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Year 8: Data Handling 2

Year 8: Data Handling 2. Dr J Frost (jfrost@tiffin.kingston.sch.uk). Learning Outcomes: To understand stem and leaf diagrams , frequency polygons , box plots and cumulative frequency graphs. Last modified: 28 th November 2013. Stem and Leaf Diagram - What is it?.

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Year 8: Data Handling 2

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  1. Year 8: Data Handling 2 Dr J Frost (jfrost@tiffin.kingston.sch.uk) Learning Outcomes: To understand stem and leaf diagrams, frequency polygons, box plots and cumulative frequency graphs. Last modified: 28th November 2013

  2. Stem and Leaf Diagram - What is it? Suppose this “stem and leaf diagram” represents the lengths of beetles. The key tells us how two digits combine. The numbers must be in order. 1 2 3 4 5 4 1 2 4 5 2 5 6 6 6 7 7 8 8 0 1 2 2 4 5 6 7 7 7 7 8 0 1 1 2 Key: 2 | 1 means 2.1cm Value represented = 4.5cm ? These numbers represent the first digit of the number. These numbers represent the second.

  3. Example Here are the weights of a group of cats. Draw a stem-and-leaf diagram to represent this data. 36kg 15kg 35kg 50kg 11kg 36kg 38kg 47kg 12kg 30kg 18kg 57kg 1 2 3 4 5 ? 1 2 5 8 0 5 6 6 8 7 0 7 ? Key: 3 | 8 means 38kg ? ? ? • What do you think are the advantages of displaying data in a stem-and-leaf diagram? • Shows how the data is spread out. • Identifies gaps in the values. • All the original data is preserved (i.e. we don’t ‘summarise’ in any way). ? ? ?

  4. Your turn Here is the brain diameter of a number of members of 8IW. Draw a stem and leaf diagram representing this data. 1.3cm 2.1cm 5.3cm 2.0cm 1.7cm 4.2cm 3.3cm 3.2cm 1.3cm 4.6cm 1.9cm 1 2 3 4 5 ? 3 3 7 9 0 1 2 3 2 6 3 ? Key: 3 | 8 means 3.8cm ? ? ? ? ? Median width = 2.1cm ? Lower Quartile = 1.7cm ? Upper Quartile = 4.2cm ?

  5. Exercises Q1 on your provided worksheet. (Ref: Yr8-DataHandlingWorksheet.doc)

  6. Frequency Diagram Suppose we wanted to plot the following data, where each value has a frequency. A suitable representation of this data would be a bar chart. ? 100 80 60 40 20 Frequency When bar charts have frequency on the y-axis, they’re known as frequency diagrams. 8 9 10 11 12 Shoe size

  7. Frequency Polygons But suppose that we had data grouped into ranges. What would be a sensible value to represent each range? 16 14 12 10 8 6 4 2 Join the points up with straight lines. Modal class interval: 100≤ x < 110 ? This is known as a frequency polygon. 90 100 110 120 130 140

  8. Frequency Polygons – Exercises on sheet Q1 Q2 ? ? b) 30 < x ≤ 40 c) 16% ? ? b) 20 < x ≤ 30 c) 16% ? ?

  9. The Whole Picture Frequency Polygon Histogram Cumulative Frequency Table Grouped Frequency Table Widths (cm): 4, 4, 7, 9, 11, 12, 14, 15, 15, 18, 28, 42 Determine Median/LQ/UQ Median/LQ/UQ class interval Cumulative Frequency Graph Estimate of Median/LQ/UQ/num values in range Box Plots

  10. Recap: Lower and Upper Quartile Suppose that we line up everyone in the school according to height. 50% We already know that the median would be the middle person’s height. 50% of the people in the school would have a height less than them. The height of the person 25% along the line is known as the: lower quartile The upper quartile is the height of the person 75% along the data. ?

  11. Check your understanding 50% of the data has a value more than the median. 75% of the data has a value less than the upper quartile. 25% of the data has a value more than the upper quartile. 75% of the data has a value more than the lower quartile. ? ? ? ? 0% 25% 50% 75% 100% LQ Median UQ

  12. Median/Quartile Revision Here are the ages of 10 people at Pablo’s party. Choose the correct value. 12, 13, 14, 14, 15, 16, 16, 17, 19, 24 (Click to vote)  15  15.5  16 Median:  13 13.5  14  Lower: 17   18 19  UQ: Interquartile Range: ? ? 3 12 Range:

  13. Quickfire Quartiles LQ Median UQ ? ? ? 1, 2, 3 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, 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.

  14. What if there’s 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 ? Use the 8 th item LQ Use item ? Median Use item Use the 16th item ? Use item UQ Use the 24th item

  15. What if there’s lots of items? Num items LQ Median UQ ? ? ? 15 4th 8th 12th ? ? ? 23 6th 12th 18th ? ? ? 39 10th 20th 30th ? ? ? 47 12th 24th 36th

  16. Box Plots Box Plots allow us to visually represent the distribution of the data. Sketch Sketch Sketch Sketch Sketch range IQR 0 5 10 15 20 25 30 How is the IQR represented in this diagram? How is the range represented in this diagram? Sketch Sketch

  17. Box Plots Sketch a box plot to represent the given weights of cats: 5lb, 6lb, 7.5lb, 8lb, 8lb, 9lb, 12lb, 14lb, 20lb ? ? ? ? ? 0 4 8 12 16 20 24 Sketch

  18. Box Plots Sketch a box plot to represent the given ages of people at Dhruv’s party: 5, 12, 13, 13, 14, 16, 22 ? ? ? ? ? 0 4 8 12 16 20 24 Sketch

  19. 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) ? ?

  20. 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 ? ? ? ? ?

  21. Cumulative Frequency Graphs Plot This graph tells us how many people had “this value or less”. Plot Plot 32 28 24 20 16 12 8 4 0 Plot Median = 10.07s ? Lower Quartile = 9.95s ? Cumulative Frequency Upper Quartile = 10.13s ? Interquartile Range = 0.18s ? 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 Time (s)

  22. 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. 26 runners Estimate how many runners had a time more than 9.95 32 – 8 = 24 runners Estimate how many runners had a time between 9.8s and 10s 11 – 3 = 8 runners 32 28 24 20 16 12 8 4 0 ? Cumulative Frequency ? ? 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 Time (s)

  23. Plot Plot Sketch Line Plot Plot Cumulative Frequency Graph Frequency Polygon 32 28 24 20 16 12 8 4 0 18 16 14 12 10 8 4 2 0 Cumulative Frequency Frequency 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 Time (s) Time (s)

  24. Worksheet Cumulative Frequency Graphs Printed handout. Q5, 6, 7, 8, 9, 10 Reference: GCSE-GroupedDataCumFreq

  25. 5 23 35 39 40 ? ? ? ? ? ? 179 ?

  26. 34 ? Lower Quartile = 16 Upper Quartile = 44.5 ? ?

  27. 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. ?

  28. ? 44 100 134 153 160 ? ? ? 30 ?

  29. C ? D ? B ? A ?

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