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KS4 Mathematics

KS4 Mathematics. D3 Presenting data. D3 Presenting data. Contents. A. D3.2 Line graphs. A. D3.1 Bar graphs. D3.3 Pie charts. A. D3.4 Stem-and-leaf diagrams. A. D3.5 Scatter graphs. A. 20 14 12 5 1 3 4 6 7 9. Bars should be separate.

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KS4 Mathematics

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  1. KS4 Mathematics D3 Presenting data

  2. D3 Presenting data Contents • A D3.2 Line graphs • A D3.1 Bar graphs D3.3 Pie charts • A D3.4 Stem-and-leaf diagrams • A D3.5 Scatter graphs • A

  3. 20 14 12 5 1 3 4 6 7 9 • Bars should be separate. • The bars must be the same width. • The gaps must be the same width. • The scales must go up by equal intervals. • The numbers on the horizontal axis must appear in the middle of the bar. • The axes must be labelled. • There should be a title. Bar graphs What is wrong with this bar graph? Make a list of mistakes.

  4. Education around the world Number of pupils per primary school teacher Uganda Tanzania Zambia United Kingdom USA Saudi Arabia Sweden 0 10 20 30 40 50 60

  5. Stacked bar graphs and drinking habits This graph shows the responses of secondary school pupils in Englandto being offered drugs. 100 80 Yes Percentage Refused 60 Never offered 40 20 0 1999 2003 What conclusions can you draw?

  6. Comparative bar graphs and world literacy Percentage of male literacy Percentage of female literacy 100 80 60 40 20 0 East Asia & Europe & Latin America & Middle East & South Asia Sub-Saharan Pacific Central Asia Caribbean North Africa Africa

  7. D3 Presenting data Contents D3.1 Bar graphs • A • A D3.2 Line graphs D3.3 Pie charts • A D3.4 Stem-and-leaf diagrams • A D3.5 Scatter graphs • A

  8. Smoking among young people These figures show the percentages of students in Years 7-10 that smoke regularly in Great Britain. What would be the most appropriate graph to illustrate this data?

  9. 16 14 12 10 Percentage of regular smokers Boys 8 Girls 6 4 2 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Regular smoking in Year 7-10 pupils Year

  10. 14 12 10 8 Percentage of regular smokers 6 4 2 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Regular smoking in Year 7-10 pupils Year

  11. Smoking among young people These figures show the percentages of Year 11 students that smoke regularly in Great Britain. How does this data compare to that of Year 7-10 students? Again we can show the data using a line graph.

  12. 35 30 25 Boys 20 Percentage of regular smokers Girls 15 10 5 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Regular smoking in Year 11

  13. 35 30 25 20 Percentage of regular smokers 15 10 5 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Regular smoking in Year 11 Year

  14. 35 30 25 20 Percentage of regular smokers 15 10 Years 7 to 10 5 Year 11 0 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year Comparing Years 7 to 10 to Year 11

  15. D3 Presenting data Contents D3.1 Bar graphs • A D3.2 Line graphs • A D3.3 Pie charts • A D3.4 Stem-and-leaf diagrams • A D3.5 Scatter graphs • A

  16. Over 10,000 Year 7-11 pupils took part in a Department of Health survey in 2003. There are about 3,000,000 people in England in this age range. Complete the table. Percentages When did you last have a drink? Percentage for 1988 Estimated total numbers for 1988 Percentage for 2003 Estimated total numbers for 2003 Last week 20% 600,000 25% 750,000 1 – 4 weeks ago 18% 540,000 15% 450,000 1 – 6 months ago 12% 360,000 12% 360,000 More than 6 months ago 11% 330,000 10% 300,000 Never 38% 1,140,000 39% 1,170,000

  17. Last week 1-4 weeks 1-6 months More than 6 months Never Compare the results for 1988 with 2003. Drinking habits among young people in 2003 1988 2003 20% 25% 39% 38% 18% 15% 11% 12% 10% 12%

  18. To convert raw data into angles for n data items: 360 ÷ n represents the number of degrees per data item. Pie charts For example, 40 people take part in a survey. What angle represents • one person? 360°÷ 40 = 9° • two people? 9° × 2 = 18° • eight people? 9° × 8 = 72° How many people are represented by an angle of 36°? There are 9°per person. 36° ÷ 9° = 4 people.

