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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. 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, then xm z s is the standardized value (a.k.a. z-score) of x. 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 variable X m Z s has the standard normal distribution. Standard Normal Table (Table A, inside cover of book or from website) Examples 1. Find the area under the standard normal curve to the left of -1.4. 2. Find the area under the N(0,1) curve between 0.76 and 1.4. 3. Find the value z of the N(0,1) which has area 0.25 to its right. 4. Suppose X~N(275, 43). What proportion of the population is greater than 200? What proportion of the population is between 200 and 375? 5. Suppose verbal SAT scores follow the N(430, 100) distribution. How high must a student score in order to place in the top 5%? 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? The Normal Quantile Plot: A Visual Test of Normality 1. Arrange the observed data values from smallest to largest and record the percentile of the data each observation occupies. 2. Compute the z-scores of the same percentiles. 3. 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. FIG. 1.34: Normal quantile plot of IQ scores for 78 seventh-graders