HANDLING DATA COURSEWORK

1. HANDLING DATACOURSEWORK World Data

2. Main Menu THE IMPORTANT STUFF What is Coursework??? Specify and Plan Collect, Process & Represent Interpret and Discuss What You Should Do? THINGS YOU NEED TO KNOW Planning the Investigation Sample Mean, Median, Mode and Range Pie Charts Bar Charts Histograms and Freq Polygons Scatter Plots Stem and Leaf Plots Cumulative Frequency Box and Whisker Plots

3. Your Task • Is given in detail on the task sheet. • Basically your task is to: “investigate what influences the amount a student drinks.” • The database has been selected for you from Rondam Secondary school.

4. What Will Happen • A MIX OF THE FOLLOWING: • Direct Teaching –statistics skills, ICT, investigation cycle • Group Work – planning, discussing, • Individual Time– writing up, working

5. Hypothesis Specify and Plan How could you make it better? Specify and plan Interpret and discuss Investigation cycle Collect, process and represent

6. What to do in this section? • Examine the Writing Frame and what decisions you must make to fill it in. • Decide on the hypothesis you are going to test. Make sure it is well explained. • Write a clear and detailed description of the task and your plan to test the hypothesis. • Do a draft first. Your final write up will come later.

7. Collect, Process and Represent Specify and Plan Hypothesis How could you make it better? Interpret and discuss Investigation cycle Specify and plan Collect, process and represent

8. What to do in this section? • Collect the data – fully explain your sampling technique and sample size. • Tabulate the data. Only include the information relevant to your hypothesis. • Using statistical and graphical methods to process and examine the data.

9. Interpret and Discuss Specify and Plan Hypothesis How could you make it better? Interpret and discuss Investigation cycle Specify and plan Collect, process and represent

10. What to do in this section? • This is the big crunch section. • Draw conclusions from all of your calculations and relate these to your initial hypothesis. • Make sure you: • Compare results to show differences/similarities. • Use facts and statistics taken directly from your calculations. • Evaluate your approach and explain any changes you would make if you were doing it again. • Consider bias in your results.

11. And Now ……. • Challenge • What will a good piece of maths investigative work look like ??? • You should consider: • What will it contain? • How will it be presented? • How will it be marked? • What will it look like? • 15 mins in groups of 5 or 6

12. Year Eleven pupils with paid jobs don’t do as well in their exams. The first step in planning a statistical enquiry is to decide what problem you want to explore. This can be done by asking questions that you want your data to answer and by stating a hypothesis. Formulating a hypothesis A hypothesis is a statement that you believe to be true but that you have not yet tested. The plural of hypothesis is hypotheses. For example,

13. “Year Eleven pupils with paid jobs don’t do as well in their exams.” How could you find out if this statement is true? Forming a hypothesis • Think about: • What data (information) would you need to collect? • How will you collect it? • Which Year Elevens does this statement cover? • How could you ensure the data you collect represents all • of these Year Elevens? • What would you do with the data? • What would you expect to find?

14. hypothesis– a statement that can be tested population– the group (often of people) referred to in the hypothesis Key vocabulary sample– a selection from the population biased sample– an unfair selection representative sample– a fair selection cross section– a selection that reflects all the subgroups within the population objective data– information that is not affected by people’s opinions

15. subjective data– information that is affected by people’s opinions primary data– information you collect yourself, by asking people, measuring, carrying out experiments, and so on Key vocabulary secondary data– information that has been collected already, that you get from books, the internet, and so on ethical issues– problems to do with confidentiality and personal questions reliable results– results that will be repeated if the experiment or survey is carried out again with a new sample

16. Extending a hypothesis Once you have collected data and drawn conclusions about your hypothesis, you could ask further questions and pursue other lines of enquiry. You will need to plan what these might be beforehand if you are carrying out a survey. For example, “People feel stressed when they have exams.” “You get less work done when it is noisy.” “Sleep deprivation affects concentration.” “Coffee can help you revise better.” “The more revision you do, the better your exam results.” How could you extend these hypotheses? What extra information might it be worth collecting?

17. Sampling – Soap Wars How are TV viewing figures compiled?

18. Television viewing figures When compiling television viewing figures, it is impractical to find out what everyone in the country is watching at a particular time. Instead, the viewing habits of a sample of households is carefully monitored and the data collected is used to compile the figures. To avoid bias, it is important that the sample is representative of all television viewing households across the country. This is done by dividing households into categories and taking samples in proportion to the size of each category. This is an example of a stratified sample.

