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Box plot Edexcel S1 Mathematics 2003 (or box and whisker plot)

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Introduction Box plot diagrams: provide a diagrammatic representation of the distribution use quartiles to divide the distribution into intervals each containing ¼ of the data values used to compare distributions used to show skewness of distribution

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Find the quartiles Use any given algorithm to calculate outliers find the values of the whiskers Draw a box plot to scale– on graph paper Stages in drawing a box plot

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers (c) Comment on the skewness of these data.

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers (c) Comment on the skewness of these data. This question uses a small number of data values for ease of calculations. The number of data values is usually larger. Very few data values can make the calculation of quartiles less meaningful.

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers (c) Comment on the skewness of these data. Any rule to identify outliers will be specified in the question. The rule provided here is a typical one.

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers (c) Comment on the skewness of these data. Make sure you use graph paper to draw a boxplot. Ask for graph paper in the module exam.

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data:

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data: Possibly use a stem and leaf diagram to re-order the data

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data: The median is the middle value: n/2 = 10/2 = th value = Q2 Whole number - so round up to th Find average of 5 th and 6 th value

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data: The median is the middle value: n/2 = 10/2 = th value = = 4.5 mins The lower quartile, Q1, is the 1/4 th value: n/4 = 10/4 = rd value = Q2 Not whole - so round up to whole Find the 3 rd value

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data: The median is the middle value: n/2 = 10/2 = th value = = 4.5 mins The lower quartile, Q1, is the 1/4 th value: n/4 = 10/4 = mins 3rd value = Q1Q2 Not whole - so round up to whole Find the 3 rd value

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data: The median is the middle value: n/2 = 10/2 = th value = = 4.5 mins The lower quartile, Q1, is the 1/4 th value: n/4 = 10/4 = mins The upper quartile, Q3, is the 3/4 th value: 3n/4 = 3x10/4 = th value = 3rd value = Q1Q2 Not whole - so round up to whole Find the 8 th value

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data: The median is the middle value: n/2 = 10/2 = th value = = 4.5 mins The lower quartile, Q1, is the 1/4 th value: n/4 = 10/4 = mins The upper quartile, Q3, is the 3/4 th value: 3n/4 = 3x10/4 = th value = 9 mins 3rd value = Q1Q3Q2 Not whole - so round up to whole Find the 8 th value

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Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range Answer re-order the data: The median is the middle value: n/2 = 10/2 = th value = = 4.5 mins The lower quartile, Q1, is the 1/4 th value: n/4 = 10/4 = mins The upper quartile, Q3, is the 3/4 th value: 3n/4 = 3x10/4 = th value = 9 mins The inter quartile range = Q3 – Q1 = 9 – 3 = 6 mins 3rd value = Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3Q2 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Check below Q1 for outliers:

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3Q2 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 Check below Q1 for outliers:

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3Q2 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers:

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Answer calculate outliers: Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: Q1Q3Q2 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 = Q1Q3Q2 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value Q1Q3Q2 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Check above Q3 for outliers: Q1Q3Q2 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Check above Q3 for outliers: Q1Q3Q2 Q3 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: Q1Q3Q2 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 = Q1Q3Q2 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is Q1Q3Q2 Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: * Q1Q3Q2

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: Q1Q3Q2 *

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: * Q1Q3Q2 Remember to use GRAPH paper

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Answer calculate outliers: Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (c) Comment on the skewness of these data Q1Q3 Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6 Q1 – 1.5 x IQR = Check below Q1 for outliers: 3 – 1.5 x 6 =-6This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1. Q x IQR = Check above Q3 for outliers: x 6 =18 This falls within the data range. So value 21 is an outlier. The right whisker is 18. Draw scale and boxplot: * Q1Q3Q2 Q3 - Q2 = 9 – 4.5 = 4.5Q2 -Q1 = 4.5 – 3 = 1.5 So Q3 – Q2 > Q2 – Q1So distribution is right (positive) skewed

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Answer (a) median = Q2 = 4.5, IQR = 6 (b) boxplot: * (c) Q3 – Q2 > Q2 – Q1So distribution is right (positive) skewed Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4 (a) Find the median and inter-quartile range An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile (b) Draw a boxplot diagram to represent these data, indicating any outliers (c) Comment on the skewness of these data.

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