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Chapter 6 The Normal Distribution Chapter 6 The Normal Distribution Major Points Distributions and area The normal distribution The standard normal distribution Setting probable limits on an observation Measures related to z 2 Chapter 6 The Normal Distribution Distributions and Area We can apply the concepts of Area Percentages Probability Addition of areas Addition of probabilities When using the normal distribution 3 4 Chapter 6 The Normal Distribution Convert Histogram to Frequency Polygon of Distribution: connect the dots…….. 10 N=50 9 8 7 6 5 4 3 2 1 0 1 to 10 11 to 20 21 to 30 31 to 40 41 to 50 51 to 60 61 to 70 71 to 80 81 to 90 91 to 100 12 Chapter 6 The Normal Distribution 5 The Normal Distribution “Normal” refers to the general shape of the distribution See next slide Remember our descriptors for distributions The formula for the normal distribution allows us to Predict a proportion of cases that fall under the normal curve between 2 given x values Cont. Chapter 6 The Normal Distribution Normal Distribution--cont. X is the value on the abscissa Y is the resulting height of the curve at X There are constants also in the formula Do you need to memorize or use the formula? No! It’s just there to show you how they arrive at the percentages discussed in a minute. 6 Chapter 6 The Normal 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, can look at the proportion of cases falling above mean and below mean 7 8 Chapter 6 The Normal Distribution Ex. of a transformed Distribution 1200 1000 800 600 400 200 Std. Dev = 1.00 Mean = -.01 N = 10000.00 0 50 3. 00 3. 50 2. 00 2. 50 1. 00 1. 0 .5 00 0. 0 -.5 0 .0 -1 0 .5 -1 0 .0 -2 0 .5 -2 0 .0 -3 0 .5 -3 0 .0 -4 X Chapter 6 The Normal Distribution z Scores Calculation of z Xμ z σ where is the mean of the population and is its standard deviation This is a simple linear transformation of X. 9 Chapter 6 The Normal Distribution 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 10 11 Chapter 6 The Normal Distribution z Table Mean to z Larger Portion Smaller Portion 0.000 .0000 .5000 .5000 0.100 .0398 .5398 .4602 0.200 .0793 .5793 .4207 1.000 .3413 .8413 .1587 1.500 .4332 .9332 .0668 1.645 .4500 .9500 .0500 1.960 .4750 .9750 .0250 z Chapter 6 The Normal Distribution 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 12 Chapter 6 The Normal Distribution 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 13 14 Chapter 6 The Normal Distribution Converting Back to X Assume = 30 and = 5 X z Therefore X z X 30 1.0 5 25 X 30 1.5 5 37.5 So 77% of the distribution is expected to lie between 25 and 37.5 Chapter 6 The Normal Distribution 15 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 (found from a z table, mean to z of 1.96 = .475, but we usually round to z = 2). X z X 30 1.96 5 39.8 X 30 1.96 5 20.2 Cont. Chapter 6 The Normal Distribution Probable Limits--cont. We have just shown that 95% of the normal distribution lies between 20.2 and 39.8, or +/1.96 SD’s Therefore the probability is .95 that an observation drawn at random will lie between those two values 16 Chapter 6 The Normal Distribution 17 Statistical Methods: The Normal Distributions Walsh & Betz (1995), p29 20 18 Chapter 6 The Normal Distribution Measures Related to z Standard score Another name for a z score, there are other standard scores, such as ….. T scores: Scores with a mean of 50 and and standard deviation of 10, no negative values Others include IQ, SAT’s, etc Percentile score/rank Percent of cases which fall at or below a given score in the norming sample- THIS is how we typically find a percentile rank, not calculating using that somewhat confusing formula Cont. Chapter 6 The Normal Distribution Related Measures--cont. Stanines Scores which are integers between 1 and 9, with mean = 5 and standard deviation = 2 Mainly used in education 19 20 Chapter 6 The Normal Distribution Review Questions Why do you suppose we call it the “normal” distribution? What do we gain by knowing that something is normally distributed? How is a “standard” normal distribution different? Cont. Chapter 6 The Normal Distribution Review Questions--cont. How do we convert X to z? How do we use the tables of z? If we know your test score, how do we calculate your percentile? What is a T score and why do we care? 21