Chapter 4 & 5

1. Chapter 4 & 5 The Normal Curve & z Scores

2. What is it? • -It’s a unimodal frequency distribution curve • *scores on X-axis & frequency on Y-axis • -distributions observed in nature usually match it • -it is a critical component for understanding inferential statistics • What sets it apart from other frequency distributions? • Most of the scores cluster around the middle of the distribution. The curve then drops down & levels out on both sides. • It is symmetrical. • The mean, median & the mode fall at the exact same point. • It has a constant relationship with the standard deviation • No matter how far out the tails are extended, they will never touch the X-axis because it is based on an infinite population. In other words, the normal curve is asymptotic to the abscissa. Normal Curve

3. Normal Curve EX: The following is a normal distribution of IQ scores M=100 SD=15

4. The “standard normal curve” is a normal curve that has been plotted using standard deviation units. • It has a mean of 0 and a standard deviation of 1.00 • The standard deviation units have been marked off in unit lengths of 1.00 on the abscissa. • The area under the curve above these units always remains the same. The Standard Normal Curve Animation

5. What is it? • -the number of standard deviations a raw score is above or below the mean • Why use it? • -to figure out how a raw score compares to a group of scores • In the previous slide we were ALSO describing z-scores • Z-score table: used to obtain the precise percentage of cases falling between any z-score and the mean. • -only shows positive z-scores because the curve is symmetrical so percentages would be the same for negative z-scores • *p. 621, Table A • Helpful hint: always draw the curve when working with z-scores • -positive z-scores go to the right of the mean, negative go to left • -the higher the z-scores the further it goes to the right, the lower the z-score the further to the left of the mean • -shade in percentage of cases you’re referring to Z-Scores

6. To find the percentage of cases between a z-score and the mean: • -look it up in the z-score table! • To find the percentage of cases above a z-score: • -if positve z-score, look up z-score in table and subtract it from 50 • -if negative z-score, look up z-score in table and add it to 50 • To find the percentage of cases below a z-score: • -if positive z-score, look up z-score in table and add it to 50 • -if negative z-score, look up z-score in table and subtract it from 50 • **this is the same way to calculate percentile • --be sure to round to the nearest percentile • --a z-score of 0 would be the 50th percentile • To find percentage of cases between z-scores: • -between a negative & postive z-score, look up both scores in table & add • -between 2 positive or 2 negative z-scores, look up both scores in table & subtract Z-Scores: Calculation • To change a raw score to a z-score: subtract the mean from the raw score & divide by the standard deviation Z-Score Practice Worksheet Z-Score Homework due Next Class

7. Gauss: Father of the Normal Curve • German Mathematician • Often referred to as the greatest mathematician of all time • “Perfect Pitch” Mathematical Prodigy • People say he did math before he could talk! • Developer of the Normal Curve or the “Gaussian Curve” Pair Share Topic: What does a Z-score tell you about a raw score? 1775-1855 Fun with Your Calculator Worksheet

8. Z-Scores Revisited • To find a raw score from a z-score: X= zSD +M • -EX: z= 2 SD=17 M=150 • **X=(2)(17) + 150 • X= 34 +150 • X= 184 • To find a raw score from a percentile • -Use Table B to find z-score • -Calculate the raw score: X= zSD +M • -EX: 72nd percentile SD=17 M=150 • **Table B: 72nd percentile is a z=0.58 • X=(0.58)(17) + 150 • X=9.86 + 150 • X=159.86 • To find a standard deviation from a z-score: • -EX: X=184 M=150 z=2 • **SD= 184-150/2 • SD= 34/2 • SD= 17 • To find a mean from a z-score: M = X – zSD • -EX: z= 2 SD=17 X=184 • **M = 184 – (2)(17) • M = 184 – 34 • M = 150

9. T-Scores • What is a T-score? • -a converted z-score with the mean always set at 50 & standard deviation at 10 • -basically, it’s just another measure of how far a raw score is from the mean • -always a positive number • Calculating T-scores: T = z(10) + 50 • -remember, the mean is always 50 & the standard deviation is always 10 • -EX: z = 2 • **T = (2)(10) +50 • T = 20 + 50 • T = 70 • From T to z to raw scores • -First, convert the T-score to a z-score: • -Second, convert the z-score to a raw score: X= zSD +M • -EX: M = 70 SD = 8.50 T = 65 • **z = 65-50/10 • z = 15/10 = 1.50 • **X= (1.50)(8.50) + 70 • X= 12.75 + 70 = 82.75 Z-Scores Revisited Worksheet

10. Other Normal Curve Transformations • Normal Curve Equivalents • -another popular standardized score (like the z-score or T-score) • -calculated by setting the mean at 50 & the SD at 21 • -larger Range than T-scores because of the larger SD • Stanines • -standardized score used mainly in educational psychology • -divides normal curve into nine units where the Z-score divides into six units • Grade Equivalent Scores • -standardized scores popular in the field of education • -based on the average score found for students in a particular grade at both the same age & time of year • -A GE=6.9 indicates a score that a 6th grader in the 9 month of the school year receives on average • -EX: a 3rd grader in the 1st month of school could take a test & get a GE=4.5 • **That would mean they tested at the level of a 4th grader in the 5th month Z-Scores Revisited Homework due Next Class