2Main Menu THE IMPORTANT STUFF THINGS YOU NEED TO KNOW What is Coursework???Specify and PlanCollect, Process & RepresentInterpret and DiscussWhat You Should Do?THINGS YOU NEED TO KNOWPlanning the InvestigationSampleMean, Median,Mode and RangePie ChartsBar ChartsHistograms and Freq PolygonsScatter PlotsStem and Leaf PlotsCumulative FrequencyBox and Whisker Plots
3“investigate what influences the amount a student drinks.” Your TaskIs 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.
4What Will HappenA MIX OF THE FOLLOWING:Direct Teaching – statistics skills, ICT, investigation cycleGroup Work – planning, discussing, plagiarism?Individual Time – writing up, working
5Specify and Plan Specify and plan Hypothesis Interpret and discuss How could you make it better?Interpret and discussSpecify and planInvestigation cycleCollect, process and represent
6What 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.
7Collect, Process and Represent Specify and PlanCollect, Process and RepresentHypothesisHow could you make it better?Interpret and discussInvestigation cycleSpecify and planCollect, process and represent
8What 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.
9Specify and Plan Interpret and Discuss Interpret and discuss HypothesisHow could you make it better?Interpret and discussInvestigation cycleSpecify and planCollect, process and represent
10What 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.
11And Now ……. Challenge 15 mins in groups of 5 or 6 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
12Formulating a hypothesis 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.A hypothesis is a statement that you believe to be true but that you have not yet tested.The plural of hypothesis is hypotheses.Discuss the first steps in planning a statistical enquiry.Define a hypothesis as a statement of something that you believe to be true but do not have any evidence to support. For example, we could hypothesize that tabloid newspapers use shorter words than broadsheet newspapers.Year Eleven pupils with paid jobs don’t do as well in their exams.For example,
13“Year Eleven pupils with paid jobs don’t do as well in their exams.” Forming a hypothesis“Year Eleven pupils with paid jobsdon’t do as well in their exams.”How could you find out if this statement is true?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 allof these Year Elevens?These questions could be discussed in groups, with each group feeding back their ideas to the class.While pupils discuss the question, issues of population, sample size and bias should arise, and you should introduce the vocabulary as it is needed. Definitions are summarized on the next two slides.Groups may decide that the population referred to in the statement is Year Elevens in their school, their area or region or the whole country.The sample should include pupils of all ability levels and equal numbers of male and female pupils (unless there is a good reason not to).Ethical issues should also arise. Some interviewees may not wish to disclose the number of hours they work; other pupils may have to work at home unpaid acting as carers. Some groups many decide to widen the statement to include unpaid work. However, people are not always good at estimating lengths of time and may exaggerate the number of hours they spend on housework, particularly if it is done unwillingly!Sample size should be taken seriously: if pupils are collecting their own data, a sample size of between 30 and 50 is probably realistic. (When using secondary data, such as information from the internet, books or school records, a larger sample is preferable.)When talking about how to make sure a sample is fair, it can be helpful to compare taking a sample to eating a slice of a pie: you don’t need to eat the whole pie to find out what it tastes like. So long as your slice includes the rim, the crust and the filling, you should get a good idea from that.Pupils are often vague about what they will do with the data once collected, and so often collect inappropriate data. For example, groups which decide to ask pupils “Do you have a job?” will have less useful data than those who ask “How many hours of paid work do you do?” They will also need to think about how to measure exam performance. Average point scores may be useful here (1 = G, 2 = F, etc).What would you do with the data?What would you expect to find?
