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Chapter 5

Chapter Five PSY201

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Kristie Dukewich

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PSY201: Chapter 5: The Normal Curve and Standard Scores Introduction: – Normal curve + a very important distribution in behavior sciences + three principal reasons why... - 1. many of the variables measured in behavioral science research have distributions that quite closely approximate the normal curve (ie: height, weight, intelligence and achievement are few examples) - 2. many of the inference tests used in analyzing experiments have sampling distributions that become normally distributed with increasing sample size. (ie: sign test & Mann-Whitney U test) - 3. many inference tests require sampling distributions that are normally distributed. The z test, Student's t test, and the F test are examples of inference tests that depend on this point → much of importance of normal curve occurs in conjunction with inferential statistics. The Normal Curve: – normal curve is a theoretical distribution of population scores. + a theoretical curve and is only approximated by real data + bell-shaped curve that is described by equation: – curve has two inflection points, one on each side of the mean + inflection points are located where the curvature changes direction + ie: inflection points are located where curve changes from being convex downward to being convex upward - if the bell-shaped curve is a normal curve, inflection points are at 1 standard deviation from the mean ( and ) - as the curve approaches the horizontal axis, it is slowly changing its Y value. - the curve never quite reaches the axis - it approaches the horizontal axis and gets closer and closer to it, but it never quite touches it. - curve is asymptotic to the horizontal axis – infection points under the curve, horizontal... in the diagram on page 97 Area Contained Under the Normal Curve: – in distributions that are normally shaped, there is a special relationship between the mean and the standard deviation with regard to the area contained under the curve – when a set of scores is normally distributed, 34.13% of the area under the curve is contained between the mean (u) and a score that is equal to u + 1o ; 13.59% of the area is contained between a score equal to + 1 and a score of u+ 2 o; 2.15%of the area is contained between scores of u+ 2o and u + 3o ; and 0.13% of the area exists beyond u+ 3o . This accounts for 50% of the area + since curve is symmetrical, same percentages hold for scores below the mean + since frequency is plotted on vertical axis, these percentages represent the percentage of scores contained within the area – ie: + have a population of 10,000 IQ scores + distribution normally shaped with u = 100 and o = 16 + since scores are normally distributed, 34.13% of scores are contained between scores of 100 and 116 ( u+ 1o = 100 + 16 = 116), 13.59% between 116 and 132 ( u+ 2o = 100+32 = 132), 2.15% between 132 and 148, and 0.13% above 148 + similarly, 34.13% of scores fall between 84 and 100, 13.59% between 68 and 84, 2.15% between 52 and 68, and 0.13% below 52. – to calculate the number of scores in each area, multiply the relevant percentage by the total number of scores → there are 34.13% x 10,000 = 3413 scores between 100 and 116, 13.59% x 10.000 = 1359 scores between 116 and 132, and 215 scores between 132 and 148; 13 scores are greater than 148. + for other half of distribution, there are 3413 scores between 84 and 100, 1359 scores between 68 and 84, and 215 scores between 52 and 68; there are 13 scores below 52. + these frequencies would be true only if distribution is exactly normally distributed + in actual practice, the frequencies would vary slightly depending on how close the distribution is to this theoretical model Standard Scores (z Scores): – IQ of 132... + a score is meaningless unless you have a reference group to compare against + without one, can't tell whether the score is high, average, or low – score is one of the 10,000 scores of distributions → gives IQ of 132 some meaning + ie: can determine the percentage of scores in distribution that are lower than 132 → determining the percentile rank of score of 132 (percentile rank of a score is defined as the percentage of scores that is below the score in question) – 132 is 2 standard deviations above the mean + in normal curve, there are 34.13 + 13.59 = 47.72% of the scores between the mean and a score that is 2 standard deviations above the mean + to fine percentile rank of 132, need to add this percentage the 50.00% that lie below the mean → 97.72% (47.72 + 50.00) of the scores fall below your IQ score of 132. + should be happy to be intelligent – to solve problem, had to determine how many standard deviations the raw score of 132 was above or below the mean + transformed the raw score into a standard score, also called a z score – a z score is a transformed score that designated how many standard deviation units the corresponding raw score is above or below the mean – process which by the raw score is altered – score transformation + z transformation results in a distribution having a mean of 0 and a standard deviation of 1 + reason z scores are called standard deviation is they are expressed relative to a distribution mean of 0 and a standard deviation of 1 – in conjunction with a normal curve, z scores allow to determine the number or percentages of scores that fall above or below any score in the distribution + z scor
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