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Lesson 5

Lesson 5. The z-Scores Distribution. Location in a Distribution. z-scores are used to describe the exact location of a score within a distribution. The sign tells whether the score is above (+) or below (-) the mean.

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Lesson 5

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  1. Lesson 5 The z-Scores Distribution Lesson 5 z-Scores

  2. Location in a Distribution z-scores are used to describe the exact location of a score within a distribution. • The sign tells whether the score is above (+) or below (-) the mean. • The number tells the distance between the score and the mean in terms of the standard deviation. • Example: A score of +1 is one standard deviation above the mean. Lesson 5 z-Scores

  3. Why? Converting a raw data set or score into a standardized format indicates • whether the raw score is below the mean or above the mean. • tells exactly how far above or below the mean the score is. • allows you to compare scores across entirely different measurements. Lesson 5 z-Scores

  4. z-Score Formula • The formula for converting a raw score from a distribution to a z-score is: Lesson 5 z-Scores

  5. Other Formula Applications • Using the same formula you can: • Find the raw score if you have its z-score, the mean, and the standard deviation (or variance). • Find the meanif you have a raw score, its z-score, and the standard deviation (or variance). • Find the standard deviation (or variance) if you have a raw score, its z-score, and the mean. Lesson 5 z-Scores

  6. Standardizing a Distribution • If we convert every raw score in a distribution to a z-score, then we have standardized the distribution. • A standardized distribution has a number of advantages: • The shape of the z-score distribution remains the same as the original. • The mean is always 0. • The standard deviation is always 1. Lesson 5 z-Scores

  7. Standardized Facts • The raw mean becomes 0 (any number subtracted from itself = 0) • The raw standard deviation becomes 1 (any number divided by itself = 1) • Every raw score X can be converted to a standardized z-score by using the z formula Lesson 5 z-Scores

  8. More Facts • All raw scores greater than the mean standardize to z-scores greater than 0. • All raw scores less than the mean • standardize to z-scores less than 0. Lesson 5 z-Scores

  9. Applications • Standardized distributions allow you to know the precise location of every score in the distribution. • Standardized distributions can be used to compare two or more dissimilar raw distributions. Lesson 5 z-Scores

  10. Picture This! • This is a normal distribution • represents a total population • bell shaped • symmetric around m • The further away from m a score is (either greater than or less than), the lower the frequency at which that score occurs. m Lesson 5 z-Scores

  11. Picture This! Because this distribution represents the exhaustive set of all possible raw scores, the total proportion of possible scores (represented by the area under the curve) is 1.00 Lesson 5 z-Scores

  12. The proportion of the area represented under the curve that is described by any single score or any set of scores is always between 0 and 1. Proportions Under the Curve Lesson 5 z-Scores

  13. Proportions Under the Curve The proportion of those scores greater than the mean is 0.5000 m Lesson 5 z-Scores

  14. Proportions Under the Curve The proportion of those scores less than the mean is 0.5000 m Lesson 5 z-Scores

  15. From Raw to Standardized • When we standardize this normal distribution, we simply exchange the X-axis from raw score terms to standardized (z) score terms. Lesson 5 z-Scores

  16. +/- 3 Standard Deviations Most z scores fall between -3 and +3. That is, within 3 standard deviations above and below the mean. A very few will fall between -4 and -3 and between +3 and +4. -3 -2 -1 0 +3 +2 +1 Lesson 5 z-Scores

  17. Finding a Raw Score Given a z-score from a standardized distribution and both the population mean and standard deviation (or the variance) for that distribution, you can use this formula to find the raw score associated with the given z-score. Lesson 5 z-Scores

  18. Finding a Raw Score • Suppose we have a z-score of -2.70. If m=100 and s = 20, then Lesson 5 z-Scores

  19. Finding a Population Mean • Given a raw score from a standardized distribution and its associated z-score along with the population standard deviation (or the variance) for that distribution, you can use the following formula to find the raw mean (m) for that population. Lesson 5 z-Scores

  20. Finding a Population Standard Deviation • Given a raw score from a standardized distribution and its associated z-score along with the population mean for that distribution, you can use the following formula to find the raw standard deviation (or variance) for that population. Lesson 5 z-Scores

  21. Finding the Area Under a Curve • We can use the standardized normal distribution to answer many important questions about a distribution. Lesson 5 z-Scores

  22. Some Common Problems • Finding the proportion (or %) of scores greater than X • When X is greater than the mean. • When X is less than the mean. • Finding the proportion (or %) of scores less than X. • When X is greater than the mean. • When X is less than the mean. Lesson 5 z-Scores

  23. More Common Problems • Finding the proportion (or %) of scores between two Xs: • When one of the Xs is the mean. • When both Xs are greater than the mean. • When both Xs are less than the mean. • When one X is greater than the mean and the other X is less than the mean. Lesson 5 z-Scores

  24. Proportion Greater than X • When X is greater than the mean: tail m X Lesson 5 z-Scores

  25. Proportion Greater than X • When X is greater than the mean: • Convert the X value to a z-score. • z = (X – m) / s > 0 • Find z in column (A). • The proportion needed is in the tail, so use Column (C). • For %, multiply by 100. Lesson 5 z-Scores