14Key vocabulary hypothesis – a statement that can be tested population – the group (often of people) referred to in the hypothesissample – a selection from the populationbiased sample – an unfair selectionrepresentative sample – a fair selectioncross section – a selection that reflects all the subgroups within the populationAll of these words should be introduced in context, for example within the discussion on the previous slide. It is useful to give pupils a sentence using each word to model its correct usage.objective data – information that is not affected by people’s opinions
15Key vocabularysubjective data – information that is affected by people’s opinionsprimary data – information you collect yourself, by asking people, measuring, carrying out experiments, and so onsecondary data – information that has been collected already, that you get from books, the internet, and so onethical issues – problems to do with confidentiality and personal questionsreliable results – results that will be repeated if the experiment or survey is carried out again with a new sample
16Extending 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.”These hypotheses have already been discussed on a previous slide, which will help to focus pupils on thinking only about extension questions rather than the many other issues involved in investigating them. This discussion is suitable for paired or group work.Suggestions for extensions might include:When else do people feel stressed? Is it before the exam more than during? Are stress levels reduced for certain types of people, such as organised people, confident people or older people?Do different types of noise have different effects, such as music with and without words, talking, outside activities such as drilling or lawn mowing?Does one night of sleep deprivation have less effect than several? Does it depend on how much sleep you need? Do people differ in how how much they need? Does this change with age? Does it depend on what you are concentrating on?What about other caffeine drinks, like tea or Coke? What about water, which is supposed to aid learning?Does it depend on when you revise (e.g. the night before or over a period of a week)? How could you measure the quality of the revision, rather than just its quantity? Does it depend on what resources you are using (e.g. websites, books, revision guides)? Does it depend on the subject or the teacher? Do people in the top sets revise more or less than those in the lower sets? Do boys revise more than girls?“The more revision you do, the better your exam results.”How could you extend these hypotheses?What extra information might it be worth collecting?
17How are TV viewing figures compiled? Sampling – Soap WarsDiscuss how pupils think television viewing figures are compiled.How are TV viewing figures compiled?
18Television 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.Discuss what categories of households would be important when monitoring viewing habits. For example, geographical location and household income levels.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.
19Different sampling methods Random samplingPeople 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 equalchance of being chosen.27Systematic samplingMembers of the population are chosen at regular intervals, such as every 100th person from a telephone directory.These kind of methods are used in research in contexts such as psychology, sociology, social policy, medicine, marketing, politics, economics. It will be useful to relate them to pupils’ own interests and future career plans.Quota samplingYou keep asking 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.
20Evaluating different sampling methods Random sampling Every member of the population has an equal chance ofbeing chosen, which makes it fair. It can be very time consuming and usually impractical.Systematic sampling You are unlikely to get a biased sample. It is not strictly random: some members of the population cannot be chosen once you have decided where to start on the list.These advantages and disadvantages are most useful in discussions in the context of particular hypotheses. Evaluating the choice of sampling method is a higher order thinking skill which is most appropriate to pupils aiming for A and A* grades. Ask pupils to come up with their own advantages and disadvantages before displaying these.
21Evaluating different sampling methods Quota sampling This is easier to manage. It could be biased. For example, if you are only askingpeople on the street or in a shop, the sample might notrepresent people at work all day.Stratified sampling It is the best way to reflect the population accurately.Ask pupils to come up with their own advantages and disadvantages before displaying these. It is time consuming and you have to limit the number ofrelevant variables to make it practical.
22The three averages and range There are three different types of average:MODEmost commonMEANsum of valuesnumber of valuesMEDIANmiddle valueThe range is not an average, but tells you how the data is spread out:RANGElargest value – smallest value
23Comparing sets of dataHere is a summary of Chris and Rob’s performance in the 200 metres over a season. They each ran 10 races.ChrisRobMean24.8 seconds25.0 secondsRange1.4 seconds0.9 secondsWhich of these conclusions are correct?Robert is more reliable.The first and the last statements are correct. The data on the next page will illustrate why the fourth statement is not always correct. The second statement is not correct because a higher mean means he is slower. The third statement is incorrect because a high range means he is inconsistent.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.
24Pie charts A pie chart is a circle divided up into sectors which are representative of the data.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.In a bar chart, the size of each category is compared with each of the others. In a pie chart, each category is compared with the whole.Point out that if the sectors are not labelled we must include a key.
