1 The Normal Distribution William P. Wattles Psychology 302

Frequency distribution n A table or graph that indicates all the values a variable can take and how often each occurs.

Density curves A density curve is a mathematical model of a distribution. It is always on or above the horizontal axis. The total area under the curve, by definition, is equal to 1, or 100%. The area under the curve for a range of values is the proportion of all observations for that range. Histogram of a sample with the smoothed density curve theoretically describing the population

Normal Distribution Gaussian Distribution n Mean=Median=Mode

Normal Distribution Normal distributions have the same general shape. They are symmetric with scores more concentrated in the middle than in the tails.

Here the means are different (  = 10, 15, and 20) while the standard deviations are the same (  = 3). Here the means are the same (  = 15) while the standard deviations are different (  = 2, 4, and 6). A family of density curves

40 Z scores and the normal curve n The rule n 68% fall within one standard deviation of the mean n 95% fall within two standard deviations of the mean n 99.7% of the observations fall with three standard deviations of the mean

Normal Curve

Because all Normal distributions share the same properties, we can standardize our data to transform any Normal curve N (  ) into the standard Normal curve N (0,1). The standard Normal distribution For each x we calculate a new value, z (called a z-score). N(0,1) => N(64.5, 2.5) Standardized height (no units)

36 Standard Scores (Z-scores) n Can use appendix A (page 690) in back of book to determine the area under the curve cut off by any Z-score.

Using Table A (…).0082 is the area under N(0,1) left of z = is the area under N(0,1) left of z = is the area under N(0,1) left of z = -2.46

47 Area under the curve n Height of young women – Mean = 64 – Standard deviation = 2.7 n What proportion of women are less than 70 inches tall?

47 Area under the curve n Height of young women – Mean = 64 – Standard deviation = 2.7 n Z score for 5’10” n Area to the left =.9868 n A woman 70 inches tall is taller than 99% of her peers.

n WAIS mean=100, SD=15 n What percent are retarded, I.e. less than 70?

n WAIS mean=100, SD=15 n What percent are MENSA eligible, I.e. greater than 130?

Area under the curve n WAIS mean=100, SD=15 n Z=X-mean/standard deviation n What percent are retarded, I.e. less than 70? n Z=70-100/15, Z=-2.00, 2.28% n What percent are MENSA eligible, I.e. greater than 130? n Z= /15 Z=+2.00, 2.28%

58 Percentile scores n The percent of all scores at or below a certain point. n The same procedure as with proportions n More commonly used than proportions

Sample Problem n SAT mean=1020, SD=207 n Division 1 athletes must have 820 to compete? Is this fair? n What percent score less than 820?

Normal Curve

n SAT mean=1020, SD=207 n What percent score less than 820?.1660

n SAT mean=1020, SD=207 n Division 1 athletes must have 720 to practice? Is this fair? n What percent score less than 720?

Normal Curve

n SAT mean=1020, SD=207 n What percent score less than 720?.0735

What is a Z score?

n A z-score tells how many standard deviations the score or observation falls from the mean and in which direction

Z-Score n A Z-score tells how many standard deviations an individual’s score lies above or below the mean.

Psy 302 Paper 1. Pick a subject that interests you. 2. Do some library research. 3. Collect data a.two groups (minimum 15 per group) b.measurement data 1. Analyze data with t-test a.SPSS b.Excel 2. Make a histogram of your results. 3. Write paper APA style per sample paper.

Houston’s G.M. Is a Revolutionary Spirit in a Risk- Averse Mind n Daryl Morey has charts and spreadsheets and clever formulas for evaluating basketball players, and a degree from M.I.T. to make sense of it all.

60 The End