  19. Converting data into angles

  20. When drawing a pie chart, it is helpful to note the following points. Displaying data as a pie chart • There should be no gaps in your pie chart. • The angles should add up to 360°. • Angles should be rounded off to the nearest degree if necessary. • If you have had to round off, the angles may add up to slightly more or less than 360°. • Each section should be labelled or a key should be used. You may want to include actual numbers or percentages. Angles are not normally included.

  21. Smoking habits Percentage Angle Regular 1 Occasional 2 Used to 2 Tried 11 Never 85 Total This data represents the smoking habits of Year 7 students in England. Calculate the angles. Solving problems with pie charts 4° 7° 7° 40° 306° 101 364° Can you explain the totals?

  22. Pie chart of Year 7 smoking habits 1% 2% 2% 11% Regular Occasional Used to Tried Never 84%

  23. Pie chart of Year 11 smoking habits • 36% of Year 11 students have never smoked. What angle represents this category? • 72° represents the Year 11 students who have tried smoking. What percentage is this? • There are approximately600,000 Year 11 students in England.22% smoke regularly. How many people is this? • One third of these regular smokers have a cigarette within 30 minutes of waking. How many is this? Regular Occasional Used to Tried Never

  24. D3 Presenting data Contents D3.1 Bar graphs • A D3.2 Line graphs • A D3.4 Stem-and-leaf diagrams D3.3 Pie charts • A • A D3.5 Scatter graphs • A

  25. Interpreting stem-and-leaf diagrams A stem-and-leaf diagramcan be used to display data items in order without grouping them. For example, this table shows how much pocket money some regular smokers in Year 11 spend on cigarettes in a fortnight.

  26. Constructing stem-and-leaf diagrams The data below represents the numbers of cigarettes smoked in a week by 22 regular smokers in Year 11. 738412220752417 15132345711173019 5103020 • Put this data into a stem-and-leaf diagram. • The stem should represent____ and the leaf should represent _____. tens units • Work out the mode, mean, median and range.

  27. Mode The mode is __ . Stem (tens) Leaf (units) Mean There are ___ people in the survey and they smoke a total of ____ cigarettes a week. 0 5 5 7 7 7 1 0 1 3 5 7 7 9 2 0 0 2 3 4 Median 3 0 0 8 The median is halfway between ___ and ___. 4 1 5 Range ___ – ___ = ___ Calculations with stem-and-leaf diagrams 7 22 426 426 ÷ 22 =___ 19 17 19 This is ___. 18 45 5 40

  28. Stem (tens) Leaf (units) 0 5 5 7 7 7 1 0 1 3 5 7 7 9 2 0 0 2 3 4 3 0 0 8 4 1 5 Solving problems with stem-and-leaf diagrams • What fraction of the group smoke more than 20 cigarettes a week? What is this as a percentage? • The mean number smoked is 19. How many smoke less than the mean? What is this as a percentage? • What percentage smoke less than 10 cigarettes? • A packet of 20 cigarettes costs about £4. Work out the average amount spent on cigarettes using the median.

  29. Investigation using stem-and-leaf diagrams • Use job advertisements in newspapers and the internet to investigate how much graduates leaving university get paid compared with school leavers of 16 or 18. • Record your results in stem-and-leaf diagrams. • Calculate the mean and median incomes for each group. • What conclusions can you draw about the financial advantages of getting a degree? An extension task could involve comparing the incomes of new graduates with graduates after ten years. Which careers offer greater opportunities for promotion or financial rewards?

