Normal Distribution CurveDate: 11/13/2000 at 10:17:09 From: Jaime Egars (Mrs. Watkins teacher) Subject: Normal Distribution Curve Dear Dr. Math, I am a seventh grader at St. James Middle School, part of Horry County Public Schools, and my question has to do with the normal distribution curve: When and where did it come into being and what are some of the applications? Date: 11/13/2000 at 10:59:37 From: Doctor Mitteldorf Subject: Re: Normal Distribution Curve Dear Jaime, A distribution curve is like a histogram where the bars get so close together that each one looks like a line rather than like a bar. You could make a histogram of all the heights of the girls in your class: How many girls are between 4'6" and 4'8", how many between 4'8" and 4'10", etc. If you look at this bar graph when you're done, it will look like a skyline, in which the tallest building is in the middle, and shorter and shorter buildings are around it. If you imagine doing the same curve for all the 7th grade girls in the United States, you could make the "width" of each bar much smaller: say, all the girls between 4'6" and 4'6.01", etc. Then you'd have a few thousand different bars, and to plot them all on the same page, you'd have to make each one a thin line. The tops of all the lines together would make a smooth curve, again with the highest point in the middle. This is the distribution curve for heights of 7th grade girls in the country. In fact, the curve you drew would look approximately like the bell-shaped normal distribution curve that is becoming familiar to you now as you study it. You can take this as an experimental fact about the world. No one can prove it, but it seems to be true that a lot of biological measures like this are approximately normally distributed over a given population. Now, there's a theorem from pure math that hints why the normal curve is so commonly seen in this and other areas. It's not a proof that heights of 7th grade girls are normally distributed - nothing of the sort. It's just an abstract mathematical theorem that seems to have something to do with a lot of real life situations. The theorem says, roughly, that if you have very many factors contributing to an outcome, and each factor contributes in the same way, then it doesn't matter how the factors are distributed: the net result will be a normal distribution. The classic example of an application of the theorem is this: Experiment A is to flip 4 coins and count how many heads you get. If you repeat Experiment A thousands of times and count how many times you get 0 heads, how many times you get 1 head, etc., and make a bar graph, the bar graph will have a nice symmetric shape, with the highest bar in the middle, at 2 heads. (There are only 5 bars in the bar graph, for 0 heads through 4 heads.) Experiment B is to flip 100 coins and count how many heads. Now there are 101 possible outcomes, and the bar graph starts to look smoother. Still, there's a tallest bar in the middle at 50 heads, flanked on either side by smaller bars. You can start to see the bell shape in the result. Remember that to collect this data you have to repeat the entire experiment very many times. Each experiment consists of 100 flips, and you have to do the 100 flips thousands of times in order to get a fair sample of the shape of the histogram. Experiment Z is to flip 100,000 coins. Repeat experiment Z many thousands of times, and make a histogram as you did before. The bars will be very, very close together now. The bars to the left of 48,000 and to the right of 52,000 will be completely empty, because it's so unlikely that you will get a result that departs very much from 50,000 heads. But the area between 48,000 heads and 52,000 heads still has 4,000 bars in the bar graph, and it will look like a very nice outline of the classic normal distribution, or bell-shaped curve. I don't recommend that you spend your day sitting at home flipping enough coins to do experiment Z. But on a computer, the equivalent experiment is quite feasible, and the programming job should be not too hard for anyone who knows a language like Basic or Pascal or C. LOGO will also work for this. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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