PSYC 214

1. PSYC 214 Normal Distributions and Standard Scores

2. Agenda • Percentiles • Distributions • Standard (-ization of) Scores • Appendix B • Examples

3. Few Reminders • Rounding • Usually round to the hundredth’s place (two numbers after decimal point e.g. 1.234 = 1.23, 1.235 = 1.24) • Below 5 round down • 5 and above round up • Greater than or less than > or < • Greater than > (e.g. 6 > 5) • Less than < (e.g. 1 < 4) • Alligator eats the biggest number

4. Percentile Rank p. 101-105 • Percentage of scores in the distribution that are at or below a specific value; Pxx • Describes the position of a score • In relation to other scores • What type of measurement is this? • Example: Baby at 90th percentile for height & weight is tall & heavy: 90% of other babies are shorter/lighter and only 10% are longer/heavier

5. 95th percentile Baby

6. Another View (7 months)

7. 70th Height and 57th Weight (4 yrs) My Nieces 55th Height and 32nd Weight (18mths)

8. 2 years 3 years 5 years

9. 9 years, 11 years, and 7 years

10. How to find Pxx • A score’s percentile: “If you were 45th from the bottom of 60 people, you’d be at the _____ percentile.” • Score position/total number of scores X 100 = percentile • 45/60 X 100 = 75th percentile or P75 • If you scored 50th from the bottom of a class of 160 people, at what percentile would your score be?

11. Practice • What’s the percentile rank for a score that’s: • 23th from the bottom of 375 scores? • 7th from the end out of 34 scores? • 95th from the bottom of 180 scores? • What’s the position (from the bottom) for a score with a percentile rank of: • P25 among 775 scores? • P90 among 425 scores?

12. Percentile Rank as a z-Score • But remember: • Pxx can reflect an ordinal measure of relative standing (position relative to the group of scores) • As a rank it does not take into consideration means or standard deviations (no math!) • Standardized scores are often used to present percentile ranks, also referred to as z-scores • They describe how far a given score is from the other scores, in terms of the z distribution

13. Purpose of z-Scores • Identify and describe location of every score in the distribution • Standardize an entire distribution • Takes different distributions and makes them equivalent and comparable

14. Two distributions of exam scores

15. Locations and Distributions • Exact location is described by z-score • Sign tells whether score is located above or below the mean • Number tells distance between score and mean in standard deviation units

16. Relationship of z-scores and locations

17. Traits of the normal distribution • Sections on the left side of the distribution have the same area as corresponding sections on the right • Because z-scores define the sections, the proportions of area apply to any normal distribution • Regardless of the mean • Regardless of the standard deviation

18. Normal Distribution with z-scores

19. z-Scores for Comparisons • All z-scores are comparable to each other • Scores from different distributions can be converted to z-scores • The z-scores (standardized scores) allow the comparison of scores from two different distributions along

20. Properties of z scores Mean ALWAYS = 0 Standard deviation ALWAYS = 1 Positive z score is ABOVE the mean Negative z score is BELOW the mean

21. Standard Scores • Give a score’s distance above or below the mean in terms of standard deviations • Positive z-scores are always above the mean • Negative z-scores are always below the mean • Negative & positive z-scores only indicate directionality on the x-axis, not necessarily a change in the value of the score

22. Standard Scores • An index of a score’s relative standing Standard (converted) scores are referred to asz-scores • x = raw score • µ = population mean • σ = population SD • Backward formula: x = z (σ) + µ • Use this one to solve for x p.105

23. Equation for z-score • Numerator is a deviation score • Denominator expresses deviation in standard deviation units

24. Determining raw score from z-score • Numerator is a deviation score • Denominator expresses deviation in standard deviation units

25. Order of Operation Backward formula Regular formula

26. Standard Normal Distribution 50% of total area under the curve +1 σ is where the line turns from concave to convex • Symmetrical • Asymptotic μ = 0 σ = 1.0 p.110, Fig. 4.1

27. Distributions • Normal distribution: all normal distributions are symmetrical and bell shaped. Thus, the mean, median & mode are all equal! • “Ideal world” bell curve • μ = 0 • σ = 1.0 • Completely symmetrical • Asymptotic

28. Standard Normal DistributionFig 4.1

29. IQ Standard Normal Distribution Genius Gifted Above average Higher average Lower average Below average Borderline low Low >144 130-144 115-129 100-114 85-99 70-84 55-69 <55 0.13% 2.14% 13.59% 34.13% 34.13% 13.59% 2.14% 0.13% 68% 95% 95% 99% 99% Adapted from http://en.wikipedia.org/wiki/IQ_test

30. Shape of a Distribution • Researchers describe a distribution’s shape in words rather than drawing it • Symmetrical distribution: each side is a mirror image of the other • Skewed distribution: scores pile up on one side and taper off in a tail on the other • Tail on the right (high scores) = positive skew • Tail on the left (low scores) = negative skew

31. Skewed Distributions • Mean, influenced by extreme scores, is found far toward the long tail (positive or negative) • Median, in order to divide scores in half, is found toward the long tail, but not as far as the mean • Mode is found near the short tail. • If Mean – Median > 0, the distribution is positively skewed. • If Mean – Median < 0, the distribution is negatively skewed

32. Distributions Positively skewed distribution: tendency for scores to cluster below the mean p.90

33. Distributions Negatively skewed distribution: tendency for scores to cluster above the mean p.90

34. Appendix B p.617 • Used to find proportions under the normal curve • We only use Columns 1, 3 and 5 • Column 3 gives proportion from the z-score towards either tail • Column 5 gives proportion between z-score and the mean z Z or z μ or μ Z

35. Standard Scores • Standard scores = z scores (in pop) • µ = mu = population mean • σ = sigma = population standard deviation • x = z (σ) + µ (Backward formula)

36. Finding a proportion from a raw score 1: Sketch the normal distribution2: Shade the general region corresponding to the required proportion3: Using the “forward” formula compute the corresponding z-score from the raw score (x). 4: Locate the proportion in the correct column of the table

37. What proportion of people have an IQ of 85 or less? What about x < 70? Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B Which column do you use? For IQ scores µ = 100 & σ =15

38. What proportion of people have an IQ of 130 or above? Of x > 120? Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B Which column do you use? For IQ scores µ = 100 & σ =15

39. Finding a z-score from a proportion 1: Sketch the normal distribution2: Shade the general region corresponding to the required proportion3: Locate the proportion in the correct column of the table4: Identify corresponding z-score in Col.1 5: Calculate raw score (x) with “backwards” formula

40. What is the lowest IQ score you can earn and still be in the top 5% of the population? Step 1: draw curve Step 2: Appendix B Which column do you use? Step 3: use formula X = (z)(σ) + µ For IQ scores µ = 100 & σ =15

41. What proportion of test takers score between 300 and 650 on the SAT? Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B Which column do you use? For SAT scores µ = 500 & σ =100

42. Applications of z-Scores • Any observation from a normal distribution can be converted to a z-score • If we have an actual, theoretical, or hypothesized normal distribution of many means we can determine the position of one particular mean relative to the other means in this distribution • This logic is the basis of many hypothesis tests

43. Z-score Practice Lecture Handout