25How many people are represented by an angle of 36°? Pie chartsTo 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 representsone person?360° ÷ 40 =9°two people?9° × 2 =18°eight people?9° × 8 =72°You may want to link this to previous work on ratio and proportion. An alternative method is to use fractions e.g. 1/40 x 360o = 9o and 36/360 =1/10; 1/10 of 40 = 4.How many people are represented by an angle of 36°?There are 9° per person.36° ÷ 9° =4 people.
26Drawing pie chartsThere are 30 people in the survey and 360º in a full pie chart.Each person is therefore represented by 360º ÷ 30 = 12ºWe can now calculate the angle for each category:NewspaperNo of peopleWorkingAngleThe Guardian8Daily Mirror7The Times3The Sun6Daily Express8 × 12º96º7 × 12º84º3 × 12º36ºTalk through the first method. This method works well if the number of people in the survey (or whatever the pie chart is being used to represent) divides exactly into 360°. Once we know how many degrees represent each person we can multiply this amount by the frequency.Stress that we should check that the angles add up to 360º.(Although, in cases where the angles have been rounded there is the possibility that the angles won’t add up to 360º.)6 × 12º72º6 × 12º72ºTotal30360º
27Drawing pie chartsOnce the angles have been calculated you can draw the pie chart.Start by drawing a circle using a compass.The Daily ExpressThe GuardianDraw a radius.Measure an angle of 96º from the radius using a protractor and label the sector.72º96º72ºThe Sun84º36ºThe Daily MirrorMeasure an angle of 84º from the the last line you drew and label the sector.The TimesRepeat for each sector until the pie chart is complete.
28Drawing bar charts When drawing bar chart remember: Give the bar chart a title.Use equal intervals on the axes.Label both the axes.Leave a gap between each bar.
29Drawing bar chartsUse 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.YearNumber of absences7748539321011Start by deciding on a suitable scale for the vertical axis. For example use each division to represent two pupils. Number this axis using the pen tool. Ask volunteers to drag each bar to the required frequency.Copy this slide and modify the table to produce more examples if required.
30Bar charts for two sets of data 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.Stress that we must include a key when more than one type of data is displayed in the same chart.Ask,What subject did most girls like the best?What subject did most boys like the best?Is it possible to tell if an equal number of boys and girls took part in the survey?
31Frequency diagramsFrequency diagrams can be used to display grouped continuous data.For example, this frequency diagram shows the distribution of heights for a group students:FrequencyHeight (cm)5101520253035150155160165170175180185Heights of studentsStress that the difference between this graph and a bar graph is that the bars are touching. Bar graphs can only be used to display qualitative data or discrete numerical data where as histograms are used to show continuous data.Strictly speaking, for a histogram we plot frequency density rather than frequency along the vertical axis. However, this is not make any difference to the graph when the class intervals are equal as in this example.This type of frequency diagram is often called a histogram.
32Drawing frequency diagrams Use the data in the frequency table to complete the frequency diagram showing the time pupils spent watching TV on a particular evening:Time spent (hours)Number of people0 ≤ h < 141 ≤ h < 262 ≤ h < 383 ≤ h < 454 ≤ h < 53h ≤ 51Start by discussing how to number the vertical axis. Use the pen tool to do this.Drag the first bar to the appropriate height. Ask a volunteer to drag the next bar and continue until the bars are complete.Finally, ask a volunteer to use the pen tool to show how the horizontal axis should be numbered.Copy this slide and modify the table to produce more examples if required.
33Histograms and Frequency Polygons We can show the trend of these graphs more clearly using a FREQUENCY POLYGON.Using a previous example, you first need to draw a histogramThen joint the midpoints of each column.FrequencyHeight (cm)5101520253035140145150155160165170175Heights of Year 8 pupilsPoint out that a frequency diagram is very similar to a bar chart except that the bars touch each other and the divisions between the bars are labelled.Ask pupils to give the modal class interval for the data on the board.Discuss the type of data that would be shown in a frequency diagram. For example, time taken to run a race, foot length, weights etc. In other words, anything that is measured.
