Normal Distributions

Normal Distributions - Page 1 of 8 Normal Distributions What makes a normal distribution normal This is just another way of saying we should see

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Normal Distributions Page 1 of 8 What makes a normal distribution normal? This is just another way of saying we should see something that looks like a bell-shaped continuous curve . Each of the curves to the right has that characteristic. If we are talking about heights of men in a certain group, say the Bambutu pygmy tribe in Africa, we are not surprised that the average height of an adult male is about 51 inches. Most adults have a height near that size while a few are somewhat taller or shorter. By contrast, in the Tutsi tribe, also called the Watusi , heights average near a spectacular 7 feet tall! Neither of these populations is normal in height to our way of thinking, but each tribe’s distribution of heights is normal . C Mathematically a curve is normal when it demonstrates a symmetry with scores more concentrated in the middle than in the tails. C Normal curves are defined by two parameters: the mean ( μ ) and the standard deviation ( σ ). C The more normal a curve is the closer the median value 1 is to the mean (or vice versa). They also get very close to the absolute maximum of the normal curve. C Many kinds of behavioral data are approximated well by a normal distribution. Many statistical tests assume a normal distribution. Most of those tests work well even if the distribution is only approximately normal as long as the distribution does not deviate greatly from normality. We should be able to find a mean ( μ ) and a standard deviation ( σ ) so that a normally distributed population can be modeled by or fitted to . 2 2 () 2 2 1 2 x fx e      That function has all the characteristics mentioned before and has become the standard equation for a normal curve . After that, everything we have ever said about a continuous probability distribution holds true. C The area under a normal curve is one. Hence, . 2 2 2 2 1 1 2 x e     C No probability of an event can exceed one and all probabilities must be positive values. Hence, where are both 2 2 2 2 1 0( ) ( ) 1 2 x bb aa Pa x b f x e  ab reasonable outcomes in the experiment. C Since is continuous, the probability of any specific result, say a height of exactly 6 feet, f x doesn’t really come from the function directly. The value describes one of infinitely many 6 x 1 This is the value that splits the data into to halves with 50% above and 50% below. Copyright 2010 - ASU School of Mathematical and Statistical Sciences (Terry Turner)
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Normal Distributions Page 2 of 8 Normal Distribution z - Table Z 0.00 0.01 0.0 0.5000 0.5040 0.1 0.5398 0.5438 0.2 0.5793 0.5832 0.3 0.6179 0.6217 0.4 0.6554 0.6591 0.5 0.6915 0.6950 0.6 0.7257 0.7291 0.7 0.7580 0.7611 0.8 0.7881 0.7910 0.9 0.8159 0.8186 1.0 0.8413 0.8438 1.1 0.8643 0.8665 1.2 0.8849 0.8869 1.3 0.9032 0.9049 1.4 0.9192 0.9207 1.5 0.9332 0.9345 1.6 0.9452 0.9463 1.7 0.9554 0.9564 1.8 0.9641 0.9649 1.9 0.9713 0.9719 2.0 0.9772 0.9778 points. So technically and is infinitesimally small realistically. We should more correctly (6 ) 0 Px  ask what is the probability of a height between say 5.999 feet and 6.001 feet. Then we can use integration to find that probability.
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This note was uploaded on 06/16/2011 for the course MAT 211 taught by Professor Seal during the Summer '08 term at ASU.

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Normal Distributions - Page 1 of 8 Normal Distributions What makes a normal distribution normal This is just another way of saying we should see

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