chapter6 - Chapter 6 The Normal Distribution The Gaussian...

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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 bell-shaped 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
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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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