Cumulative Frequency Objectives: level :9 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: Cumulative means adding up, so a cumulative frequency diagram requires a running total of the frequency. weight (g) Frequency (f) Cumulative Frequency 0 < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤

Cumulative Frequency Weight (g)Frequency (f) Cumulative Frequency 0 < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ < w ≤ x x x x x x x x x x The point are now joined with straight lines 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 Cumulative frequency Weight (g)

Cumulative Frequency From this graph we can now find estimates of the median, and x x x x x x x x x x Cumulative frequency Weight (g) upper and lower quartiles There are 80 pieces of data The middle is the 40th Median position Read across, then Down to find the median weight Upper 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 38gUpper 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

Cumulative Frequency activity In an international competition 60 children from Britain and France Did the same Maths test. The results are in the table below: MarksBritain FrequencyBritainc.f.France FrequencyFrancec.f

Cumulative Frequency activity Marks Britain Frequency Britain c.f. France Frequency France c.f 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 x5x5 Britain France France Median = LQ = Britain LQ = Median = UQ = IQR = 9 UQ = IQR = 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