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The Normal Distribution

The Normal Distribution. The Distribution. The Standard Normal Distribution. We simply transform all X values to have a mean = 0 and a standard deviation = 1 Call these new values z Define the area under the curve to be 1.0. z Scores. Calculation of z

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The Normal Distribution

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  1. The Normal Distribution

  2. The Distribution

  3. The Standard Normal Distribution • We simply transform all X values to have a mean = 0 and a standard deviation = 1 • Call these new values z • Define the area under the curve to be 1.0

  4. z Scores • Calculation of z • where  is the mean of the population and  is its standard deviation • This is a simple linear transformation of X.

  5. Tables of z • We use tables to find areas under the distribution • A sample table is on the next slide • The following slide illustrates areas under the distribution

  6. z = 1.6454545 Area = .05 .05

  7. Using the Tables • Define “larger” versus “smaller” portion • Distribution is symmetrical, so we don’t need negative values of z • Areas between z = +1.5 and z = -1.0 • See next slide

  8. Calculating areas • Area between mean and +1.5 = 0.4332 • Area between mean and -1.0 = 0.3413 • Sum equals 0.7745 • Therefore about 77% of the observations would be expected to fall between z = -1.0 and z = +1.5

  9. Converting Back to X • Assume  = 30 and  = 5 • 77% of the distribution is expected to lie between 25 and 37.5

  10. Probable Limits • X =  + z   • Our last example has  = 30 and  = 5 • We want to cut off 2.5% in each tail, so • z = + 1.96 Cont.

  11. Probable Limits--cont. • We have just shown that 95% of the normal distribution lies between 20.2 and 39.8 • Therefore the probability is .95 that an observation drawn at random will lie between those two values

  12. Measures Related to z • Standard score • Another name for a z score • Percentile score • The point below which a specified percentage of the observations fall • T scores • Scores with a mean of 50 and a standard deviation of 10 Cont.

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