What if.... You recently finished taking a test that you received a score of 90 It was out of 200 points The highest score was 110 The average score was 95 The lowest score was 90
Z-score A mathematical way to modify an individual raw score so that the result conveys the score’s relationship to the mean and standard deviation of the other scores
Z-score Ingredients: X Raw score Mean of scores S The standard deviation of scores
Z-score
What it does x - Tells you how far from the mean you are and if you are > or < the mean S Tells you the “size” of this difference
Example Sample 1: X = 8 = 6 S = 5
Example Sample 1: X = 8 = 6 S = 5 Z score =.4
Example Sample 1: X = 8 = 6 S = 1.25
Example Sample 1: X = 8 = 6 S = 1.25 Z-score = 1.6
= 6 S = 5
= 6 S = 5 = 6 S = 1.25
= 6 S = 5 = 6 S = 1.25
X X = 6 S = 5 X = 8 Z =.4 = 6 S = 1.25 X = 8 Z=1.6
Practice SAT and GRE Mean = 500 S = 100 Your scores:Verbal = 400 Analyt = 500 Quant = 700 How did you do on each section?
Practice SAT and GRE Mean = 500 S = 100 Your scores:Verbal = 400Z = -1 Analyt = 500 Z = 0 Quant = 700Z = 2 How did you do on each section?
Practice The history teacher Mr. Hand announced that the lowest test score for each student would be dropped. Jeff scored a 85 on his first test. The mean was 74 and the SD was 4. On the second exam, he made 150. The class mean was 140 and the SD was 15. On the third exam, the mean was 35 and the SD was 5. Jeff got 40. Which test should be dropped?
Practice Test #1 Z = ( ) / 4 = 2.75 Test #2 Z = ( ) / 15 =.67 Test #3 Z = ( ) / 5 = 1.00
Practice
Did Ross do worse in the endurance challenge then in the throwing challenge? Did Monica do better in the throwing challenge than the endurance?
Practice = 34.6 = 7.4 S = 7.12 S = 2.15
Practice = 34.6 = 7.4 S = 7.12 S = 2.15
Practice = 34.6 = 7.4 S = 7.12 S = 2.15
Ross did worse in the throwing challenge than the endurance and Monica did better in the endurance than the throwing challenge. = 34.6 = 7.4 S = 7.12 S = 2.15
Z-Scores A distribution of scores has a standard deviation = 10. Find the z-score corresponding to each of the following values. A score that is 20 points above the mean A score 10 points below the mean A score 15 points above the mean A score 30 points below the mean
Z-Scores A distribution of scores has a standard deviation = 10. Find the z-score corresponding to each of the following values. A score that is 20 points above the mean = 2.00 A score 10 points below the mean = A score 15 points above the mean = 1.50 A score 30 points below the mean = -3.00
Z-Scores A score that is 12 points above the mean corresponds to a Z-score of Z = What is the standard deviation for this population?
Z-Scores A score that is 12 points above the mean corresponds to a Z-score of Z = What is the standard deviation for this population? 12 / y = 2 y = 6
One-Step Beyond For a population of exam scores, a score of X = 58 corresponds to Z =.50 and a score of X = 46 corresponds to Z = Find the mean and standard deviation for the population.
Z-Scores 1. Sketch out the distribution to help 2. Notice that the difference between the two raw scores (X = 58 and X = 46) is 12 raw units. 3. Notice the difference between the two raw scores is 1.5 SD.
Z-Scores 4. Thus, 1.5 (SD) = SD = 8 6. Plug in the SD into either Z-score formula
Z-Scores 7. Z score = 46 - y / 8 = Z score = / 8 = Mean = 54! SD = 8
Effect Size Index On average, males are taller than females. On average, females are less extraverted than males. On average, middle children tend to be more rebellious than first born children.
Effect Size Estimate Gives a mathematical way to answer the question: How much more?
Effect Size Estimate Ingredients: Two different means an estimated SD of both samples
Effect Size Estimate * Note: This is used to examine differences between MEANS
Effect Size Estimate ** When calculating put the bigger mean first (so d will always be positive)
d can range from 0 upward Small effectd =.20 Medium effectd =.50 Large effectd =.80
Heights Do you think the difference in males and females heights is small, medium, or large? Women = 64.6 inches Men = 69.8 inches = 2.8 inches
= 1.86
Rebelliousness Do you think the difference in rebelliousness between middle and first born children is large? Middle = First = 9.88 = 26.68
=.36
= 1.86 =.36 Heights Rebelliousness The mean difference for rebelliousness is 9.57 The mean difference for height is 5.2 Why does Height have a bigger effect size, but smaller mean difference?
Practice Jeff Labowski recently did a study on bowling. He was interested in how much his consumption of alcohol affected his score. He bowled 12 games sober and 12 games intoxicated. Let = 60. Sober: 160, 145, 180, 155, 150, 145, 195, 190, 200,185, 160, 155 Intoxicated: 100, 101, 156, 140, 135, 112, 90, 104, 140,100, 89, 99
Practice Sober: Mean = Median = (6.5) = 160 Min = 145Max = th = (3) = 15075th = 190 Sober: 145, 145, 150, 155, 155, 160, 160, 180, 185, 190, 195, 200, Intoxicated: 89, 90, 99, 100, 100, 101,104, 112,135, 140, 140, 156
Practice Intoxicated Mean = Median = (6.5) =102.5 Min = 89Max = th = (3) = 9975th = 140 Sober: 145, 145, 150, 155, 155, 160, 160, 180, 185, 190, 195, 200, Intoxicated: 89, 90, 99, 100, 100, 101,104, 112,135, 140, 140, 156
Practice Sober Intoxicated Score
Practice Jeff Labowski recently did a study on bowling. He was interested in how much his consumption of alcohol affected his score. He bowled 12 games sober and 12 games intoxicated. Let = 60. ( ) / 60 =.91
Laverne and Shirley always compete against each other. Recently Laverne bragged that she was the fastest bottler at Scholtz. She could bottle 20 beers in one hour while her 4 coworkers could bottle 15, 14, 18, and 19 bottles in this same time frame. Shirley bragged that she was the fastest runner out of all of her friends. She could run 14 mph while her 4 friends could only run 13, 8, 4, and 12 mph. Who actually has the most bragging rights?
Practice Bottle M = 17.2; S = 2.31; Z = 1.21 Speed M = 10.2; S = 3.71; Z = 1.02
Practice What is the Z score for a raw score of 6?
Mean = 4.42S = 1.57
Z score of 6 = ( ) / 1.57 = 1.01
Review Bring cookbook to class on Wednesday!