not be the best use of time. Reading a book or two about how to work the Web can be very helpful, but sometimes it's more cost effective to buy the information needed. Working harder is often the only option available. But when possible, work smarter, which often means hiring services. In the end you'll have more fun and rake in greater profits sooner. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Bob McElwain Want to build a winning site? Improve one you already have? Fix one that's busted? Get ANSWERS. Subscribe to "STAT News" now! mailto:join-stat@lyris.dundee.net Web marketing and consulting since 1993 Site: Phone: 209-742-6349 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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.regression.doc with the full text and graphics. Dear Statman- I have often heard the word "regression" used when people talk about fitting data to a straight line. Where does that word come from? Signed, Curious Dear Curious- The term regression has its origin in the 19th century with the British anthropologist Sir Francis Galton. He looked at the relationship between the sizes of parents and their children. This he did this for people, sweet peas, etc. He found something interesting. From his analyses, it appeared that the size of children tended to be more towards the overall average of the population than that of the parents. For example, suppose two tall parents had children. Their offspring would tend to be taller than average, but shorter than the parents. Similarly, short parents would tend to have short children, but they would tend to be a little taller than the parents. Thus, it appeared that offspring were "regressing to the mean" or average. The implication was that with each successive generation, people would become closer and closer to the same height (or sweet peas would eventually all have the same diameter). It turns out that Galton's analysis was flawed and we can learn a little by examining why. Let's look at Galton's data on heights of people. The data are graphed in Figure 1. Here, the average children's height is on the y-axis and the average parents' height is on the x-axis. It appears that the data are reasonably well approximated by a straight line, as demonstrated by the fitted line. Also included in the plot is a horizontal dashed line showing the overall average height of the children (68.2 inches). Notice that when the parental height is small, so is the average height of their offspring and when the parental height is large, the offspring are also tall. Further, these data seem to support the idea of regression toward the mean. For example, in one case the average parental height was 73.6 inches, while the average height of the children was 72.2 inches. So, while the children are tall, they are closer to the average (mean) height of 68.2 inches than the parents. In another case, the parent's average height was 63.6 inches, while the children's average height was 65 inches. So, while the children were short, they were closer to the overall average than the parents. Several important points need to be made. First, consider the slope of the line of offspring vs. parent height. If the slope of the line were 1, then the predicted height of the children would be the same as that of the parents (except for a constant). Thus, for every 5 inches increase in the parents' size, there should be a corresponding increase in the children's size. Now, if the offspring's size really was "regressing", then the slope of the line should be less than 1. This is because the fitted line should be tilted toward the overall average (dashed line in Figure 1). The actual slope turns out to be 0.61, in agreement with our hypothesis. What about the opposite case? Suppose we tried to predict the size of the parents from the size of the offspring? Then we would have the parents' height on the y-axis and the children's height on the x-axis. If regression were a real phenomenon, then the slope of the line should be greater than 1. That's because the average size of the parents should be tilted away from the overall average. Figure 2 shows the data with the axes switched. Here, the slope is only 0.29! Thus, our hypothesis is not supported by the data. What is going on? It turns out that what Galton discovered was an artifact of how the line is fitted to the data, rather than anything special in the data itself. When a regression is performed, it means that a line is fit using the method of least squares. Figure 1 shows a "residual". The residual is the distance from the data point to the fitted line. The method of least squares finds the equation of the line that minimizes the sum of the squares of residuals for all the data. Note that the residuals are a vertical distance. So, the fitted lines in Figures 1 and 2 are solutions to two different problems since the vertical distances for the two cases are different. Thus, the X and Y data are not symmetric. You cannot find the slope for the Figure 2 data by taking the reciprocal of the slope for the Figure 1 data. You need to solve each problem independently. This fact comes from considering the vertical distance. Another important point to make is that Galton considered the average size of the offspring. Having read many Statman articles before, you might remember that data have variation so the average is only half the story. The heights of the offspring have a lot of variation around that average, about the same as the parents. It turns out that the overall distribution of the parents' heights is about the same as that of the children. So, no "regression" is actually occurring. I hope this answers your question. 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, Bob McElwain, 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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