  26. Proportion Greater than X • Example when X is greater than the mean • Let m = 100, s = 20, and X=150 • Find z = (X – m) / s which is (150-100)/20 • So z = 50/20 = 2.5 • Double check—is z greater than 0? Yes. Lesson 5 z-Scores

  27. Proportion Greater than X • Find z=2.50 in Column (A) • Find the correct proportion in the tail in Column (C). So the area is .0062 Lesson 5 z-Scores

  28. Proportion Greater than X • When X is less than the mean: body X Lesson 5 z-Scores

  29. Proportion Greater than X • When X is less than the mean: • Convert the X value to a z-score. • z = (X – m) / s < 0 • Find z in column A (ignore – sign). • The proportion needed is in the body, so use Column (B). • For %, multiply by 100. Lesson 5 z-Scores

  30. Proportion Greater than X • Example when X is less than the mean • Let m = 100, s = 20, and X=85 • Find z = (X – m) / s which is (85-100)/20 • So z = -15/20 = -0.75 • Double check—is z less than 0? Yes. Lesson 5 z-Scores

  31. Proportion Greater than X • Find z=-0.75 in Column (A) • Find the correct proportion in the body in Column (B). So the area is .7734 Lesson 5 z-Scores

  32. Proportion Less than X • When X is greater than the mean: body X Lesson 5 z-Scores

  33. Proportion Less than X • When X is greater than the mean: • Convert the X value to a z-score. • z = (X – m) / s > 0 • Find z in column A. • The proportion needed is in the body, so use Column (B). • For %, multiply by 100. Lesson 5 z-Scores

  34. Proportion Less than X • When X is less than the mean: tail X Lesson 5 z-Scores

  35. Proportion Less than X • When X is less than the mean: • Convert the X value to a z-score. • z = (X – m) / s < 0 (ignore – sign) • Find z in column A. • The proportion needed is in the tail, so use Column (C). • For %, multiply by 100. Lesson 5 z-Scores

  36. Proportion Between Two Xs • When one of the Xs is the mean X Lesson 5 z-Scores

  37. Proportion Between Two Xs • When one of the Xs is the mean • Convert the X value to a z-score. • z = (X – m) / s > 0 or z = (X – m) / s < 0 • Find z in column A (ignore sign). • The proportion needed is in the “Proportion Between Mean and z” column, which is Column (D). • For %, multiply by 100. Lesson 5 z-Scores

  38. Proportion Between Two Xs • When both Xs are greater than the mean X1 X2 Lesson 5 z-Scores

  39. Proportion Between Two Xs • When both Xs are greater than the mean • X1 < X2 • z1 = (X1 – m) / s > 0 • z2 = (X2 – m) / s > 0 • z1 < z2 • Find z1 in column (A). Use proportion in Column (D) as p1. • Find z2 in column (A). Use proportion Column (D) as p2. • Then… Lesson 5 z-Scores

  40. Proportion Between Two Xs • When both Xs are greater than the mean (continued) • p = p2 – p1 - = p2 p p1 Lesson 5 z-Scores

  41. Proportion Between Two Xs • Example when both Xs are greater than the mean • Let m = 100, s = 20, X1=120, and X2=140 • Find z1 = (X1 – m) / s which is (120-100)/20 • So z1 = 20/20 = 1.00 • Find z2=(X2- m)/ s which is (140-100)/20 • So z2 = 40/20 = 2.00 Lesson 5 z-Scores

  42. Proportion Between Two Xs • Example when both Xs are greater than the mean • Then p1 = .3413 and p2 . - = p=.1359 p1= .3413 p2= .4772 Lesson 5 z-Scores

  43. Proportion Between Two Xs • When both Xs are less than the mean X1 X2 Lesson 5 z-Scores

  44. Proportion Between Two Xs • When both Xs are less than the mean • X1 > X2 • z1 = (X1 – m) / s < 0 • z2 = (X2 – m) / s < 0 • z1 > z2 • Find z1 in column (A). Use proportion in the “Between mean and z” column (D) as p1. • Find z2 in column A. Use proportion in the “Between mean and z” column (D) as p2. • Then… Lesson 5 z-Scores

  45. Proportion Between Two Xs • When both Xs are less than the mean (continued) • p = p2 – p1 - = p2 p p1 Lesson 5 z-Scores

  46. Proportion Between Two Xs • When one X is less than the mean and the other X is greater than the mean X2 X1 Lesson 5 z-Scores

  47. Proportion Between Two Xs • When one X is less than the mean and the other X is greater than the mean • X1 < X2 ; X1 < m ; X2 > m • z1 = (X1 – m) / s < 0 • z2 = (X2 – m) / s > 0 • z1 < z2 ; z1 < 0; z2 > 0 • Find z1 in column (A). Use proportion in the “Between mean and z” column (D) as p1. • Find z2 in column A. Use proportion in the “Between mean and z” column (D) as p2. Lesson 5 z-Scores

  48. Proportion Between Two Xs • When one X is less than the mean and the other X is greater than the mean (continued) • p = p2 + p1 + = p2 p p1 Lesson 5 z-Scores

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