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Quantiles

Quantiles. Edexcel S1 Mathematics 2003. D1. D2. D3. D4. D5. D6. D7. D8. D9. Introduction- what is a quantile?. Quantiles are used to divide data into intervals containing an equal number of values. For example:. Quartiles Q1, Q2, Q3 divide data into 4 parts.

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Quantiles

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  1. Quantiles Edexcel S1 Mathematics 2003

  2. D1 D2 D3 D4 D5 D6 D7 D8 D9 Introduction- what is a quantile? Quantiles are used to divide data into intervals containing an equal number of values. For example: • Quartiles Q1, Q2, Q3 divide data into 4 parts • Deciles D1, …, D9 divide data into 10 parts • Percentiles P1, …, P100 divide into 100 parts . . . . . .. . . … . . .. .. ... ... . .. ... ...... .... .. .. . .. . … … . . . . . .

  3. Introduction – Grouped / Ungrouped data Ungrouped data • Treat data as individual values • Use textbook method of rounding to next value or next .5 th value Grouped data • Use linear interpolation to estimate quantile. • Treat data as continuous within each group / class • Assumes values are evenly distributed within each class.

  4. Example –Ungrouped data Question: The number of appointments at a doctors surgery for each of 18 days were: 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 15, 16 Find the median, and first and ninth deciles of the number of appointments Answer: 6 7 7 8 8 9 9 10 10 11 11 11 11 12 12 13 15 16 median The median is the middle value: n/2 = 18/2 = 9  9.5th value = = 10.5 appointments Whole number- so round up to .5th Find average of 9th and 10th value

  5. Example –Ungrouped data Question: The number of appointments at a doctors surgery for each of 18 days were: 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 15, 16 Find the median, and first and ninth deciles of the number of appointments Answer: 6 7 7 8 8 9 9 10 10 11 11 11 11 12 12 13 15 16 D1 median The median is the middle value: n/2 = 18/2 = 9  9.5th value = = 10.5 appointments The first decile, D1, is the 1/10th value:  2nd value = 7 appointments n/10 = 18/10 = 1.8 Not whole - so round up to whole Find the 2nd value

  6. Example –Ungrouped data Question: The number of appointments at a doctors surgery for each of 18 days were: 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 15, 16 Find the median, and first and ninth deciles of the number of appointments Answer: 6 7 7 8 8 9 9 10 10 11 11 11 11 12 12 13 15 16 D1 median D9 The median is the middle value: n/2 = 18/2 = 9  9.5th value = = 10.5 appointments The first decile, D1, is the 1/10th value:  2nd value = 7 appointments n/10 = 18/10 = 1.8 The ninth decile, D9, is the 9/10th value: 9n/4 = 9x18/10 = 16.2  17th value = 15 appointments Not whole - so round up Find the 17th value to whole

  7. Example –Ungrouped data Question: The number of appointments at a doctors surgery for each of 18 days were: 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 15, 16 Find the median, and first and ninth deciles of the number of appointments Answer: 6 7 7 8 8 9 9 10 10 11 11 11 11 12 12 13 15 16 D1 median D9 The median is the middle value: n/2 = 18/2 = 9  9.5th value = = 10.5 appointments The first decile, D1, is the 1/10th value:  2nd value = 7 appointments n/10 = 18/10 = 1.8 The ninth decile, D9, is the 9/10th value: 9n/4 = 9x18/10 = 16.2  17th value = 15 appointments

  8. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: The median is the middle value: lies in class 15 - 19 n/2 = 100/2 = 50th value No rounding as interpolation is being used 9 + 15 + 20 = 44 so 50th value is not in first 3 classes

  9. 15 – 19 class 15 44 14.5 19.5 median Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: The median is the middle value: lies in class 15 - 19 n/2 = 100/2 = 50th value Cumulative frequency to lcb Median position (19.5 – 14.5) median = 14.5 + lcb ucb Lower class boundary (lcb) class frequency

