PSC 41 – Lecture 16 – Standardized Scores PDF

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PSC 41 – Lecture 16 – Standardized Scores – 05/23/2019 A. The Standard Normal Distribution a. Standard Normal Distribution shows a specific relationship between the mean and the standard deviation. b. It is symmetric about the mean. 50% of the scores are below the mean and the other 50% are above the mean. c. 68% of the population fall between ±1 standard deviation, 95% of the population fall between ± 2 standard deviations, and 99% of thee population fall between ±3 standard deviations.

https://www.statology.org/the-normal-distribution/

B. Average Height of Americans by Gender a. Suppose we have two different standard normal distributions. b. The probability that a man would have a “female” height and the probability that a woman would have a “male” height is low. c. The difference in the drawn height of the distribution is meaningful due to variation in standard deviation. d. There are more women at the mean height for women than men at the mean height for men. This is because female height has a smaller standard deviation than male height.

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C. Standardized Scores Example a. Suppose we have mean 18 and standard deviation of 4 for test A. A score of 17 is below the mean so it would have a negative z score. b. For test B, the mean is 25 and standard deviation is 3. ´ +z (SD ) . c. The formula to convert a data value to a z-score is x= M d. For the example of score 17 on test A, we have 17=18 + z (4) and −1 =−0.25 . doing the algebra, we get z= 4 −8 e. For test B, we have 17= 25 + z (3) , resulting in z= . 3 f. Alternatively, we can solve for z from the formula ´ ´ + z (SD ) . In this case, we get z= x− M . x= M SD g. A score of 34 on Test B is equivalent to what score on Test A? To answer this question convert 34 into a z score, (34 −25 ) 9 z= = =3 . 3 3 D. Standardized Scores a. SAT had a mean of 1518 and a standard deviation of 325. If John scored 1518, what is his z score? It would be zero since 1518−1518 =0 . This tells us that John scored right at the z= 325 mean. b. SAT had a mean of 1518 and a standard deviation of 325. ACT has a mean of 21.1 and a standard deviation of 4.8. Which is relatively better? 1030 on the SAT or 14.0 on the ACT? For the SAT, 1030−1518 14−21.1 z= =−1.5 . For the ACT, z= =−1.479 . The 325 4.8 score that will most likely get you into university is ACT score. E. Key Points a. Social Scientists are interested in many constructs that can’t be directly measured. b. We invent clever ways to measure these constructs. c. The measurement is often on an arbitrary scale d. Using what we know about normal distributions, we can standardize the scores. e. Z-scores allow us to easily compare scores on any scale. F. Review Concepts a. The descriptive statistic that tells you the typical value of a data set is central tendency.

b. Standard deviation, variance, and interquartile range are all measures of variance. c. If a group of scores is tightly clustered around the mean, they will have a small standard deviation. d. The most appropriate measure of central tendency for a continuous variable that is not skewed is the mean. G. Interpreting Continuous Data with Two Predictor Variables a. Two predictor variables indicate a factorial design. b. Is there a main effect of Variable A? If the average of variable A and variable B are less than 1 point apart, then we can conclude that there is no main effect of variable A on the outcome. c. Is there a main effect of Variable B? If the average of variable A and variable B is more than 1 point apart, than there is a main effect of variable B on the outcome. d. The variable (A or B) is on the horizontal axis and the outcome is on the vertical axis of the graph. e. When there is no interaction, the main effect of variable on the xaxis has a slope. The main effect of variable has a gap. f. In an additive relationship the lines look parallel while in an interactive relationship, the lines are not parallel. g. Example 1: Anxiety has no effect on self-esteem or task difficulty. There was no interaction. h. Example 2: There is a main effect of task difficulty. When completing an easy task, participants rated their anxiety lower than when completing a difficult task....