Discrete and continuous data Numerical data can come in a wide variety of different forms. Discrete data is data that can only take certain values, such as the number of spectators at a sporting event. Continuous data can take any value and still have a meaning, such as the height of different trees in a forest. Teacher notes Encourage students to think of more examples of continuous and discrete data. This could include shoe size (discrete), number of goals scored in a football match (discrete), number of pets owned (discrete), speed of a car (continuous), temperature outside (continuous), a student’s height and weight (continuous), etc. Rational (not whole) numbers are allowed for continuous data, but not for discrete. Discrete data can be categorical, such as colour, so is not necessarily a number/value. Photo credits: (crowd) © Just ASC, Shutterstock.com (forest) © Chaikovskiy Igor, Shutterstock.com Can you think of other examples of discrete and continuous data?

Continuous data Don Lippincott was the holder of the first world 100 metre sprint record. In 1912, he ran 100 m in 10.6 seconds. The second fastest woman in the world in the 100 metre sprint in 2009 was Carmelita Jeter. She ran 100 m in a time of 10.64 seconds. Is it possible to tell who was the fastest? What unit has each time been rounded to? Would it be possible for someone to get a time rounded to the nearest thousandth of a second if we had sensitive enough equipment? Teacher notes Students may think that Don Lippincott is fastest as his time appears to be quicker. However, they should note that his time has only been specified to 1 d.p., whereas Carmelita Jeter’s time has been quoted to 2 d.p. With this information, it is impossible to tell which of the two was the fastest. Advances in technology since 1912 have allowed us to be more accurate in our time recordings. It would be possible for someone to get their time recorded to the nearest thousandth of a second if we had sensitive enough equipment. This is because time is a continuous measurement. If possible, we could measure time to a millionth of a second. Photo credit: © Sandra van der Steen, Shutterstock.com

Analysing data Tom regularly takes part in downhill cycle races. He records the race times of all the competitors in a race on a spreadsheet. Tom’s best time is 101.6 seconds. How accurately has he measured this time? Is the data continuous or discrete? Teacher notes Tom has measured the time to a tenth of a second. The data, because it deals with time, will be a continuous measure. The real data used in this presentation was collected by Tom Pye at a race at Aston Hill, Bucks. Encourage pupils to refer back to previous work on averages, graphs and types of data. Emphasise the size of the data set; guide them towards grouping the data. Remind pupils that the larger the number, the slower the speed. Photo credit: © Michael Woodruff, Shutterstock.com

Analysing data Here are the race times in seconds from a downhill race event. 88.4 91.5 92.1 93.3 93.9 94.7 95.0 95.3 95.5 95.6 95.6 96.3 96.5 96.9 97.0 97.0 97.0 97.3 97.4 97.4 97.7 97.8 98.0 98.2 98.2 98.4 98.4 98.5 98.9 99.0 99.1 99.6 99.6 99.8 100.0 100.6 100.6 101.1 101.4 101.4 101.5 101.6 101.6 101.8 101.9 102.1 102.5 102.6 102.7 103.1 103.1 103.1 104.1 105.0 105.2 105.6 105.6 105.7 105.8 105.9 If you wanted to analyze the performances, what could you do with the data? How easy is the data to analyze in this format? Can you draw any conclusions? Teacher notes Currently, it’s quite difficult to analyse the data in this format. In order to analyze the performances more accurately, represent the data on some kind of graph. This would make it far more accessible. Encourage students to try and make some conclusions based upon the data. Accept any answers and then split the class into groups and get them to investigate whether these conclusions are accurate or not. You may wish to print out slides with large sets of data for ease of use by pupils. Encourage pupils to refer back to previous work on averages and graphs if appropriate. Emphasise the size of the data set; guide them towards grouping the data. Remind pupils that the larger the number, the slower the speed.

Obtaining information Teacher notes Question 4 of this activity asks the students to calculate the mean time of the riders in the race. Obviously, this is a massive data set and there is a fair chance of miscalculation along the way. You could split the class into 6 groups and get each group to total up a row to minimize the risk of any calculations being entered incorrectly. Alternatively, you can supply your students with row by row data of the total times: row 1 total: 935.3; row 2 total: 968.4; row 3 total: 983.1; row 4 total: 1003.3; row 5 total: 1023; row 6 total: 946 Combined total: 5859.1 Number of participants: 59 Mean race time: 99.3 s.

Choosing the right graph Teacher notes This slide acts as a quick recap of the more obvious types of graph. Students should realize that the bar charts, line graphs and pie charts would not really be appropriate for displaying this data. Nor would a pictogram or a scatter graph. A frequency diagram would be a far better tool to use to display the data. For more details on frequency diagrams, see the presentation ‘Drawing frequency diagrams’.

Choosing the right graph A student used a spreadsheet program to produce a graph of the race data. Here, you can see the bar chart he printed to display the data. 80.0 85.0 90.0 95.0 100.0 105.0 110.0 What labels could be added to the axes? What does the graph show? Is it an appropriate graph? Teacher notes Pupils should realize that each bar represents one data item. Since the data is in ascending order, the bars gradually increase in height. The horizontal axis represents the riders 1 – 60; the vertical axis the race times. The graph should illustrate the problems of using a spreadsheet without full understanding. Suggest that grouping the data might help to see general trends and an overall picture. Ask what the class intervals should be. Point out that when the data is grouped, the times will go on the horizontal axis and frequency on the vertical axis.

