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Assignment 9: Standard Deviation and Standard Scores Psych 6250 Spring, 2010 Lesson: Standard Deviation and Standard Scores 1. The variance of some sample equals 64, the standard deviation of the sample=_____. 2. One study of the heights of trees in a forest used inches as the measure of height, another study of the very same trees used feet as the measure of height. Would the two studies arrive at (identical/different) values for the standard deviation? Circle one. 3. The _________ __________ theorem implies that if scores are the result of the sum (or the mean) of several independent, non-normal, measures, then the distribution of those scores will likely be at least roughly normally distributed. 4. The greater the number of measures that are summed to create a score, the (more/less) likely the scores will be normally distributed. (select one) 5. The (smaller/larger) the variability in the population the more likely it is to get a nonrepresentative sample (select one). 6. What percent of a normal curve falls in the designated area of Figure 1? ______ 7. What percent of a normal curve falls in the designated area of Figure 2? ______ 8. What percent of a normal curve falls in the designated area of Figure 3? ______ 9. What percent of a normal curve falls in the designated area of Figure 4? ______ Figure 1 Figure 2 Figure 3 Figure 4 10. For all normal curves approximately _______ % of the scores fall within 1 standard deviation of the mean (round to the nearest percent). 11. For all normal curves approximately _______ % of the scores fall within 2 standard deviations of the mean (round to the nearest percent). 12. For all normal curves over _______ % of the scores fall within 3 standard deviations of the mean (round to the nearest percent). 13. If a population is normally distributed, with μ=150 and σ=10, we know that approximately _____% of the scores fall between 130 and 170. 14. If a population is normally distributed, with μ=150 and σ=10, we know that over ____ % of the scores fall between 120 and 180. 15. If a population is normally distributed, with μ=150 and σ=10, we know that approximately ____% of the scores fall between 140 and 160. 16. Estimate the standard deviation of the following curve (you will be given some leeway in grading your answer)._________ 17. Estimate the standard deviation of the following curve (you will be given some leeway in grading your answer)._________ 18. The score that reflects how many standard deviations above or below the mean a particular score falls is called the _________ __________ 19. Look at the graph available through the image button. Estimate (by looking at the graph) the standard score for a score of 60 in that population.________ 20. Using the formula, compute the exact value of the standard score for a score of 60 in the population above.______ Timmy comes home from school with grades from exams in the following subjects. Math Y=145, z= -1.89 English Y=32, z= 2.5 Geography Y=10, z= 0.2 21. In which class did he do very well compared to other students in the class?__________ 22. In which class did he do very poorly compared to other students in the class? __________ 23. In which class did he do about average compared to other students in the class? __________ For the following questions please do not round your answers, copy the answer given by the Normal Distribution Tool to the full number of decimal places it provides. 24. In a population that is normally distributed, what PROPORTION of the scores will fall between z=0 and z= -2.22? __________ 25. In a population that is normally distributed, what PERCENT of the scores will fall between z=0 and z= -2.22? This question is just to make sure you can go from proportion to percent._________ 26. In a population that is normally distributed, what proportion of the scores will fall within 2.22 standard deviations of the mean (i.e. between z=-2.22 and z= 2.22)?_____ 27. In a population that is normally distributed, what proportion of the scores will fall 2.22 standard deviations or more away from the mean (in both directions)? _______ 28. In a population that is normally distributed with a mean of 66 and a standard deviation of 5, what proportion of the scores will fall between the mean and 7 above the mean? ________ 29. In a population that is normally distributed with a mean of 66 and a standard deviation of 5, what proportion of the scores will be at least 7 above the mean (i.e. Y≥73)? ______ 30. In a population that is normally distributed with a mean of 66 and a standard deviation of 5, what proportion of the scores will be at least 7 away from the mean in either direction (i.e. Y<= 59 or Y>=73)? _______ SPSS In SPSS open up the data file we used for the previous assignment on the ages of a sample of CEO's from small companies ('age.sav'). Go to Graphs>>Legacy Dialogs (or the graph menu of your choice) and have SPSS display a histogram of the data (you learned how to get SPSS to do histograms in an earlier assignment). Just for practice, and to help you make sure you understand how to do this, guess the standard deviation of that curve just by looking at it. The actual value of the standard deviation is given next to the histogram, see if your guess was within 2 or 3 of the correct answer. Note, you are guessing the standard deviation of what is presented in the histogram (i.e. the sample) while SPSS is giving you the estimate of the standard deviation of the population (which would be tad larger). Now, we want SPSS to transform all of our raw Y scores into standard scores. In which menu do you think we could find that function? Why, in the transform menu of course... not. It's not there. How about the Data>>Recode menu? Nope. Don't ask me why, but to get SPSS to recode our data as standard scores we need to go to the Analyze>>Descriptive Statistics>>Descriptives and click on the 'Save standardized values as variables' box....do that. Now look at the data window to see that the ages have been turned into standard scores. Sort the variables by Z scores ascending, you learned how to sort data in an earlier assignment, and you will see that this also sorts them by age, which should make sense. 31. What is the standard score for the CEO who was 38 years old? (you can follow our normal rounding rules)___________. Tying back to the lecture, a score of 38 is roughly one and a half standard deviations below the mean. 32. Have SPSS compute the mean of the z scores____________ 33. Have SPSS estimate the standard deviation of the population of z scores________ (remember this is what SPSS gives you when it says 'Std. Deviation'). It is worthy to note that the answers to 32 and 33 will always be those values when you compute the mean and standard deviation of a set of scores z scores computes from a set of raw scores. To see why this makes sense (optional) try computing the z score of a z score of 1 (plugging in the z score, the mean of the z scores, and the standard deviation of the z scores into the z score formula).