34What does this scatter graph show? 50556065707580852040100120Number of cigarettes smoked in a weekLife expectancyThis data is fictional. However, there is a variety of research linking smoking to a number of fatal diseases such as cancer. For further details, see the ASH website (www.ash.co.uk).It shows that life expectancy decreases as the number of cigarettes smoked increases.This is called a negative correlation.
35Interpreting 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 variableStress that when there in no correlation between the variables it does not necessarily mean that there is no relationship, only that there is no linear relationship between them.negative correlation: as one variable increases, the other variable decreaseszero correlation: no linear relationship between the variables.Correlation can be weak or strong.
36The line of best fitThe 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,510152025Strong positive correlationWeak negative correlation510152025Strong negative correlationWeak positive correlationNotice 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.
37Line of best fitWhen 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.
38Constructing stem-and-leaf diagrams The data below represents the numbers of cigarettes smoked in a week by regular smokers in Year 11.Put this data into a stem-and-leaf diagram.The stem should represent ____ and the leaf should represent _____.tensThis data is fictional. Ask questions such as “How many people took part in the survey altogether?” “How many spent under £5/ £7 etc?” “What is the mean/ median/ modal amount spent?”unitsWork out the mode, mean, median and range.
39Calculations with stem-and-leaf diagrams ModeThe mode is __ .1 54321Leaf (units)Stem (tens)7MeanThere are ___ people in the survey and they smoke a total of ____ cigarettes a week.22427427 ÷ 22 =___19MedianThe median is halfway between ___ and ___.Ask questions such as “How many people took part in the survey altogether?” “How many spent under £5/ £7 etc?” “What percentage smoke more than 20 a fortnight?” as well as the averages and range.1719This is ___.18Range___ – ___ = ___45540
40Solving 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.1 54321Leaf (units)Stem (tens)8 ÷ 22 x 100 = …… = 36%11 ÷ 22 = 50%5 ÷ 22 x 100 = 22.72…. = 23%Median = ÷ £20 = £0.20 per cigarette. 18 x £0.20 = £3.60. In practice, you would probably smoke the whole packet though!You could then work out how much would be spent in a year for a selection of the smokers, and discuss why people choose to spend their money in this way.
41Cumulative 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,The results will be continuous data. You will have to decide how accurately to measure the times. The nearest second is most appropriate, but intervals of 5 seconds would provide adequate information. It depends on what the data is used for. For example, medical research might require more accurate information.to the nearest tenth of a second?to the nearest second?to the nearest five seconds?
42Holding Your BreathYou 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.Verify that there are no gaps or overlaps for the two class intervals given.This is usually written as 30 < t ≤ 35, where t is the time in seconds.The next class interval would be _________.35 < t ≤ 40
43Cumulative frequencyCumulative 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:Cumulative frequency1650 < t ≤ 551155 < t ≤ 60930 < t ≤ 351235 < t ≤ 402440 < t ≤ 452845 < t ≤ 50Time in secondsFrequency90 < t ≤ 35= 210 < t ≤ 40= 450 < t ≤ 45The right hand column represents the cumulative frequency. For example, the second category of 0 < t < 40 means 21 people held their breath for 40 seconds or less. Ask pupils for the cumulative frequencies and to explain what they mean e.g. “What does the cumulative frequency 45 mean?” Answer: “45 pupils held their breath for 45 seconds or less.” Ask “what percentage of the class can hold their breath for 45 seconds or less?” and “How many can hold their breath for 60 seconds?”Discuss how you could find an estimate for the mean and median from the grouped data; and the modal group.= 730 < t ≤ 50= 890 < t ≤ 55= 1000 < t ≤ 60
44Plotting a cumulative frequency graph Time in secondsCumulative frequency303540455055601020708090100The 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.Note that the first point that is plotted is the lower boundary of the first class interval which has a cumulative frequency of 0.Point out the characteristic S-shape of the cumulative frequency curve.Ask pupils to use the graph to estimate the number of seconds the middle person held their breath for. This is technically the ( ) ÷ 2th person, but since the graph is not accurate enough to measure this, it is appropriate to use the 50th person.This gives us a median time of 47 seconds.
