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z – Scores and Probability

z – Scores and Probability. z - Score. A standardized value A number of standard deviations a given value, x, is above or below the mean z = (score (x) – mean)/s (standard deviation) A positive z-score means the value lies above the mean A negative z-score means the value lies below the mean

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z – Scores and Probability

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  1. z – Scores and Probability

  2. z - Score • A standardized value • A number of standard deviations a given value, x, is above or below the mean • z = (score (x) – mean)/s (standard deviation) • A positive z-score means the value lies above the mean • A negative z-score means the value lies below the mean • Round to 2 decimals

  3. z – Score Example • The mean for IQ scores is 50 with a standard deviation of 5 with a normal distribution • What is the probability that scores will be between 45 and 55? • Calculate z-score first • Use the Normal Distribution (z) statistical table

  4. z – Score Example • Score (x) = 45 • Mean = 50 • s = 5 • z = (score (x) – mean)/s (standard deviation) • z = (45 - 50)/5 (standard deviation) • = -5/5, = -1 • Wait! There’s a second value to consider.

  5. z – Score Example • Score (x) = 55 • Mean = 50 • s = 5 • z = (score (x) – mean)/s (standard deviation) • z = (55 - 50)/5 (standard deviation) • = 5/5, = 1

  6. z – Score Example • Using the normal distribution (z) statistical table: • Determine the area from the mean: • 1 s up (mean to z) = .3413 • 1 s down (mean to z) = .3413 • Add the 2 values together • .3413 + .3413 = .6826 * 100% = 68.26% • So, the probability that a score will be between 45 and 55 is 68.26%!

  7. z – Score Example • The mean for IQ scores is 50 with a standard deviation of 5 with a normal distribution • What is the probability that an IQ score will be between 55 and 60? • Calculate z-score first • Use the Normal Distribution (z) statistical table

  8. z – Score Example • Score (x) = 55 • Mean = 50 • s = 5 • z = (score (x) – mean)/s (standard deviation) • z = (55 - 50)/5 (standard deviation) • = 5/5, = 1 • Wait! There’s a second value to consider.

  9. z – Score Example • Score (x) = 60 • Mean = 50 • s = 5 • z = (score (x) – mean)/s (standard deviation) • z = (60 - 50)/5 (standard deviation) • = 10/5, = 2

  10. z – Score Example • Using the normal distribution (z) statistical table: • Determine the area from the mean: • 1 s up (mean to z) = .3413 • 2 s up (mean to z) = .4772 • Subtract the 2 values (because we only want the distance from 1z to 2z, not the mean to 2z) • .4772 - .3413 = .1359 * 100% = 13.59% • So, the probability that an IQ score will be between 55 and 60 is 13.59%!

  11. z – Score Example • The mean for IQ scores is 100 with a standard deviation of 15 with a normal distribution • What percentage of scores will lie below 100? • Calculate z-score first • Use the Normal Distribution (z) statistical table

  12. z – Score Example • Score (x) = 100 • Mean = 100 • s = 15 • z = (score (x) – mean)/s (standard deviation) • z = (100 - 100)/15 (standard deviation) • = 0/15 = 0

  13. z – Score Example • Using the normal distribution (z) statistical table: • Determine the area from the mean: • 0 s down (larger portion) = .5000 • So, the probability that a student’s IQ score will be below 100 is 50%.

  14. z – Score Example • The mean for IQ scores is 100 with a standard deviation of 15 with a normal distribution • What percentage of scores will lie below 115? • Calculate z-score first • Use the Normal Distribution (z) statistical table

  15. z – Score Example • Score (x) = 115 • Mean = 100 • s = 15 • z = (score (x) – mean)/s (standard deviation) • z = (115 - 100)/15 (standard deviation) • = 15/15 = 1

  16. z – Score Example • Using the normal distribution (z) statistical table: • Determine the area from the mean: • 1 s up (larger portion) = .8413 • So, the percentage of scores that will be below 115 is 84.13%

  17. z – Score Example • The mean for IQ scores is 100 with a standard deviation of 15 with a normal distribution • What percentage of scores will lie above 115? • Calculate z-score first • Use the Normal Distribution (z) statistical table

  18. z – Score Example • Score (x) = 115 • Mean = 100 • s = 15 • z = (score (x) – mean)/s (standard deviation) • z = (115 - 100)/15 (standard deviation) • = 15/15 = 1

  19. z – Score Example • Using the normal distribution (z) statistical table: • Determine the area from the mean: • 1 s up (smaller portion) = .1587 • So, the probability that a student’s IQ score will be above 115 is 15.87% • Note: this area of 15.87% plus the area of scores below 115, 84.13%, equal 100%.

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