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Unformatted text preview: Chapter 6 The Normal Distribution The Gaussian distribution, which we have mentioned a number of times already, is perhaps the most common and most important of the continuous distributions treated in statistics. The real reason for the importance of the bellshaped normal curve lies in the central limit theory which states that the sum of a large number of independent random variables will be approximately normally distributed almost regardless of their individual distributions; any random variable which can be regarded as the sum of a large number of small, independent contributions is thus likely to follow the normal distribution approximately. 1 A continuous random variable is said to be normally distributed with mean μ and variance σ 2 if its probability density function (p.d.f) is: f ( x ) = 1 σ √ 2 π e 1 2 ‡ ( x μ ) σ · 2 (6.1) Two parameters fully specify the normal distribution, its mean usually denoted μ and its variance denoted σ 2 . Hence, the mathematical symbol for the normal curve is often written as N ( μ,σ 2 ). Note that the normal distribution ranges from∞ to + ∞ , and is sym metric about its mean. It is in fact a histogram, albeit a rather special one, 1 More of this in later chapters. 1 2 CHAPTER 6. THE NORMAL DISTRIBUTION Figure 6.1: The normal distribution and empirical rule and hence the total area under the curve is equal to 1 or 100%. You will find that it is easy to work with the normal distribution using diagrams and tables, without mathematically manipulating the menacing looking equation above....
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 Spring '10
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