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Chapter 4: Standardized Scores and the Normal Distribution

Chapter 4: Standardized Scores and the Normal Distribution

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Chapter 4: Standardized Scores and the Normal Distribution

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  1. Chapter 4: Standardized Scores and the Normal Distribution The simplest standardized scores are called z scores. They tell you the distance of a raw score from the mean in terms of standard deviations (the sign of the z score tells you whether a score is above or below the mean). Here is the formula: Properties of z scores: • The mean of a complete set of z scores is always 0, and its standard deviation will always be 1.0. • Converting a set of raw scores into z scores will not change the shape of the distribution at all. • However, z scores are especially helpful when used in conjunction with a normal distribution. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  2. Try This Example of Computing z Scores: Jan just received three midterm grades (see table below). Which grade is asso-ciated with the highest z score? To get you started, here is the calculation of the first z score: For Explaining Psychological Statistics, 4th ed. by B. Cohen

  3. T Scores • Transform z scores into Tscores, which follow a distribution with mean = 50 and σ = 10, using this formula: • In the table below, first find the z score for each subject, and then the T score. We will get you started by calculating z and T for Alice: For Explaining Psychological Statistics, 4th ed. by B. Cohen

  4. SAT Scores • Transform z scores into SAT scores, which follow a distribution with mean = 500 and σ = 100, using this formula: • In the table below, find the SAT score for each subject. You can use the z scores you calculated for the previous slide. • Note that, traditionally, T scores have been used as a way to standardize the scores from psychological tests, whereas SAT and related scores have been used by ETS, and other companies that administer standardized aptitude and achievement/ability tests. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  5. IQ Scores • You can transform z scores into Stanford-Binet IQ scores, which follow a distribu-tion with mean = 100 and σ = 16, using this formula: • In the table below, find the IQ score for each subject. You can use the z scores you calculated for the T or SAT scores. • To find WAIS IQ scores instead, multiply the z score by 15, instead of 16. For Alice, the WAIS IQ score would be 118.9. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  6. The Normal Distribution • It is the most commonly used theo-retical distribution in psychological research. • It is unimodal and symmetric, and its tails extend infinitely in both the positive and negative directions. • It is described as bell-shaped, but it is defined by a specific mathemati-cal equation. • It arises frequently, in an approxi-mate form, in nature. • When the area under the normal curve is used to determine proport-ions of the population, the whole area is defined as 1.0 (or 100%). For Explaining Psychological Statistics, 4th ed. by B. Cohen

  7. Parameters of the Normal Distribution The normal distribution is actually a family of distributions each defined by two parameters: the mean and the standard deviation. • Two normal distributions can have different means and the same standard deviation; the same mean and different standard deviations; or they can differ on both parameters. • However, all normal distributions have the same shape. • The standard normal distribution has the following parameters: μ = 0, and σ = 1. It arises whenever you trans-form all the scores in any normal distribution into z scores. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  8. Areas Under the Normal Distribution • There is a table of the standard normal distribution that tells us the exact area under the normal curve, as a proportion of the total, between the mean and any z score, to two decimal places. • Inthe figure below, you can see the areas corresponding to z scores of 1, 2, and 3, on both sides of the distribution. • There is a table to determine the exact area under the curve • This area is the probability of a particular event z scores For Explaining Psychological Statistics, 4th ed. by B. Cohen

  9. Finding the Area Above a Positive z Score or Below a Negative z Score: • Find the percent area between the mean and the given z score in Table A of your text. • Subtract that percentage from 100. • Divide by 100 to convert to a proportion or probability. Finding the Area Belowa Positive z Score or Abovea Negative z Score: • Find the percent area between the mean and the given z score in Table A of your text. • Add 50 to that percentage. • Divide by 100 to convert to a proportion or probability. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  10. Finding the z Score Corresponding to a Given Area in the Positive Tail of the Normal Distribution • Ex., What z score does a student need in order to be among the highest 12% of scores? • The area you want is not between the mean and z, it is abovez, so subtract the given percent from 50.00. For this example, the area you are looking for is 50.00 – 12.00 = 38.00. • Look in the body of Table A for the percent area that comes closest to the area calculated in the previous step, and read off the corresponding z score. • For this example, the z score that cuts off the top 12% falls midway between 1.17 and 1.18, so the answer is that the student needs a z score that is at least 1.175 to be in the top 12% of the distri-bution. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  11. Finding the z Score Corresponding to a Given Area in the Negative Tail of the Normal Distribution • Ex., Below what z score does the lowest 8% of the normal distribution fall? • The area you want is not between the mean and z, it is belowz, so subtract the given percent from 50.00. For this example, the area you are looking for is 50.00 – 8.00 = 42.00. • For this example, the z score that cuts off the top 8% falls midway between 1.40 and 1.41, so the answer for the top 8% would be 1.405. • Because you are looking for the bottom 8%, place a minus sign in front of the z score you found in the previous step – i.e., 8% of the normal distribution is below a z score of –1.405. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  12. Areas in the Middle of the Normal Distribution • It is often useful to know the z scores that enclose a given amount of area in the middle of the normal distribution— e.g., the middle 80%. • The area you need to find in the body of Table A is half of that area — in this case, 40.00. • The z score that comes closest to containing 40.00% between the mean and z is 1.28. • Because of the symmetry of the normal distribution, the two z scores that enclose the middle 80% are (approximately) –1.28 and +1.28. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  13. Sampling Distribution of the Mean • The mean of the sampling distribution of the mean is the same as the population mean. • The standard deviation of the sampling distribution of the mean: standard error of the mean • Is smaller than the standard deviation of the population distribution • Extreme scores are more likely than extreme means, so the distribution of means will be less variable than the population. • As N increases, sample means are clustered more closely and the standard error gets smaller. For Explaining Psychological Statistics, 4th ed. by B. Cohen

  14. Sampling Distribution Versus Population Distribution • The sampling distribution is normal if the population distribution is normal. • The sampling distribution will approach normal even if the population distribution is not normal (if N is large enough— Central Limit Theorem). • Mean will be the same as population distribution, but the variability is less (smaller standard error). For Explaining Psychological Statistics, 4th ed. by B. Cohen

  15. Describing Groups • You can use zscores to find the location of one group with respect to all other groups of the same size. • Must use the sampling distribution of the mean for groups. • Need to determine the zscore and then find the area (proportion). For Explaining Psychological Statistics, 4th ed. by B. Cohen

  16. Ex., Suppose we know that the average weight of 6th grade boys is normally distributed with  = 99 lbs and  = 4 lbs. What is the probability of obtaining a sample mean (with N = 25) of less than 98 lbs? • Ex., A beverage company sets its bottling machinery to put a  = 16.5 oz of ice tea in each bottle, with a  = 0.5 oz. The ice tea is sold by in 6-packs. We assume that each of the bottles in a 6-pack is a random sample. What percentage of 6-packs have a mean bottle amount of 17 oz or more? • These are questions about means, not individuals. For Explaining Psychological Statistics, 4th ed. by B. Cohen

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