# Dr Frost Cumulative Frequency Graphs Starter: Problems involving Dr Frost Cumulative Frequency Graphs Starter: Problems involving mean The mean height of a group of eight girls is 1.56m. When another girl joins the group the mean height is 1.55m. Work out the height of this girl. Answer = 1.47m ? [Harder] The mean weight of cats is 1.5kg. When an obese cat of weight 14.3kg comes into the room, the new average weight is 1.7kg. Find the original number of cats . ? Girl The Whole Picture... Histogram Widths (cm): 4, 4, 7, 9, 11, 12, 14, 15, 15, 18, 28, 42 Determine Median/LQ/UQ Grouped Frequency Table Width (cm)

Frequency Polygon Frequency Width (cm) Cumulative Frequency Table Cum Freq 0 < w < 10 4 0 < w < 10 4 10 < w < 25 6 0 < w < 25 10 25 < w < 60 2 0 < w < 60

12 Median/LQ/UQ class interval Box Plots Cumulative Frequency Graph Estimate of Median/LQ/UQ/num values in range Median/Quartile Revision Here are the ages of 10 people at Pablos party. Choose the correct value. 12, 13, 14, 14, 15, 16, 16, 17, 19, 24 (Click to vote) Median: 15 15.5 16

LQ: 13 13.5 14 UQ: 17 18 19 Interquartile Range: 3? Range: ? 12 Quickfire Quartiles

LQ Median UQ 1? 2? 3? 1, 2, 3, 4 ? 1.5 ? 2.5 ? 3.5 1, 2, 3, 4, 5 1.5 ? 2? 4.5 ?

2? 3.5 ? 5? 1, 2, 3 1, 2, 3, 4, 5, 6 Rule for lower quartile: Even num of items: find median of bottom half. Odd num of items: throw away middle item, find medium of remaining half. What if theres lots of items? There are 31 items, in order of value. What items should we use for the median and lower/upper quartiles? 0 1 1 2 4 5 5 6 7 8 10 10 14 14 14 14 15 16 17 29 31 31 37 37 38 39 40 40 41 43 44 LQ Use the 8? th item Use item Median ? th item Use the 16

Use item UQ ? th item Use the 24 Use item What if theres lots of items? Num items LQ Median UQ 15 4th? 8?th 12? th 23 6th? 12?th

18? th 39 10?th 20? th 30? th 47 12?th 24? th 36? th Box Plots Box Plots allow us to visually represent the distribution of the data. Minimum Maximum Median Lower Quartile Upper Quartile 3 27 17

15 Sketch Sketch Sketch 22 Sketch Sketch range IQR 0 5 How is the IQR represented in this diagram? 10 Sketch 15 20 25

30 How is the range Sketch represented in this diagram? Box Plots Sketch a box plot to represent the given weights of cats: 5lb, 6lb, 7.5lb, 8lb, 8lb, 9lb, 12lb, 14lb, 20lb Minimum 5 ? Maximum Median ? 20 0 ? 8 4 Lower Quartile Upper Quartile

8 6.75 12 Sketch ? 16 13 20 ? 24 Worksheet Printed handout. Q1 Reference: GCSE-BoxPlotsQuartileStemLeaf Box Plots If you only had the diagram, how could we interpret the distribution of data? What we observe What we can deduce The second box is wider than the first.

There is a greater spread of weights in the top ? half. This is known as positive skew. The length of the second whisker (pun intended) is quite long. The fattest cat has a weight that is an ? the weight is far extreme value, because above the Upper Quartile. 0 4 8 12 16 20 24 Comparing Box Plots Box Plot comparing house prices of Croydon and Kingston-upon-Thames. Croydon Kingston

100k 150k 200k 250k 300k 350k 400k 450k Compare the prices of houses in Croydon with those in Kingston. (2 marks) For 1 mark, one of: In interquartile range of house prices in Kingston is greater than Croydon. The range of house prices in Kingston is greater than Croydon. ? For 1 mark: The median house price in Kingston was greater than that in Croydon. (Note that in old mark schemes, comparing the minimum/ maximum/quartiles would have been acceptable, but currently, you MUST compare the median) ?

Worksheet Printed handout. Q7 Reference: GCSE-BoxPlotsQuartileStemLeaf Stem and Leaf 3 3 7 8 4 1 4 6 ? Median/ Quartiles? ? ? 5 1 1 2 4 ? 5 5 6 0 2 3

? 1|3 means 13 ? ? Worksheet Printed handout. Q3 Reference: GCSE-BoxPlotsQuartileStemLeaf Recap: Frequency Tables (f) (x) How would we usually calculate the mean from a list of items? Total of?values Mean of a list = Num ? values Mode ?

