Quantiles Edexcel S1 Mathematics 2003

Introduction- what is a quantile? Quantiles are used to divide data into intervals containing an equal number of values. For example: Deciles D1, …, D9 divide data into 10 parts Quartiles Q1, Q2, Q3 divide data into 4 parts Percentiles P1, …, P100 divide into 100 parts … … … D1D2D3D4D5D6D7D8D9

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. Introduction – Grouped / Ungrouped data

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 The median is the middle value: n/2 = 18/2 = th value = = 10.5 appointments Whole number- so round up to.5 th Answer: median Find average of 9 th and 10 th value

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 The median is the middle value: n/2 = 18/2 = th value = = 10.5 appointments The first decile, D1, is the 1/10 th value: n/10 = 18/10 = appointments 2 nd value = Not whole - so round up to whole Find the 2 nd value Answer: D1median

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 The median is the middle value: n/2 = 18/2 = th value = = 10.5 appointments The first decile, D1, is the 1/10 th value: n/10 = 18/10 = appointments The ninth decile, D9, is the 9/10 th value: 9n/4 = 9x18/10 = th value = 15 appointments 2 nd value = Not whole - so round up to whole Find the 17 th value Answer: D1medianD9

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 The median is the middle value: n/2 = 18/2 = th value = = 10.5 appointments The first decile, D1, is the 1/10 th value: n/10 = 18/10 = appointments The ninth decile, D9, is the 9/10 th value: 9n/4 = 9x18/10 = th value = 15 appointments 2 nd value = Answer: D1medianD9

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

Example – G rouped data Waiting times frequency Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Answer: Estimate the median and interquartile range of waiting times The median is the middle value: n/2 = 100/2 = 50 th value Lower class boundary (lcb) class frequenc y median = (19.5 – 14.5) Median position Cumulative frequency to lcb uc b lcb median – 19 class lies in class

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

Example – G rouped data Waiting times frequency Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Answer: Estimate the median and interquartile range of waiting times The median is the middle value: n/2 = 100/2 = 50 th value Lower class boundary (lcb) class frequenc y median = Frequency in class up to median 96 median – 19 class Linear interpolation: Assume 15 values are evenly distributed in class lies in class (19.5 – 14.5) uc b lcb

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

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

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

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

Example – G rouped data Waiting times frequency Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Answer: Estimate the median and interquartile range of waiting times The Q1 value is the 1/4 th value: n/4 = 100/4 = 25 th value Lower class boundary (lcb) class frequenc y Q1 = (14.5 – 9.5) Q1 position Cumulative frequency to lcb uc b lcb Q – 14 class Q1 lies in class

Example – G rouped data Waiting times frequency Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Answer: Estimate the median and interquartile range of waiting times The Q1 value is the 1/4 th value: n/4 = 100/4 = 25 th value Q1 = (5) Q – 14 class Q1 lies in class = 9.75 Lower class boundary (lcb) class frequenc y Frequency in class up to Q1 class width

Example – G rouped data Waiting times frequency Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Answer: Estimate the median and interquartile range of waiting times The Q1 value is the 1/4 th value: n/4 = 100/4 = 25 th value Q1 = (5) Q – 14 class Q1 lies in class = 9.75

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

Example – G rouped data Waiting times frequency Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Answer: Estimate the median and interquartile range of waiting times The Q3 value is the 3/4 th value: n/4 = 100/4 = 75 th value Lower class boundary (lcb) class frequenc y Q3 = (19.5 – 29.5) Q3 position Cumulative frequency to lcb uc b lcb Q – 29 class Q3 lies in class Q1 = 9.75

Example – G rouped data Waiting times frequency Question: Waiting times, to the nearest minute, at a doctors surgery for 100 patients were recorded: Answer: Estimate the median and interquartile range of waiting times The Q3 value is the 3/4 th value: n/4 = 100/4 = 75 th value Lower class boundary (lcb) Q3 = (10) Q – 29 class Q3 lies in class Q1 = 9.75 class frequenc y Frequency in class up to Q3 class width = 27.5

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

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