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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Section 2-6 Measures of Position. M ARIO F . T RIOLA. E IGHTH. E DITION. Measures of Position. z Score (or standard score) the number of standard deviations that a given value x is above or below the mean. Measures of Position. Sample.

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Section 2-6 Measures of Position MARIO F. TRIOLA EIGHTH EDITION

  2. Measures of Position

  3. z Score(or standard score) the number of standard deviations that a given value x is above or below the mean Measures of Position

  4. Sample Measures of Position z score x - x z = s

  5. Sample Measures of Position z score Population x - µ x - x z = z =  s

  6. Sample Measures of Position z score Population x - µ x - x z = z =  s Round to 2 decimal places

  7. FIGURE 2-16 Interpreting Z Scores Unusual Values Ordinary Values Unusual Values - 3 - 2 - 1 0 1 2 3 Z

  8. Measures of Position Quartiles, Deciles, Percentiles

  9. Quartiles

  10. Quartiles Q1, Q2, Q3

  11. Quartiles Q1, Q2, Q3 divides ranked scores into four equal parts

  12. Quartiles Q1, Q2, Q3 divides ranked scores into four equal parts 25% 25% 25% 25% Q1 Q2 Q3

  13. Quartiles Q1, Q2, Q3 divides ranked scores into four equal parts 25% 25% 25% 25% Q1 Q2 Q3 (minimum) (maximum) (median)

  14. Deciles D1, D2, D3, D4, D5, D6, D7, D8, D9 divides ranked data into ten equal parts

  15. 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% D1 D2 D3 D4 D5 D6 D7 D8 D9 Deciles D1, D2, D3, D4, D5, D6, D7, D8, D9 divides ranked data into ten equal parts

  16. 99 Percentiles Percentiles

  17. Quartiles, Deciles, Percentiles Fractiles

  18. Quartiles, Deciles, Percentiles Fractiles (Quantiles) partitions data into approximately equal parts

  19. Finding the Percentile of a Given Score

  20. Finding the Percentile of a Given Score number of scores less than x Percentile of score x = • 100 total number of scores

  21. Finding the Score Given a Percentile

  22. k L = • n 100 Finding the Score Given a Percentile n total number of values in the data set kpercentile being used L locator that gives the position of a value Pkkth percentile

  23. ) k ( 100 Start Finding the Value of the kth Percentile Sort the data. (Arrange the data in order of lowest to highest.) Compute L = nwhere n = number of values k = percentile in question The value of the kth percentile is midway between the Lth value and the next value in the sorted set of data. Find Pk by adding the L th value and the next value and dividing the total by 2. Is L a whole number ? Yes No Change L by rounding it up to the next larger whole number. Figure 2-17 The value of Pk is the Lth value, counting from the lowest

  24. Q1 = P25 Q2 = P50 Q3 = P75 Quartiles

  25. Q1 = P25 Q2 = P50 Q3 = P75 D1 = P10 D2 = P20 D3 = P30 • • • D9 = P90 Deciles Quartiles

  26. Interquartile Range (or IQR): Q3 - Q1

  27. Interquartile Range (or IQR): Q3 - Q1 Semi-interquartile Range: Q3 - Q1 2

  28. Interquartile Range (or IQR): Q3 - Q1 Semi-interquartile Range: Midquartile: Q3 - Q1 2 Q1 + Q3 2

  29. Interquartile Range (or IQR): Q3 - Q1 Semi-interquartile Range: Midquartile: 10 - 90 Percentile Range: P90 - P10 Q3 - Q1 2 Q1 + Q3 2

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