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Psych 230 Psychological Measurement and Statistics

Psych 230 Psychological Measurement and Statistics. Pedro Wolf September 16, 2009. Today…. Symbols and definitions reviewed Understanding Z-scores Using Z-scores to describe raw scores Using Z-scores to describe sample means. Symbols and Definitions Reviewed.

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Psych 230 Psychological Measurement and Statistics

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  1. Psych 230Psychological Measurement and Statistics Pedro Wolf September 16, 2009

  2. Today…. • Symbols and definitions reviewed • Understanding Z-scores • Using Z-scores to describe raw scores • Using Z-scores to describe sample means

  3. Symbols and Definitions Reviewed

  4. Definitions: Populations and Samples • Population : all possible members of the group of interest • Sample : a representative subset of the population

  5. Symbols and Definitions: Mean • Mean • the most representative score in the distribution • our best guess at how a random person scored • Population Mean = x • Sample Mean = X

  6. Symbols and Definitions • Number of Scores or Observations = N • Sum of Scores = ∑X • Sum of Deviations from the Mean = ∑(X-X) • Sum of Squared Deviations from Mean = ∑(X-X)2 • Sum of Squared Scores = ∑X2 • Sum of Scores Squared = (∑X)2

  7. Symbols and Definitions: Variability • Variance and Standard Deviation • how spread out are the scores in a distribution • how far the is average score from the mean • Standard Deviation (S) is the square root of the Variance (S2) • In a normal distribution: • 68.26% of the scores lie within 1 std dev. of the mean • 95.44% of the scores lie within 2 std dev. of the mean

  8. Symbols and Definitions: Variability • Population Variance = 2X • Population Standard Deviation = X • Sample Variance = S2x • Sample Standard Deviation = Sx • Estimate of Population Variance = s2x • Estimate of Population Standard Deviation = sx

  9. Normal Distribution and the Standard Deviation Mean=66.57 Var=16.736 StdDev=4.091 58.38 74.75 62.48 70.66

  10. Normal Distribution and the Standard Deviation • IQ is normally distributed with a mean of 100 and standard deviation of 15 13% 13% 70 85 100 115 130

  11. Understanding Z-Scores

  12. The Next Step • We now know enough to be able to accurately describe a set of scores • measurement scale • shape of distribution • central tendency (mean) • variability (standard deviation) • How does any one score compare to others in the distribution?

  13. The Next Step • You score 82 on the first exam - is this good or bad? • You paid $14 for your haircut - is this more or less than most people? • You watch 12 hours of tv per week - is this more or less than most? • To answer questions like these, we will learn to transform scores into z-scores • necessary because we usually do not know whether a score is good or bad, high or low

  14. Z-Scores • Using z-scores will allow us to describe the relative standing of the score • how the score compares to others in the sample or population

  15. Frequency Distribution of Attractiveness Scores

  16. Frequency Distribution of Attractiveness Scores Interpreting each score in relative terms: Slug: below mean, low frequency score, percentile low Binky: above mean, high frequency score, percentile medium Biff: above mean, low frequency score, percentile high To calculate these relative scores precisely, we use z-scores

  17. Z-Scores • We could figure out the percentiles exactly for every single distribution • e ≈ 2.7183, π≈ 3.1415 • But, this would be incredibly tedious • Instead, mathematicians have figured out the percentiles for a distribution with a mean of 0 and a standard deviation of 1 • A z-distribution • What happens if our data doesn’t have a mean of 0 and standard deviation of 1? • Our scores really don’t have an intrinsic meaning • We make them up • We convert our scores to this scale - create z-scores • Now, we can use the z-distribution tables in the book

  18. Z-Scores • First, compare the score to an “average” score • Measure distance from the mean • the deviation, X - X • Biff: 90 - 60 = +30 • Biff: z = 30/10 = 3 • Biff is 3 standard deviations above the mean.

  19. Z-Scores • Therefore, the z-score simply describes the distance from the score to the mean, measured in standard deviation units • There are two components to a z-score: • positive or negative, corresponding to the score being above or below the mean • value of the z-score, corresponding to how far the score is from the mean

  20. Z-Scores • Like any score, a z-score is a location on the distribution. A z-score also automatically communicates its distance from the mean • A z-score describes a raw score’s location in terms of how far above or below the mean it is when measured in standard deviations • therefore, the units that a z-score is measured in is standard deviations

  21. Raw Score to Z-Score Formula • The formula for computing a z-score for a raw score in a sample is:

  22. Z-Scores - Example • Compute the z-scores for Slug and Binky • Slug scored 35. Mean = 60, StdDev=10 • Slug: = (35 - 60) / 10 = -25 / 10 = -2.5 • Binky scored 65. Mean = 60, StdDev=10 • Binky: = (65 - 60) / 10 = 5 / 10 = +0.5

  23. Z-Scores - Your Turn • Compute the z-scores for the following heights in the class. Mean = 66.57, StdDev=4.1 • 65 inches • 66.57 inches • 74 inches • 53 inches • 62 inches

  24. Z-Scores - Your Turn • Compute the z-scores for the following heights in the class. Mean = 66.57, StdDev=4.1 • 65 inches: (65 - 66.57) / 4.1 = -1.57 / 4.1 = -0.38 • 66.57 inches: (66.57 - 66.57) / 4.1 = 0 / 4.1 = 0 • 74 inches: (74 - 66.57) / 4.1 = 7.43 / 4.1 = 1.81 • 53 inches: (53 - 66.57) / 4.1 = -13.57 / 4.1 = -3.31 • 62 inches: (62 - 66.57) / 4.1 = -4.57 / 4.1 = -1.11