19. 27 Random sampling People are chosen at random e.g. names picked from a hat or using a random number generator on a calculator. Every member of the population has an equal chance of being chosen. Different sampling methods Systematic sampling Members of the population are chosen at regular intervals, such as every 100th person from a telephone directory. Quota sampling You keepasking until you have enough people from each category. An example would be a survey in the street where you stop when you have enough people from each age category.

20. Random sampling Every member of the population has an equal chance of being chosen, which makes it fair. Evaluating different sampling methods  It can be very time consuming and usually impractical. Systematic sampling You are unlikely to get a biased sample.  It is notstrictly random: some members of the population cannot be chosen once you have decided where to start on the list.

21. Quota sampling This is easier to manage.  It could be biased. For example, if you are only asking people on the street or in a shop, the sample might not represent people at work all day. Evaluating different sampling methods Stratified sampling It is the best way to reflect the population accurately.  It is time consuming and you have to limit the number of relevant variables to make it practical.

22. MEAN MODE most common MEDIAN middle value sum of values number of values RANGE largest value – smallest value There are three different types of average: The three averages and range The range is not an average, but tells you how the data is spread out:

23. Here is a summary of Chris and Rob’s performance in the 200 metres over a season. They each ran 10 races. Comparing sets of data Which of these conclusions are correct? • Robert is more reliable. • Robert is better because his mean is higher. • Chris is better because his range is higher. • Chris must have run a better time for his quickest race. • On average, Chris is faster but he is less consistent.

24. A pie chart is a circle divided up into sectors which are representative of the data. Pie charts In a pie chart, each category is shown as a fraction of the circle. For example, in a survey half the people asked drove to work, a quarter walked and a quarter went by bus.

25. To convert raw data into angles for n data items: 360 ÷ n represents the number of degrees per data item. For example, 40 people take part in a survey. What angle represents Pie charts • 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.

26. Total There are 30 people in the survey and 360º in a full pie chart. Each person is therefore represented by 360º ÷ 30 = 12º We can now calculate the angle for each category: Drawing pie charts 96º 8 × 12º 84º 7 × 12º 36º 3 × 12º 72º 6 × 12º 72º 6 × 12º 30 360º

27. Once the angles have been calculated you can draw the pie chart. Start by drawing a circle using a compass. Drawing pie charts The Daily Express The Guardian Draw a radius. 72º Measure an angle of 96º from the radius using a protractor and label the sector. 96º 72º 84º The Sun 36º The Daily Mirror Measure an angle of 84º from the the last line you drew and label the sector. The Times Repeat for each sector until the pie chart is complete.

28. Give the bar chart a title. Use equal intervals on the axes. Label both the axes. Leave a gap between each bar. When drawing bar chart remember: Drawing bar charts

29. Year Number of absences 7 74 8 53 9 32 10 11 11 10 Use the data in the frequency table to complete a bar chart showing the the number of children absent from school from each year group on a particular day. Drawing bar charts

30. Two or more sets of data can be shown on a bar chart. For example, this bar chart shows favourite subjects for a group of boys and girls. Bar charts for two sets of data

31. Heights of students 35 30 25 20 Frequency 15 10 5 0 155 160 165 170 175 180 185 150 Height (cm) Frequency diagrams Frequency diagrams can be used to display grouped continuous data. For example, this frequency diagram shows the distribution of heights for a group students: This type of frequency diagram is often called a histogram.

32. Time spent (hours) Number of people 0 ≤ h < 1 4 1 ≤ h < 2 6 2 ≤ h < 3 8 3 ≤ h < 4 5 4 ≤ h < 5 3 h ≤ 5 1 Use the data in the frequency table to complete the frequency diagram showing the time pupils spent watching TV on a particular evening: Drawing frequency diagrams

33. Heights of Year 8 pupils 35 30 25 20 Frequency 15 10 5 0 145 150 155 160 165 170 175 140 Height (cm) We can show the trend of these graphs more clearly using a FREQUENCY POLYGON. Using a previous example, you first need to draw a histogram Histograms and Frequency Polygons Then joint the midpoints of each column.

34. Scatter Graphs 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.

35. 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.