  10. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Linear interpolation: Assume 15 values are evenly distributed in class Answer: The median is the middle value: lies in class 15 - 19 n/2 = 100/2 = 50th value Cumulative frequency to lcb Median position 15 – 19 class 6 9 44 (19.5 – 14.5) median = 14.5 + lcb ucb 14.5 19.5 Lower class boundary (lcb) class frequency median

  11. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Linear interpolation: Assume 15 values are evenly distributed in class Answer: The median is the middle value: lies in class 15 - 19 n/2 = 100/2 = 50th value Frequency in class up to median 15 – 19 class 6 9 44 (19.5 – 14.5) median = 14.5 + lcb ucb 14.5 19.5 Lower class boundary (lcb) class frequency median

  12. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Linear interpolation: Assume 15 values are evenly distributed in class Answer: The median is the middle value: lies in class 15 - 19 n/2 = 100/2 = 50th value Frequency in class up to median 15 – 19 class 6 9 44 . (5) median = 14.5 + class width 14.5 19.5 5 Lower class boundary (lcb) class frequency median

  13. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Linear interpolation: Assume 15 values are evenly distributed in class Answer: The median is the middle value: lies in class 15 - 19 n/2 = 100/2 = 50th value 15 – 19 class 6 9 44 median = 14.5 + 2 14.5 19.5 2 3 median

  14. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Linear interpolation: Assume 15 values are evenly distributed in class Answer: The median is the middle value: lies in class 15 - 19 n/2 = 100/2 = 50th value 15 – 19 class 6 9 44 median = 14.5 + 2 = 16.5 minutes 14.5 19.5 median = 16.5

  15. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: The Q1 value is the 1/4th value: Q1 lies in class 10 - 14 n/4 = 100/4 = 25th value

  16. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: The Q1 value is the 1/4th value: Q1 lies in class 10 - 14 n/4 = 100/4 = 25th value Cumulative frequency to lcb Q1 position 10 – 14 class 1 19 24 (14.5 – 9.5) Q1 = 9.5 + lcb ucb 9.5 14.5 Lower class boundary (lcb) class frequency Q1

  17. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: The Q1 value is the 1/4th value: Q1 lies in class 10 - 14 n/4 = 100/4 = 25th value Frequency in class up to Q1 10 – 14 class 1 19 24 . (5) = 9.75 Q1 = 9.5 + class width 9.5 14.5 Lower class boundary (lcb) class frequency Q1

  18. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: The Q1 value is the 1/4th value: Q1 lies in class 10 - 14 n/4 = 100/4 = 25th value 10 – 14 class 1 19 24 . (5) = 9.75 Q1 = 9.5 + 9.5 14.5 Q1

  19. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: Q1 = 9.75 The Q3 value is the 3/4th value: Q3 lies in class 20 - 29 n/4 = 100/4 = 75th value

  20. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: Q1 = 9.75 The Q3 value is the 3/4th value: Q3 lies in class 20 - 29 n/4 = 100/4 = 75th value Cumulative frequency to lcb Q3 position 20 – 29 class 16 4 59 (19.5 – 29.5) Q3 = 19.5 + lcb ucb 19.5 29.5 Lower class boundary (lcb) class frequency Q3

  21. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: Q1 = 9.75 The Q3 value is the 3/4th value: Q3 lies in class 20 - 29 n/4 = 100/4 = 75th value Frequency in class up to Q3 20 – 29 class 16 4 59 . (10) = 27.5 Q3 = 19.5 + class width 19.5 29.5 Lower class boundary (lcb) class frequency Q3

  22. Example – Grouped data Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Estimate the median and interquartile range of waiting times Answer: Q1 = 9.75 Q3 = 27.5 IQR = Q3 – Q1 = 27.5 – 9.75 = 17.75 minutes

  23. Grouped data - summary • Use linear interpolation to estimate quantile. • Treat data as continuous within each group / class • Assumes values are evenly distributed within each class. Quantile = lcb + .(ucb – lcb) . class width = lcb + class freq in class up to quantile rest of class freq cum. freq. to lcb lcb ucb Quantile

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