  25. Z-Score to Raw Score Formula • When a z-score and the associated Sx and X are known, we can calculate the original raw score. The formula for this is:

  26. Z-Score to Raw Score : Example • Attractiveness scores. Mean = 60, StdDev=10 • What raw score corresponds to the following z-scores? • +1 : X = (1)(10) + 60 = 10 + 60 = 70 • -4 : X = (-4)(10) + 60 = -40 + 60 = 20 • +2.5: X = (2.5)(10) + 60 = 25 + 60 = 85

  27. Z-Score to Raw Score : Your Turn • Height in class. Mean=66.57, StdDev=4.1 • What raw score corresponds to the following z-scores? • +2 • -2 • +3.5 • -0.5

  28. Z-Score to Raw Score : Your Turn • Height in class. Mean=66.57, StdDev=4.1 • What raw score corresponds to the following z-scores? • +2: X = (2)(4.1) + 66.57 = 8.2 + 66.57 = 74.77 • -2: X = (-2)(4.1) + 66.57 = -8.2 + 66.57 = 58.37 • +3.5: X = (3.5)(4.1) + 66.57 = 14.35 + 66.57 = 80.92 • -0.5: X = (-0.5)(4.1) + 66.57 = -2.05 + 66.57 = 64.52

  29. Using Z-scores

  30. Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means

  31. Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means

  32. Z-Distribution • A z-distribution is the distribution produced by transforming all raw scores in the data into z-scores • This will not change the shape of the distribution, just the scores on the x-axis • The advantage of looking at z-scores is the they directly communicate each score’s relative position • z-score = 0 • z-score = +1

  33. Distribution of Attractiveness Scores Raw scores

  34. Z-Distribution of Attractiveness Scores Z-scores

  35. Z-Distribution of Attractiveness Scores Z-scores In a normal distribution, most scores lie between -3 and +3

  36. Characteristics of the Z-Distribution • A z-distribution always has the same shape as the raw score distribution • The mean of any z-distribution always equals 0 • The standard deviation of any z-distribution always equals 1

  37. Characteristics of the Z-Distribution • Because of these characteristics, all normal z-distributions are similar • A particular z-score will be at the same relative location on every distribution • Attractiveness: z-score = +1 • Height: z-score = +1 • You should interpret z-scores by imagining their location on the distribution

  38. Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means

  39. Using Z-Scores to compare variables • On your first Stats exam, you get a 21. On your first Abnormal Psych exam you get a 87. How can you compare these two scores? • The solution is to transform the scores into z-scores, then they can be compared directly • z-scores are often called standard scores

  40. Using Z-Scores to compare variables • Stats exam, you got 21. Mean = 17, StdDev = 2 • Abnormal exam you got 87. Mean = 85, StdDev = 3 • Stats Z-score: (21-17)/2 = 4/2 = +2 • Abnormal Z-score: (87-85)/2 = 2/3 = +0.67

  41. Comparison of two Z-Distributions Stats: X=30, Sx=5 Millie scored 20 Althea scored 38 English: X=40, Sx=10 Millie scored 30 Althea scored 45

  42. Comparison of two Z-Distributions

  43. Uses of Z-Scores • Describing the relative standing of scores • Comparing scores from different distributions • Computing the relative frequency of scores in any distribution • Describing and interpreting sample means

  44. Using Z-Scores to compute relative frequency • Remember your score on the first stats exam: • Stats z-score: (21-17)/2 = 4/2 = +2 • So, you scored 2 standard deviations above the mean • Can we compute how many scores were better and worse than 2 standard deviations above the mean?

  45. Proportions of Area under the Standard Normal Curve

  46. Relative Frequency • Relative frequency can be computed using the proportion of the total area under the curve. • The relative frequency of a particular z-score will be the same on all normal z-distributions. • The standard normal curve serves as a model for any approximately normal z-distribution

  47. Z-Scores • z-scores for the following heights in the class. • Mean = 66.57, StdDev=4.1 • 65 inches: (65 - 66.57) / 4.1 = -1.57 / 4.1 = -0.38 • 66.57 inches: (66.57 - 66.57) / 4.1 = 0 / 4.1 = 0 • 74 inches: (74 - 66.57) / 4.1 = 7.43 / 4.1 = 1.81 • 53 inches: (53 - 66.57) / 4.1 = -13.57 / 4.1 = -3.31 • 62 inches: (62 - 66.57) / 4.1 = -4.57 / 4.1 = -1.11

  48. Z-Scores • z-scores for the following heights in the class. • Mean = 66.57, StdDev=4.1 • 65 inches: (65 - 66.57) / 4.1 = -1.57 / 4.1 = -0.38 • 66.57 inches: (66.57 - 66.57) / 4.1 = 0 / 4.1 = 0 • 74 inches: (74 - 66.57) / 4.1 = 7.43 / 4.1 = 1.81 • 53 inches: (53 - 66.57) / 4.1 = -13.57 / 4.1 = -3.31 • 62 inches: (62 - 66.57) / 4.1 = -4.57 / 4.1 = -1.11 • What are the relative frequencies of these heights?

  49. Z-Scores • How can we find the exact relative frequencies for these z-scores? • 65 inches: z = -0.38 • 66.57 inches: z = 0 • 74 inches: z = 1.81 • 53 inches: z = -3.31 • 62 inches: z = -1.11

  50. Z-Scores • How can we find the exact relative frequencies for these z-scores? • 65 inches: z = -0.38 • 66.57 inches: z = 0 • 74 inches: z = 1.81 • 53 inches: z = -3.31 • 62 inches: z = -1.11

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