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Stat6_Normal_Curve.doc College Math Natural Phenomena many times result in "Normal" Curves Line Graph Ch 5 - The Normal Curve 55 46 37 28 19 10 1 Prob. of x Heads Probabilities 0.6 Many phenomena, especially natural phenomena, result in 0.4 probability histograms that are bell shaped, symmetric, and Series1 would "fit" the equation for a "normal distribution". By 0.2 repeatedly decreasing the class sizes of a histogram, and 0 then connecting the midpoints of the pedestals, we can obtain (in some cases) curves that are very close to the Lengths of Gobies normal distribution. Centimeters The example graphs below and Microsoft Excel 97 Line Graph to the right show the results of Graph A tossing n = 6 to 200 coins with the relative frequency of the The sum of the areas of the rectangles is one. number of heads as the tops of The area under the curve is close to one. As the numbers of rectangles increase, the area under the pedestals. These relative .2 the curve gets closer and closer to one. We can frequencies are, as usual, the then think of probabilities as the same as the empirical probabilities that a areas under the curve. certain number of heads will result from n tosses of the coin. In the lower right examples, only the tops of the pedestals were 0 1 2 3 4 5 6->x graphed (as little boxes for simplicity) and a smooth curve was graphed through them. In all of the graphs, the "pedestals" would have widths of one (1). The heights of the pedestals (boxes) are the probabilities P(x) - see graph A to the right. When we use the formula A = W x L for the area of The experiments below consist of flipping increasing numbers of coins and determining the probabilities of a rectangle to calculate getting 1, 2, 3, etc. heads. As the number of flips increases, the curve becomes "more" normal. the areas, with W = 1 and L = P(x), each rectangle then has an area equal to the probability of the class. Adding all of the areas will then give us one The vertical scales (probabilities) vary from box to box (as do the "Windows"). (1), the sum of the The horizontal scales represent the number of heads…from 0 to n All the areas are 1.0. probabilities (which is always one (1) for the sums of all the probabilities for a sample space). In this manner, we associate the areas of the rectangles with the probabilities of the numbers of heads. Then we can find any probability geometrically by adding the appropriate areas and we can consider the area as equal (or synonymous) to the probability. The Normal distributions at the left are both normal. They have been superimposed on each other. They are both bell-shaped but have different means and standard deviations. The wide one has a standard deviation of 5. The narrow one is a STANDARD NORMAL DISTRIBUTION. It has a mean of 0 and a standard deviation of 1. The areas between one standard deviation on either side of the means are .6827 for both of them. Because of the complexity of calculating the areas (and hence probabilities) under the curves, we "standardize" all normal distributions with a simple formula, and then use one table to look up the probabilities. For the same number of standard deviations, the areas will be the same for all normal distributions. Total area under any probability curve is 1, regardless of the shape. Page 1 of 1D:\My Documents 2000\Word\Math\xReference\LA_Ref\Stat6_Normal_Curve.doc Created on 7/19/98 8:53 AM Last printed 3/29/02 7:00 AM R Mower, Instructor