? = ? Frequency Polygons With a bar chart, wed plot each value with its frequency. But weve now grouped the data. We all each IQ range a class interval. What could we use as a representative value for each class interval? IQ (x) Frequency 16 14 12 10 8 6 Modal class interval: ? 4 2 90 100 110 120

130 140 Worksheet Frequency Polygons Printed handout. Q5, 8 Reference: GCSE-BoxPlotsQuartileStemLeaf Recap: Grouped Frequency Tables 570 = =19 ? 30 Question: Why is our mean going to be an estimate? Because we dont know the exact ? times within each range. Write Down The Greek letter capital sigma, means sum of. frequencies =

midpoint of range Worksheet Grouped Frequency Tables Printed handout. Q1, 2 Reference: GCSE-GroupedDataCumFreq Median from Grouped Frequency Tables IQ (x) Frequency Lower Quartile class interval: Median class interval: Upper Quartile class interval: ? ? ? We will see soon that we can actually estimate a value (rather than just give a range) for the median, using something called a cumulative frequency graph. The Whole Picture... Frequency Polygon Histogram

Grouped Frequency Table Width (cm) Widths (cm): 4, 4, 7, 9, 11, 12, 14, 15, 15, 18, 28, 42 Determine Median/LQ/UQ Frequency Width (cm) Cumulative Frequency Table Cum Freq 0 < w < 10 4 0 < w < 10 4 10 < w < 25 6 0 < w < 25

10 25 < w < 60 2 0 < w < 60 12 Median/LQ/UQ class interval Box Plots Cumulative Frequency Graph Estimate of Median/LQ/UQ/num values in range 100m times at the 2012 London Olympics Modal class interval 10.05 < t ?10.2 ?

Median class interval 10.05 < t ?10.2 Estimate of mean 10.02 Time (s) Frequency Cum Freq 9.6 < t 9.7 1 1? 9.7 < t 9.9 4 5? 9.9 < t 10.05 10 15? 10.05 < t 10.2 17

32? TOTAL ? 32 Time (s) Frequency Cum Freq 9.6 < t 9.7 1 1 Plot 9.7 < t 9.9 4 5 Plot 9.9 < t 10.05

10 15 Plot 10.05 < t 10.2 17 32 Plot Cumulative Frequency Graphs This graph tells us how many people had this value or less. Cumulative Frequency 28 Median = 10.07s ? 24 Lower Quartile = 9.95s ? 20

16 Upper Quartile = 10.13s ? 12 8 Interquartile Range = 0.18s ? 4 0 9.5 9.6 9.7 9.8 9.9 10.0 Time (s) 10.1 10.2 10.3

A Cumulative Frequency Graph is very useful for finding the number of values greater/smaller than some value, or within a range. Cumulative Frequency Graphs Estimate how many runners had a time less than 10.15s. 32 Cumulative Frequency 28 26 ? runners 24 Estimate how many runners had a time more than 9.95 20 16 32 ? 8 = 24 runners 12 Estimate how many

runners had a time between 9.8s and 10s 8 4 11 ?3 = 8 runners 0 9.5 9.6 9.7 9.8 9.9 10.0 Time (s) 10.1 10.2 10.3 Time (s) Frequency Cum Freq

9.6 < t 9.7 1 1 Plot 9.7 < t 9.9 4 5 Plot 9.9 < t 10.05 17 22 Plot 10.05 < t 10.2 10 32 Plot Sketch Line

Cumulative Frequency Cumulative Frequency Graph Frequency Polygon 18 28 16 Frequency 32 24 14 20 12 16 10 12 8 8

4 4 2 0 9.5 9.6 9.7 9.8 9.9 Time (s) 10.0 10.1 10.2 0 10.3 9.5 9.6 9.7

9.8 9.9 Time (s) 10.0 10.1 10.2 Worksheet Cumulative Frequency Graphs Printed handout. Q5, 6, 7, 8, 9, 10 Reference: GCSE-GroupedDataCumFreq 5? 23 ? 35 ? 39 ? 40 ? ? 179 ?

34 ? Lower Quartile = 16 ? Upper Quartile = 44.5 ? We previously found: Minimum = 9, Maximum = 57, LQ = 16, Median = 34, UQ = 44.5 ? 1 mark: Range/interquartile range of boys times is greater. 1 mark: Median of boys times ? is greater. 25< ? 35 44 100 134 153 160 ? 30 ? ? 40.9 24.1=16.8

? C? D? B? A? Summary You use a time machine (you made in DT) to travel forward time to when youre doing your GCSE, in order to give yourself 2 things not to forget when drawing a cumulative frequency diagram. What do you tell yourself? Plot the point with 0 cumulative frequency at the START of your first range, ? not necessarily at the start of the x-axis. Use the END of each range when plotting points, so that your cumulative ? frequency includes all the people in the range. The Whole Picture... Frequency Polygon Histogram Grouped Frequency Table

Width (cm) Widths (cm): 4, 4, 7, 9, 11, 12, 14, 15, 15, 18, 28, 42 Determine Median/LQ/UQ Frequency Width (cm) Cumulative Frequency Table Cum Freq 0 < w < 10 4 0 < w < 10 4 10 < w < 25 6 0 < w < 25 10

25 < w < 60 2 0 < w < 60 12 Median/LQ/UQ class interval Box Plots Cumulative Frequency Graph Estimate of Median/LQ/UQ/num values in range ? Description ? Cumulative Frequency Graph

? Box Plot ? Description ? Cumulative Frequency Graph ? Box Plot ? Description ? Cumulative Frequency Graph ? Box Plot ? Description ? Cumulative Frequency Graph ? Box Plot

CARD SORT SOLUTIONS ? Description ? Cumulative Frequency Graph ? Box Plot ? Description ? Cumulative Frequency Graph ? Box Plot ? Description ? Cumulative Frequency Graph ? Box Plot ?

Description ? Cumulative Frequency Graph ? Box Plot