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Hand out z tables

This course provides an introduction to statistics for social science students. Topics covered include deviation scores, standard deviation, mean, variance, and normal distribution.

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Hand out z tables

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  1. Hand out z tables

  2. Introduction to Statistics for the Social SciencesSBS200, COMM200, GEOG200, PA200, POL200, or SOC200Lecture Section 001, Spring 2015Room 150 Harvill Building8:00 - 8:50 Mondays, Wednesdays & Fridays. Welcome http://courses.eller.arizona.edu/mgmt/delaney/d15s_database_weekone_screenshot.xlsx

  3. Lab sessions Labs continue this week finalizing Project 1

  4. Schedule of readings Before next exam (March 6th) Please read chapters 5, 6, & 8 in Ha & Ha Please read Chapters 10, 11, 12 and 14 in Plous Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness

  5. Homework due – Wednesday (February 25th ) On class website: Please print and complete homework worksheet #10 Calculating z-score, raw scores and areas under normal curve

  6. Writing Assignment – Revisit our old Pop Quiz Review • 1. What is a “deviation score” • 2. Preston has a deviation score of 2: What does that tell us about Preston? • Is he taller or shorter than the mean? And by how much? • Are most people in the group taller or shorter than Preston • Mike has a deviation score of -4: What does that tell us about Mike? • Is he taller or shorter than the mean? And by how much? • Are most people in the group taller or shorter than Mike • Diallo has a deviation score of 0: What does that tell us about Diallo? • Is he taller or shorter than the mean? And by how much? • Are most people in the group taller or shorter than Diallo? • Please write the formula for the standard deviation of a population • Please draw 3 curves showing 1, 2 & 3 standard deviations from mean How far away is each score from the mean? Preston is 2” taller than the mean (taller than most) Mike is 4” shorter than the mean (shorter than most) Diallo is exactly same height as mean (half taller half shorter)

  7. Writing Assignment – Revisit our old Pop Quiz Review The standard deviation (population) 7. What does this symbol refer to? sigma • What is it called? • What does it mean? • Is it referring to a sample or population? population The mean (population) 8. What does this symbol refer to? mu • What is it called? • What does it mean? • Is it referring to a sample or population? population The mean (sample) 9. What does this symbol refer to? x-bar • What is it called? • What does it mean? • Is it referring to a sample or population? sample The standard deviation (sample) 10. What does this symbol refer to? s • What is it called? • What does it mean? • Is it referring to a sample or population? sample 11. What does this symbol refer to? Each individual score

  8. Writing Assignment – Revisit our old Pop Quiz Review population sample 12. What does this refer to? Variance • What are they called? • What do they refer to? • How are they different S squared Sigma squared 13. What does this refer to? sample population Deviation scores • What are they called? • How are they different population sample 14. What do these two refer to? Sum of squares • What are they called? • How are they different 15. What does this refer to? Degrees of freedom • What is it called? • Use it for sample data or population?

  9. Standard Deviation Writing Assignment – Revisit our old Pop Quiz Review 16. What does this refer to? • What are they called? • What do they refer to? • How are they different population sample Variance 17. What does this refer to? • What are they called? • What do they refer to? • How are they different population sample

  10. Scores, standard deviations, and probabilities The normal curve always has the same shape. They differ only by having different means and standard deviation

  11. Scores, standard deviations, and probabilities Actually 68.26 Actually 95.44 To be exactly 95% we will use z = 1.96

  12. Scores, standard deviations, and probabilities What is total percent under curve? What proportion of curve is above the mean? .50 100% The normal curve always has the same shape. They differ only by having different means and standard deviation Review

  13. Scores, standard deviations, and probabilities What score is associated with 50th percentile? What percent of curve is below a score of 100? 50% median Mean = 100 Standard deviation = 5 The normal curve always has the same shape. They differ only by having different means and standard deviation Review

  14. Raw scores, z scores & probabilities Distance from the mean (z scores) convert convert Proportion of curve (area from mean) Raw Scores (actual data) 68% We care about this! What is the actual number on this scale?“height” vs “weight” “pounds” vs “test score” We care about this! “percentiles” “percent of people” “proportion of curve” “relative position” z = -1 z = -1 z = 1 z = 1 68% Proportion of curve (area from mean) Raw Scores (actual data) Distance from the mean (z scores) convert convert Review

  15. Normal distribution Raw scores z-scores probabilities Z Scores Have z Find raw score Have z Find area z table Formula Have area Find z Area & Probability Raw Scores Have raw score Find z Review

  16. 50 60 Find the area under the curve that falls between 50 and 60 34.13% 1) Draw the picture 2) Find z score 3) Go to z table - find area under correct column 4) Report the area z-table (from z to area) Distance from the mean ( from raw to z scores) Raw Scores (actual data) Proportion of curve (area from mean) z = 1 50 60 10 Review

  17. Writing AssignmentLet’s do some problems Mean = 50Standard deviation = 10

  18. Let’s do some problems ? Mean = 50Standard deviation = 10 60 Find the area under the curve that falls below 60 means the same thing as Find the percentile rank for score of 60 Problem 1

  19. Let’s do some problems ? 60 Mean = 50Standard deviation = 10 Find the percentile rank for score of 60 z-table (from z to area) Distance from the mean ( from raw to z scores) .3413 .5000 Raw Scores (actual data) Proportion of curve (area from mean) 1) Find z score z score = 60 - 50 10 = 1 2) Go to z table - find area under correct column (.3413) 3) Look at your picture - add .5000 to .3413 = .8413 4) Percentile rank or score of 60 = 84.13% Problem 1 Hint always draw a picture!