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 The line of best fit is drawn by eye so that there are roughly an equal number of points below and above 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. 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. Constructing stem-and-leaf diagrams The data below represents the numbers of cigarettes smoked in a week by 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.

39. 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 22 427 427 ÷ 22 =___ 19 17 19 This is ___. 18 45 5 40

40. 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.

41. Cumulative Freq - Choosing class intervals You are going to record how long each member of your class can keep their eyes open without blinking. How could this information be recorded? What practical issues might arise? Time is an example of continuous data. You will have to decide how accurately to measure the times, • to the nearest tenth of a second? • to the nearest second? • to the nearest five seconds?

42. Holding Your Breath You will also have to decide what size class intervals to use. When continuous data is grouped into class intervals it is important that no values are missed out and that there are no overlaps. For example, you may decide to use class intervals with a width of 5 seconds. If everyone holds their breath for more than 30 seconds the first class interval would be more than 30 seconds, up to and including 35 seconds. This is usually written as 30 < t≤ 35, where t is the time in seconds. The next class interval would be _________. 35 < t≤ 40

43. Time in seconds Frequency Cumulative frequency Time in seconds 30 < t ≤ 35 9 35 < t ≤ 40 12 40 < t ≤ 45 24 45 < t ≤ 50 28 50 < t ≤ 55 16 55 < t ≤ 60 11 Cumulative frequency Cumulative frequency is a running total. It is calculated by adding up the frequencies up to that point. Here are the results of 100 people holding their breath: 9 0 < t ≤ 35 9 + 12 = 21 0 < t ≤ 40 21 + 24 = 45 0 < t ≤ 45 45 + 28 = 73 0 < t ≤ 50 73 + 16 = 89 0 < t ≤ 55 89 + 11 = 100 0 < t ≤ 60

44. 100 90 80 70 60 Cumulative frequency 50 40 30 20 10 0 30 35 40 45 50 55 60 Time in seconds Plotting a cumulative frequency graph The upper boundary for each class interval is plotted against its cumulative frequency. A smooth curve is then drawn through the points. We can use the graph to estimate the median by finding the time for the 50th person. This gives us a median time of 47 seconds.

45. The interquartile range Remember, the rangeis a measure of spread. It is the difference between the highest value and the lowest value. When the range is affected by outliers it is often more appropriate to use the interquartile range. The interquartile rangeis the range of the middle 50% of the data. The lower quartile is the data item ¼ of the way along the list. The upper quartile is the data item ¾ of the way along the list. interquartile range = upper quartile – lower quartile

46. 100 90 80 70 60 Cumulative frequency 50 40 30 20 10 0 30 35 40 45 50 55 60 Time in seconds Finding the interquartile range The cumulative frequency graph can be used to locate the upper and lower quartiles and so find the interquartile range. The lower quartile is the time of the 25th person. 42 seconds The upper quartile is the time of the 75th person. 51 seconds The interquartile range is the difference between these two values. 51 – 42 = 9 seconds

47. 100 90 80 70 60 50 Cumulative frequency 40 30 20 10 0 30 35 40 45 50 55 60 Time in seconds A box-and-whisker diagram A box-and-whisker diagram, or boxplot, can be used to illustrate the spread of the data in a given distribution using the median, the lower quartile and the upper quartile. These values can be found from a cumulative frequency graph. For example, for this cumulative frequency graph showing the results of 100 people holding their breath, Minimum value = 30 Lower quartile = 42 Median = 47 Upper quartile = 51 Maximum value = 60

48. Minimum value Median Maximum value Lower quartile Upper quartile 30 42 47 51 60 A box-and-whisker diagram The corresponding box-and-whisker diagram is as follows:

49. th 378 + 1 value ≈ 2 Lap times James takes part in karting competitions and his Dad records his lap times on a spreadsheet. One of the karting tracks is at Shenington. In 2004, 378 of James’ lap times were recorded. The track is 1108 metres long. James’ fastest time in a race was 51.8 seconds. In which position in the list would the median lap time be? There are 378 lap times and so the median lap time will be the 190th value

50. th 3 × th value ≈ 378 + 1 378 + 1 value ≈ 4 4 Lap times In which position in the list would the lower quartile be? There are 378 lap times and so the lower quartile will be the 95th value In which position in the list would the upper quartile be? There are 378 lap times and so the upper quartile will be the 284th value