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Cumulative Frequency Objectives: B Grade Construct and interpret a cumulative frequency diagram Use a cumulative frequency diagram to estimate the median and interquartile range

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Cumulative Frequency A cumulative frequency diagram is a graph that can be used to find estimates of the median and upper and lower quartiles of grouped data. The median is the middle value when the data has been placed in order of size The lower quartile is the ‘median’ of the bottom half of the data set and represents the value ¼ of the way through the data. The upper quartile is the ‘median’ of the top half of the data set and represents the value ¾ of the way through the data.

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Cumulative Frequency A pet shop owner weighs his mice every week to check their health. The weights of the 80 mice are shown below: weight (g) Frequency (f) 0 < w ≤ 103 10 < w ≤ 205 20 < w ≤ 305 30 < w ≤ 409 40 < w ≤ 5011 50 < w ≤ 6015 60 < w ≤ 7014 70 < w ≤ 808 80 < w ≤ 906 90 < w ≤1004 Cumulative means adding up, so a cumulative frequency diagram requires a running total of the frequency. Cumulative Frequency 3 8 13 22 33 48 62 70 76 80

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Cumulative Frequency Weight (g) Frequency (f) 0 < w ≤ 103 10 < w ≤ 205 20 < w ≤ 305 30 < w ≤ 409 40 < w ≤ 5011 50 < w ≤ 6015 60 < w ≤ 7014 70 < w ≤ 808 80 < w ≤ 906 90 < w ≤1004 The cumulative frequency (c.f.) can now be plotted on a graph taking care to plot the c.f. at the end of each class interval. Cumulative Frequency 3 8 13 22 33 48 62 70 76 80 This is because we don’t know where in the class interval 0 < w ≤ 10, the values are, but we do know that by the end of the class interval there are 3 pieces of data 80 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 x x x x x x x x x x The point are now joined with straight lines The resulting graph should look like this and is sometimes called an ‘S’ curve. Cumulative frequency Weight (g) The line always starts at the bottom of the first class interval

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Cumulative Frequency From this graph we can now find estimates of the median, and upper and lower quartiles There are 80 pieces of data 80 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 x x x x x x x x x x Cumulative frequency Weight (g) The middle is the 40th Median position Read across, then Down to find the median weight Median weight is 54g The lower quartile is the 20 th piece of data ¼ of the total pieces of data Lower quartile Lower quartile is 38g The upper quartile is the 60 th piece of data ¾ of the total pieces of data Upper quartile Upper quartile is 68g

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Cumulative Frequency The upper and lower quartiles can now be used to find what is called The interquartile range and is found by: Upper quartile – Lower quartile In this example: The interquartile range (IQR) = 68 – 38 = 30g Lower quartile is 38g Upper quartile is 68g Because this has been found by the top ¾ subtract the bottom ¼ ½ of the data (50%) is contained within these values So we can also say from this that half the mice weigh between 38g and 68g

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Cumulative Frequency In an international competition 60 children from Britain and France Did the same Maths test. The results are in the table below: Using the same axes draw the cumulative frequency diagram for each country. Find the median mark and the upper and lower quartiles for both countries and the interquartile range. Make a short comment comparing the two countries Marks Britain Frequency Britain c.f. France Frequency France c.f 1 - 51 2 6 - 102 5 11 - 154 11 16 - 208 16 21 - 2516 10 26 - 3019 8 31 - 3510 8

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Marks Britain Frequency Britain c.f. France Frequency France c.f 1 - 51 2 6 - 102 5 11 - 154 11 16 - 208 16 21 - 2516 10 26 - 3019 8 31 - 3510 8 Cumulative Frequency 1 3 7 15 31 50 60 2 7 18 34 44 52 60 50 40 30 20 10 0 0 5 10 15 20 25 30 35 Cumulative frequency Marks Both have 60 pieces of data Median position is 30 Lower quartile position is 15 Upper quartile position is 45 x x x x x x x x x x x x x Britain France Britain France Median = 25 Median = 19 LQ = 20 LQ = 13.5 UQ = 29 UQ = 26 IQR = 9 IQR = 12.5 The scores in Britain are higher with less variation

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Cumulative Frequency Summary B Grade Construct and interpret a cumulative frequency diagram Use a cumulative frequency diagram to estimate the median and interquartile range Make a running total of the frequency Plot the points at the end of the class interval Put the end points not the class interval on the x axis Join the points with straight lines – if it is not an ‘S’ curve ****Check your graph**** Find the median by drawing across from the middle of the cumulative frequency axis Find the LQ and UQ from ¼ and ¾ up the c.f. axis

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