  20. ? 75 Mean = 50Standard deviation = 10 Find the percentile rank for score of 75 .4938 1) Find z score z score = 75 - 50 10 z score = 25 10 = 2.5 2) Go to z table Problem 2 Hint always draw a picture!

  21. ? 75 Mean = 50Standard deviation = 10 Find the percentile rank for score of 75 .4938 .5000 1) Find z score z score = 75 - 50 10 z score = 25 10 = 2.5 2) Go to z table 3) Look at your picture - add .5000 to .4938 = .9938 4) Percentile rank or score of 75 = 99.38% Problem 2 Hint always draw a picture!

  22. ? 45 Mean = 50Standard deviation = 10 Find the percentile rank for score of 45 z-table (from z to area) Distance from the mean ( from raw to z scores) Raw Scores (actual data) Proportion of curve (area from mean) 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table Problem 3

  23. ? 45 Mean = 50Standard deviation = 10 Find the percentile rank for score of 45 .1915 ? 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table Problem 3

  24. ? 45 Mean = 50Standard deviation = 10 Find the percentile rank for score of 45 .1915 z-table (from z to area) Distance from the mean ( from raw to z scores) .3085 Raw Scores (actual data) Proportion of curve (area from mean) 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table 3) Look at your picture - subtract .5000 -.1915 = .3085 Problem 3 4) Percentile rank or score of 45 = 30.85%

  25. ? Mean = 50Standard deviation = 10 Find the percentile rank for score of 55 z-table (from z to area) Distance from the mean ( from raw to z scores) 55 Raw Scores (actual data) Proportion of curve (area from mean) 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table Problem 4

  26. ? Mean = 50Standard deviation = 10 Find the percentile rank for score of 55 .1915 55 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table Problem 4

  27. ? Mean = 50Standard deviation = 10 Find the percentile rank for score of 55 .1915 z-table (from z to area) Distance from the mean ( from raw to z scores) .5 55 Raw Scores (actual data) Proportion of curve (area from mean) 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table 3) Look at your picture - add .5000 +.1915 = .6915 4) Percentile rank or score of 55 = 69.15% Problem 4

  28. Find the score for z = -2 ? Mean = 50Standard deviation = 10 30 Hint always draw a picture! Find the score that is associated with a z score of -2 z-table (from z to area) Distance from the mean ( from raw to z scores) raw score = mean + (z score)(standard deviation) Raw Scores (actual data) Proportion of curve (area from mean) Raw score = 50 + (-2)(10) Raw score = 50 + (-20) = 30 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion

  29. ? .7700 ? Mean = 50Standard deviation = 10 Find the score for percentile rank of 77%ile z-table (from z to area) Distance from the mean ( from raw to z scores) Raw Scores (actual data) Proportion of curve (area from mean) Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 5

  30. ? .7700 ? Mean = 50Standard deviation = 10 .27 Find the score for percentile rank of 77%ile .5 .5 + .27 = .77 .5 .27 1) Go to z table - find z score for for area .2700 (.7700 - .5000) = .27 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion area = .2704 (closest I could find to .2700) z = 0.74 Problem 5

  31. ? .7700 ? Mean = 50Standard deviation = 10 .27 Find the score for percentile rank of 77%ile .5 x = 57.4 .5 .27 2) x = mean + (z)(standard deviation) x = 50 + (0.74)(10) x = 57.4 x = 57.4 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 5

  32. ? .5500 ? Mean = 50Standard deviation = 10 Find the score for percentile rank of 55%ile z-table (from z to area) Distance from the mean ( from raw to z scores) Raw Scores (actual data) Proportion of curve (area from mean) Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 6

  33. ? .5500 ? Mean = 50Standard deviation = 10 .05 Find the score for percentile rank of 55%ile .5 .5 + .05 = .55 .5 .05 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .05 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion area = .0517 (closest I could find to .0500) z = 0.13 Problem 6

  34. ? .5500 ? Mean = 50Standard deviation = 10 .05 Find the score for percentile rank of 55%ile .5 .5 .05 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .05 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion area = .0517 (closest I could find to .0500) z = 0.13 Problem 6

  35. ? .5500 ? Mean = 50Standard deviation = 10 .05 Find the score for percentile rank of 55%ile .5 x = 51.3 .5 .05 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .0500 area = .0517 (closest I could find to .0500) z = 0.13 2) x = mean + (z)(standard deviation) x = 50 + (0.13)(10) x = 51.3 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion x = 51.3 Problem 6

  36. Thank you! See you next time!!

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