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

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.

A pet shop owner weighs his mice every week to check their health. 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 ≤ 10 3 10 < w ≤ 20 5 20 < w ≤ 30 30 < w ≤ 40 9 40 < w ≤ 50 11 50 < w ≤ 60 15 60 < w ≤ 70 14 70 < w ≤ 80 8 80 < w ≤ 90 6 90 < w ≤100 4 Cumulative Frequency   3 8 13 22 33 48 62 70 76 80 Cumulative means adding up, so a cumulative frequency diagram requires a running total of the frequency.

The point are now joined with straight lines Cumulative Frequency x 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100 Weight (g) Frequency (f) 0 < w ≤ 10 3 10 < w ≤ 20 5 20 < w ≤ 30 30 < w ≤ 40 9 40 < w ≤ 50 11 50 < w ≤ 60 15 60 < w ≤ 70 14 70 < w ≤ 80 8 80 < w ≤ 90 6 90 < w ≤100 4 Cumulative Frequency   x x 3 8 13 22 33 48 62 70 76 80 x x Cumulative frequency x x x x x Weight (g) The point are now joined with straight lines 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. The line always starts at the bottom of the first class interval 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 The resulting graph should look like this and is sometimes called an ‘S’ curve.

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

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: Lower quartile is 38g Upper quartile is 68g The interquartile range (IQR) = 68 – 38 = 30g 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

In an international competition 60 children from Britain and France Cumulative Frequency In an international competition 60 children from Britain and France Did the same Maths test. The results are in the table below: Marks Britain Frequency Britain c.f. France Frequency France c.f 1 - 5 1   2 6 - 10 5 11 - 15 4 11 16 - 20 8 16 21 - 25 10 26 - 30 19 31 - 35 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

Both have 60 pieces of data Cumulative Frequency Both have 60 pieces of data Marks Britain Frequency Britain c.f. France Frequency France c.f 1 - 5 1   2 6 - 10 5 11 - 15 4 11 16 - 20 8 16 21 - 25 10 26 - 30 19 31 - 35 1 2 Median position is 30 3 7 7 18 Lower quartile position is 15 15 34 31 44 Upper quartile position is 45 50 52 60 60 Britain 60 50 40 30 20 10 0 5 10 15 20 25 30 35 Cumulative frequency Marks x x France Britain France LQ = 20 LQ = 13.5 Median = 25 Median = 19 UQ = 26 UQ = 29 IQR = 12.5 IQR = 9 The scores in Britain are higher with less variation

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 Put the end points not the class interval on the x axis Plot the points at the end of the class interval 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