  30. D3 Presenting data Contents D3.1 Bar graphs • A D3.2 Line graphs • A D3.5 Scatter graphs D3.3 Pie charts • A D3.4 Stem-and-leaf diagrams • A • A

  31. 60 55 Weight (kg) 50 45 40 140 150 160 170 180 190 Height (cm) Scatter graphs What does this scatter graph show about the relationship between the height and weight of twenty Year 10 boys? As height increases, weight increases. This is called apositive correlation.

  32. 85 80 75 Life expectancy 70 65 60 55 50 0 20 40 60 80 100 120 Number of cigarettes smoked in a week Scatter graphs What does this scatter graph show? It shows that life expectancy decreases as the number of cigarettes smoked increases. This is called anegative correlation.

  33. Interpreting scatter graphs Scatter graphs can show a relationship between two variables. This relationship is called correlation. Correlation is a general trend. Some data items will not fit this trend, as there are often exceptions to a rule. They are called outliers. Scatter graphs can show: • positive correlation: as one variable increases, so does the other variable • negative correlation: as one variable increases, the other variable decreases • zero correlation: no linear relationship between the variables. Correlation can be weak or strong.

  34. A D B C E F Identifying correlation from scatter graphs Decide whether each of the following graphs shows, • strong positive correlation • strong negative correlation • zero correlation • weak positive correlation • weak negative correlation.

  35. Relationships between two variables

  36. 25 25 20 20 15 15 10 10 5 5 0 0 5 10 15 20 25 Strong negative correlation 0 0 5 10 15 20 25 Strong positive correlation Weak positive correlation Weak negative correlation The line of best fit Lines of best fit are drawn on graphs by eye so that there are a roughly equal number of points above and below the line. Look at these examples, Notice that the stronger the correlation, the closer the points are to the line. If the gradient is positive, the correlation is positive and if the gradient is negative, then the correlation is also negative.

  37. The line of best fit When drawing the line of best fit remember the following points, • The line does not have to pass through the origin. • For an accurate line of best fit, find the mean for each variable. This forms a coordinate, which can be plotted. The line of best fit should pass through this point. • The line of best fit can be used to predict one variable from another. • It should not be used for predictions outside the range of data used. • The equation of the line of best fit can be found using the gradient and intercept.

  38. 85 80 75 Life expectancy 70 65 60 55 50 0 20 40 60 80 100 Number of cigarettes smoked in a week The line of best fit This graph shows the relationship between life expectancy and the number of cigarettes smoked in a week.

  39. Solving problems with lines of best fit • Work out an estimate for the equation of the line of best fit using the gradient and intercept. • Use the equation to estimate the life expectancy for someone who smokes 10 cigarettes a day. • Why would an estimate of the number of cigarettes smoked for a life expectancy of 40 years not be reliable? • Can you explain why there are so many outliers for this data?

  40. The local newspaper reports, “Buying a new car can help you get pregnant!” Cause and effect A study finds a positive correlation between the number of cars in a town and the number of babies born. Does the study support this conclusion? Correlation does not necessarily imply that there is a causal relationship between the two variables. There may be some other cause. What might this be in the example above?

  41. The local newspaper reports, “If you want it to snow, go out and buy a sledge!” Cause and effect A study finds a negative correlation between the number of sledges sold and the temperature. Does the study support this conclusion? Explain.

  42. Cause and effect • Discuss these headlines. • “Taller students do better in new Maths test!” • “The more coffee you drink, the more stressed you are.” • “Chocolate causes lower grades at university.” • “Counselling can make you depressed.” • “New exercise regime causes dramatic rise in injuries at local hospital.”

  43. Review • List the types of graph you have covered in this topic. • What kind of data would suit each kind of graph? • What possible mistakes could you make when drawing a bar graph? • If you are investigating a relationship between two variables, what kind of graph would you use? • How do you calculate the angles in a pie chart if you know how many data items there are altogether?

  44. Review • What other problems can you solve with a pie chart? Give examples and outline the method for each. • How would you calculate the three averages from a stem and leaf diagram? • What are the different types of correlation? Give examples. • What is a line of best fit and how would you draw one?

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