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The 68-95-99.7 Rule

The 68-95-99.7 Rule. In any normal distribution: 68 % of the individuals fall within 1 s of m . 95 % of the individuals fall within 2 s of m . 99.7 % of the individuals fall within 3 s of m . How can we make a valid comparison of observations from two distributions?.

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The 68-95-99.7 Rule

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  1. The 68-95-99.7 Rule • In any normal distribution: • 68 % of the individuals fall within 1s of m. • 95 % of the individuals fall within 2s of m. • 99.7 % of the individuals fall within 3s of m.

  2. How can we make a valid comparison of observations from two distributions? • By standardizing the values of the observations with respect to the distributions from which they come. • If x is an observation from a distribution with mean m and standard deviation s, thenis the standardized value (a.k.a. z-score) of x.

  3. Standard Normal Distribution The standard normal distribution is the normal distribution N(0,1) with mean 0 and standard deviation 1. If a variable X has a N(m,s) distribution, the standardized variablehas the standard normal distribution.

  4. Standard Normal Table(Table A, inside cover of book or from website)

  5. Examples • Find the area under the standard normal curve to the left of -1.4. • Find the area under the N(0,1) curve between 0.76 and 1.4. • Find the value z of the N(0,1) which has area 0.25 to its right. • Suppose X~N(275, 43). What proportion of the population is greater than 200? What proportion of the population is between 200 and 375? • Suppose verbal SAT scores follow the N(430, 100) distribution. How high must a student score in order to place in the top 5%?

  6. Testing for Normality • The normal distribution provides a good model for many real data distributions. • Furthermore, the normal distribution is a nice model to work with mathematically. • However, we need to be cautious when assuming normality of data. Are we sure our data are normally distributed?

  7. The Normal Quantile Plot: A Visual Test of Normality • Arrange the observed data values from smallest to largest and record the percentile of the data each observation occupies. • Compute the z-scores of the same percentiles. • Plot each data point against the corresponding z-score. If the points on a normal quantile plot lie close to a straight line then the plot indicates the data are normal.

  8. FIG. 1.34: Normal quantile plot of IQ scores for 78 seventh-graders

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