Cumulative Frequency Objectives B Grade Construct and interpret

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.

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) 0 < w ≤ 10 10 < w ≤ 20 20 < w ≤ 30 30 < w ≤ 40 40 < w ≤ 50 50 < w ≤ 60 60 < w ≤ 70 70 < w ≤ 80 80 < w ≤ 90 90 < w ≤ 100 Frequency Cumulative (f) Frequency 3 3 5 8 5 13 22 9 33 11 15 48 14 62 8 70 76 6 80 4 Cumulative means adding up, so a cumulative frequency diagram requires a running total of the frequency.

Cumulative Frequency 0 < w ≤ 10 10 < w ≤ 20 20 < w ≤ 30 30 < w ≤ 40 40 < w ≤ 50 50 < w ≤ 60 60 < w ≤ 70 70 < w ≤ 80 80 < w ≤ 90 90 < w ≤ 100 Frequency Cumulative (f) Frequency 3 3 5 8 5 13 22 9 33 11 15 48 62 14 8 70 76 6 80 4 80 x x x 70 x 60 Cumulative frequency Weight (g) 50 x 40 x 30 x 20 10 0 x x x 0 10 20 30 40 50 60 70 80 90 100 Weight (g) The point are now joined with straight lines The cumulative frequency (c. f. ) can now be plotted on a graph The line always starts at the taking care to plot the c. f. at the end of each class interval. bottom of the first class interval This is because we don’t know where in the class interval The resulting graph should look like this and is sometimes called an 0 < w ≤ 10, the values are, but we do know that by the end of ‘S’ curve. the class interval there are 3 pieces of data

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

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

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 1 - 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 31 - 35 Britain Frequency 1 2 4 8 16 19 10 Britain France c. f. Frequency c. f 2 5 11 16 10 8 8 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

Cumulative Frequency Marks 1 - 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 31 - 35 Britain Frequency 1 2 4 8 16 19 10 Both have 60 pieces of data Britain France c. f. Frequency c. f 2 1 2 7 3 5 11 18 7 34 15 16 44 31 10 8 52 50 8 60 60 Britain Median position is 30 Lower quartile position is 15 Upper quartile position is 45 x 60 Britain LQ = 20 Median = 25 UQ = 29 IQR = 9 Cumulative frequency France xx 50 40 x 30 20 x 10 0 xx x France x LQ = 13. 5 Median = 19 UQ = 26 IQR = 12. 5 x x 0 5 10 15 20 25 30 35 Marks 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