45The interquartile range Remember, the range is 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 range is the range of the middle 50% of the data.The lower quartile is the data item ¼ of the way along the list.Link:D2.5 Comparing data.The upper quartile is the data item ¾ of the way along the list.interquartile range = upper quartile – lower quartile
46Finding the interquartile range Time in secondsCumulative frequency303540455055601020708090100The 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 secondsThe upper quartile is the time of the 75th person.51 secondsThe lower quartile for 100 people is technically the ( ) ÷ 4th person and the upper quartile the 3( ) ÷ 4th person. The graph is not accurate enough to measure this and so it is appropriate to use the 25th and 75th person.Remind pupils this represents the range of the middle half of the data. Compare this with the range.Link:D2.5 Comparing data.The interquartile range is the difference between these two values.51 – 42 = 9 seconds
47A 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.Time in secondsCumulative frequency303540455055601020708090100For example, for this cumulative frequency graph showing the results of 100 people holding their breath,Minimum value = 30The minimum and maximum values are not available from the grouped data. The lower and upper bounds from the lowest and highest groups respectively have been used here.Lower quartile = 42Median = 47Upper quartile = 51Maximum value = 60
48A box-and-whisker diagram The corresponding box-and-whisker diagram is as follows:30Minimum value47Median60Maximum value42Lower quartile51Upper quartileNote that the boxes are drawn to scale, so that the relative positions of the lower quartile, median and upper quartile can be seen clearly. Discuss the position of the median: it is not halfway within the interquartile range, as it was within the full range. It is closer to the upper quartile than the lower quartile. The interquartile range is clearly much smaller than the full range.The class could draw a box-and-whisker diagram for the data collected on slide 25.
49In which position in the list would the median lap time be? Lap timesJames 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?Sometimes the number of data items does not divide equally into 4, and the position of the median, lower quartile and upper quartile have to be rounded. This should be discussed.The median would be 379/2 = = 190thThe lower quartile would be 379/4 = = 95thThe upper quartile would be 379/4 × 3 = = 284th.There are 378 lap times and so the median lap time will be the2thvalue ≈190th value
50Lap times In which position in the list would the lower quartile be? There are 378 lap times and so the lower quartile will be the4thvalue ≈95th valueIn which position in the list would the upper quartile be?Sometimes the number of data items does not divide equally into 4, and the position of the median, lower quartile and upper quartile have to be rounded. This should be discussed.The median would be 379/2 = = 190thThe lower quartile would be 379/4 = = 95thThe upper quartile would be 379/4 × 3 = = 284th.There are 378 lap times and so the upper quartile will be the4thvalue ≈3 ×284th value
51Lap times at Shenington karting circuit James’ lap times are displayed in the following cumulative frequency graph.Lap times in secondsCumulative frequency52545658606264666870727476788082848688909250100150200250300350400This data was collected for James Peace (www.54racing.com) during 2004 at the Shenington karting circuit. The data has been grouped in two-second intervals. Point out that faster lap times are smaller, ie 52 is his best lap time (to the nearest second) and 92 his worst. Most of his lap times are between 52 and 57 seconds. The graph tails off at the end because there are fewer slower lap times. Discuss what might cause these e.g. the weather conditions. Pupils would benefit from a print out of the graph. Discuss appropriate levels of accuracy for reading from the graph. It is not possible to read more accurately than the nearest 0.5 for the lap times, and the nearest 10 for the cumulative frequency.
52Box and whisker plot for James’ race times Minimum valueMaximum valueLower quartileMedianUpper quartileDiscuss the position of the median within the data and within the interquartile range. It is much closer to the faster times (since the data is skewed). The interquartile range is clearly much smaller than the full range, and there are a minority of lap times that distort the data overall, demonstrating the usefulness of the interquartile range and the advantage of the median over the mean.5254589153What conclusions can you draw about James’ performance?