The best graph? Teacher notes This slide should generate a lot of discussion as there is not just one answer for each scenario. Students may have different ideas to the sample answers given below. The favourite colour of a Year 3 class: BAR CHART The test results in science and maths of a Year 10 class: SCATTER GRAPH The results of a pocket money survey for a student newspaper: PICTOGRAM Comparing the cricket scores of 2 opening batsmen for a whole season: FREQUENCY POLYGON, BOX PLOTS The weight of all the apples from one apple tree: HISTOGRAM The revision time for 150 Year 11 students in one week: CUMULATIVE FREQUENCY CURVE The proportions of Years 7-11 in a school: PIE CHART The scores in a maths exam: HISTOGRAM How people in a town voted in the last General Election: PIE CHART If students are uncertain how best to display any of the data above, refer them back to the previous activity for a reminder of the main purpose of the different data representation methods.

Grouping data A list of results is called a data set and it is often easier to analyze a large data set if the data is put into groups. The widths of the groups are called class intervals. You can then use this information to draw a frequency diagram or a histogram. Before drawing the diagrams, you need to decide on the size of each class interval so that there are between 5 and 10 class intervals. Teacher notes On the next page the results are shown again to aid discussion. Ask pupils what the shortest and fastest times are. It is appropriate for the interval to be a multiple of 5 or 10 if possible. The best interval here would be 5 seconds. Photo credit: © grynold, Shutterstock.com What is the best size for the class intervals for the race times data?

Class intervals Here are the race times in seconds from a downhill race event. 88.4 91.5 92.1 93.3 93.9 94.7 95.0 95.3 95.5 95.6 95.6 96.3 96.5 96.9 97.0 97.0 97.0 97.3 97.4 97.4 97.7 97.8 98.0 98.2 98.2 98.4 98.4 98.5 98.9 99.0 99.1 99.6 99.6 99.8 100.0 100.6 100.6 101.1 101.4 101.4 101.5 101.6 101.6 101.8 101.9 102.1 102.5 102.6 102.7 103.1 103.1 103.1 104.1 105.0 105.2 105.6 105.6 105.7 105.8 105.9 The times range from about 85 to about 110 seconds: 110 – 85 = 25 seconds. Photo credit: © grynold, Shutterstock.com We could use class intervals with a width of 5 seconds: 25 ÷ 5 = 5 class intervals.

Notation for class intervals Tom decides to create his own groups and draws a table with class intervals that he thinks fit the race data. What is wrong with this table? How should the class intervals be written down? How can your knowledge of inequalities help you to create better class intervals? Times in seconds Frequency 85 – 90 90 – 95 95 – 100 100 – 105 105 – 110 Teacher notes Students should see that some data types could have multiple entries on the table. If students do not see this, encourage them to work through the data list putting the times in the appropriate space. Discuss where 90.0, 100.0 and 105.0 should go. The table is ambiguous. For discrete data it would be possible to edit the table to say 86 – 90, 91 – 95, 96 – 100, 101 – 105 etc (or 85 – 89, 90 – 94 etc) but for continuous data this would not work. The students should use inequalities to firm up the boundaries of the data intervals.

Notation for class intervals Teacher notes Discuss more about the meaning of inequalities and how we can use them. Ask the students to tell you verbally what is meant by: 85 ≤ t < 90. Discuss where 90, 95 etc would go in this table. Represent the inequality on a number line. Ask pupils where numbers such as 99.9999999 would go. Note that the data has been rounded off to 1 d.p. so it could be argued that we could write the intervals as 85.0 – 89.9, 90.0 – 94.9 etc but this would imply that the data is discrete, so this would be incorrect.

Notation for class intervals 85 ≤ t < 90 will include times that are ‘larger than or equal to 85 seconds and less than 90 seconds’. Another way to say this is that this class interval includes times ‘from 85 seconds up to, but not including, 90 seconds” What times will be included within this class interval: 90 ≤ t < 95? What times will be included within this class interval: 105 ≤ t < 110? Teacher notes Ask students to verbally tell you what times would be included in the class interval: 90 ≤ t < 95. They should answer something along the lines of: “this interval includes times larger than or equal to 90 seconds and less than 95 seconds” or “this interval includes times from 90 up to, but not including, 95 seconds”. Ask students to verbally tell you what times would be included in the class interval: 105 ≤ t < 110. They should answer something along the lines of: “this interval includes times larger than or equal to 105 seconds and less than 110 seconds” or “this interval included times from 105 up to, but not including, 110 seconds”. Pupils could work together in pairs to practise the correct use of vocabulary.

Notation for class intervals Teacher notes This activity involves pupils deciding which class interval a number belongs in. The numbers should be dragged into the right interval. The flying saucers will only disappear when they are dropped into the correct hole. Each time the reset button is pressed, a new set of parameters will be generated. 18

Class intervals 100 ≤ t < 105 105 ≤ t < 110 95 ≤ t < 100 88.4 91.5 92.1 93.3 93.9 94.7 95.0 95.3 95.5 95.6 95.6 96.3 96.5 96.9 97.0 97.0 97.0 97.3 97.4 97.4 97.7 97.8 98.0 98.2 98.2 98.4 98.4 98.5 98.9 99.0 99.1 99.6 99.6 99.8 100.0 100.6 100.6 101.1 101.4 101.4 101.5 101.6 101.6 101.8 101.9 102.1 102.5 102.6 102.7 103.1 103.1 103.1 104.1 105.0 105.2 105.6 105.6 105.7 105.8 105.9 100 ≤ t < 105 105 ≤ t < 110 95 ≤ t < 100 90 ≤ t < 95 85 ≤ t < 90 Time in seconds Frequency Use the data from Tom’s race times to fill in the table. What graph would you use to represent this data? 1 Teacher notes Pupils would benefit from having a print out of the data to avoid missing out data items. A tally could be used if required. They should check they have 60 items in total in the frequency column. Encourage students to think about what graph they could use to represent this data. An appropriate graph for this data is shown in the presentation ‘Drawing frequency diagrams’ 5 28 19 7