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Topic 17

Topic 17. Standard Deviation, Z score, and Normal Distribution. Standard Deviation. Compute the standard deviation of 6, 1, 2, 11 Compute the mean first = (6 + 1 + 2 + 11) / 4 = 5 Now compute the squared deviations (1–5) 2 = 16, (2–5) 2 = 9, (6–5) 2 = 1, (11–5) 2 = 36

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Topic 17

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  1. Topic 17 Standard Deviation, Z score, and Normal Distribution

  2. Standard Deviation • Compute the standard deviation of 6, 1, 2, 11 • Compute the mean first = (6 + 1 + 2 + 11) / 4 = 5 • Now compute the squared deviations (1–5)2 = 16, (2–5)2 = 9, (6–5)2 = 1, (11–5)2 = 36 • Average the squared deviations (16 + 9 + 1 + 36) / 3 = 20.7 • Taking the square root of 20.7 = 4.55 Standard deviation = 4.55 Standard deviation is not a resistant measurement of the spread.

  3. Empirical Rule

  4. Empirical Rule • The empirical rule • If the distribution is roughly bell shaped, then • The empirical rule • If the distribution is roughly bell shaped, then • Approximately 68% of the data will lie within 1 standard deviation of the mean • The empirical rule • If the distribution is roughly bell shaped, then • Approximately 68% of the data will lie within 1 standard deviation of the mean • Approximately 95% of the data will lie within 2 standard deviations of the mean • The empirical rule • If the distribution is roughly bell shaped, then • Approximately 68% of the data will lie within 1 standard deviation of the mean • Approximately 95% of the data will lie within 2 standard deviations of the mean • Approximately 99.7% of the data (i.e. almost all) will lie within 3 standard deviations of the mean

  5. Empirical Rule • For a variable with mean 17 and standard deviation 3.4 • For a variable with mean 17 and standard deviation 3.4 • Approximately 68% of the values will lie between(17 – 3.4) and (17 + 3.4), i.e. 13.6 and 20.4 • For a variable with mean 17 and standard deviation 3.4 • Approximately 68% of the values will lie between(17 – 3.4) and (17 + 3.4), i.e. 13.6 and 20.4 • Approximately 95% of the values will lie between(17 – 2  3.4) and (17 + 2  3.4), i.e. 10.2 and 23.8 • For a variable with mean 17 and standard deviation 3.4 • Approximately 68% of the values will lie between(17 – 3.4) and (17 + 3.4), i.e. 13.6 and 20.4 • Approximately 95% of the values will lie between(17 – 2  3.4) and (17 + 2  3.4), i.e. 10.2 and 23.8 • Approximately 99.7% of the values will lie between(17 – 3  3.4) and (17 + 3  3.4), i.e. 6.8 and 27.2 • For a variable with mean 17 and standard deviation 3.4 • Approximately 68% of the values will lie between(17 – 3.4) and (17 + 3.4), i.e. 13.6 and 20.4 • Approximately 95% of the values will lie between(17 – 2  3.4) and (17 + 2  3.4), i.e. 10.2 and 23.8 • Approximately 99.7% of the values will lie between(17 – 3  3.4) and (17 + 3  3.4), i.e. 6.8 and 27.2 • A value of 2.1 and a value of 33.2 would both be very unusual

  6. Z score • z-scores can be used to compare the relative positions of data values in different samples • z-scores can be used to compare the relative positions of data values in different samples • Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12 • z-scores can be used to compare the relative positions of data values in different samples • Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12 • Pat received a grade of 72 on her biology exam where the mean grade was 65 and the standard deviation was 10 • z-scores can be used to compare the relative positions of data values in different samples • Pat received a grade of 82 on her statistics exam where the mean grade was 74 and the standard deviation was 12 • Pat received a grade of 72 on her biology exam where the mean grade was 65 and the standard deviation was 10 • Pat received a grade of 91 on her kayaking exam where the mean grade was 88 and the standard deviation was 6

  7. Z score • Statistics • Grade of 82 • z-score of (82 – 74) / 12 = .67 • Statistics • Grade of 82 • z-score of (82 – 74) / 12 = .67 • Biology • Grade of 72 • z-score of (72 – 65) / 10 = .70 • Statistics • Grade of 82 • z-score of (82 – 74) / 12 = .67 • Biology • Grade of 72 • z-score of (72 – 65) / 10 = .70 • Kayaking • Grade of 81 • z-score of (91 – 88) / 6 = .50 • Statistics • Grade of 82 • z-score of (82 – 74) / 12 = .67 • Biology • Grade of 72 • z-score of (72 – 65) / 10 = .70 • Kayaking • Grade of 81 • z-score of (91 – 88) / 6 = .50 • Biology was the highest relative grade

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