Graphs, charts and tables!

1. G D A Graphs,charts andtables! L Q1 Q2 Q3 H

2. Mean is sum of scores number of scores Some reminders … Scores are the wee numbers Median is the middle score. Mode is the score which occurs most often Range is highest score – lowest score

3. Relative Frequency Frequency is a measure of how often something occurs. Relative Frequency is a measure of how often something occurs compared to the total amount. Relative Frequency is given by frequency divided by the number of scores. Relative Frequency is always less than 1.

4. Example: A supermarket keeps a record of wine sales, noting the country of origin of each bottle. The frequency table shows one day’s sales. Draw a relative frequency table for the wine sales. 120  240 = 0.5 30  240 = 0.125 27  240 = 0.1125 24  240 = 0.1 18  240 = 0.075 = 0.0875 21  240 1 Note: The total of the relative frequencies is always 1. This is a useful check.

5. Country Frequency Relative Frequency France 120 120  240 = 0.5 Australia 30 30  240 = 0.125 Italy 27 27  240 = 0.1125 Spain 24 24  240 = 0.1 18  240 Germany 18 = 0.075 21  240 Others 21 = 0.0875 1 Total 240 If the supermarket wishes to order 1000 bottles of wine they may start by assuming that the relative frequencies are fixed … French wines = 0.5 x 1000 = 500 bottles Australian wines = 0.125 x 1000 = 125 bottles. Relative frequencies can be used as a measure of the likelihood of some event happening, e.g. when a customer comes in for wine, half of the time you would expect them to ask for French wine. P138/139 Ex1 (omit questions 3b, 5b)

6. Page 140, 141 Ex 2 Won Drawn Lost Reading Pie Charts A pie chart is a graphical representation of information. … … however, a pie chart can be used to calculate accurate data. Example Newton Wanderers have played 24 games. The pie chart shows how they got on. A full circle represents 24 games. Using a protractor we can measure the angles at the centre. (u estimate angles) A full circle is 360 = 8 games  24 Won: 120 90 150 = 6 games  24 Drawn: = 10 games  24 Lost: (Check that 8 + 6 + 10 = 24)

7. Example Constructing Pie Charts A geologist examines pebbles on a beach to study drift. She counts the types and makes a table of information. Draw a pie chart to display this information. 360 Now we draw the pie chart ...

8. Geology Survey Step 1: Title. Step 2: Draw a circle. Limestone Granite Step 3: Draw in start line. 58° Step 4: Using a protractor draw in the other lines. 103° 74° Sandstone 125° (you do not need to write the angles) Step 5: Label the sectors. Dolerite P141/142Ex 3

9. Grade Frequency 1 0 2 2 3 4 4 10 5 11 6 10 7 6 8 4 9 2 10 1 Example Cumulative Frequency Fifty maths students are graded 1 to 10 where 10 is the best grade. The grades and frequencies are shown below. A third column has been created which keeps a running total of the frequencies. These figures are called cumulative frequencies. Cumulative Frequency 0 2 6 16 The cumulative frequency of grade 7 is 43. 27 37 This can be interpreted as … ‘43 candidates are graded 7 or less’. 43 P143/144 Ex4 47 49 50

10. Grade Frequency Cumulative Frequency Information gathered 1 0 0 2 2 2 3 4 6 CumulativeFrequency 4 10 16 Fixed before gathering data 5 11 27 6 10 37 7 6 43 8 4 47 9 2 49 Grade 10 1 50 Fixed before gathering data Cumulative Frequency Diagrams Using the previous example we can draw a cumulative frequency diagram. We make line graph of cumulative frequency (vertical) against grade (horizontal). Maths Students Grades

11. Maths Students Grades Cumulative Frequency Grade P145,146 Ex 5 Using the diagram only … How many pupils were grade 6 or less ? 37 At least 25 pupils were less than grade 5.

12. Dotplots It is useful to get to get a ‘feel’ for the location of a data set on the number line. A good way to achieve this is to construct a dotplot. Example A group of athletes are timed in a 100m sprint. Their times, in seconds, are … 10.8 10.9 11.2 11.5 11.6 11.6 11.6 11.9 12.2 12.2 12.8 Each piece of data becomes a data point sitting above the number line

13. Some features of the distribution of figures become clearer … ● the lowest score is 10.8 seconds ● the highest score is 12.8 seconds ● the mode (most frequent score) is 11.6 seconds ● the median (middle score) is 11.6 seconds ● the distribution is fairly flat

14. Here are some expressions commonly used to describe distributions P147/148 EX 6

15. The Five-Figure Summary When a list of numbers is put in order it can be summarised by quoting five figures: H Highest number L Lowest number Q2 Median of the full list (middle score) Q1 Lower quartile – the median of the lower half Q3 Upper quartile – the median of the upper half

16. Example Make a five-figure-summary for the following data ... 6 3 7 8 11 8 6 10 9 8 5 3 5 6 6 7 8 8 8 9 10 11 Q3 Q2 Q1 L = Q1 = Q2 = Q3 = H = 3 8 9 11 6

17. Example Make a five-figure-summary for the following data. 6 3 7 8 11 6 10 9 8 5 3 5 6 6 7 8 8 9 10 11 Q1 Q2 Q3 L = Q1 = Q2 = Q3 = H = 3 7.5 9 11 6

18. Example Make a five-figure-summary for the following data. 6 3 7 8 11 6 10 9 5 3 5 6 6 7 8 9 10 11 Q1 Q3 Q2 L = Q1 = Q2 = Q3 = H = 3 7 9.5 11 5.5 P151: Ex 7

19. Q2 Q3 Q1 H L A suitable scale Boxplots A boxplot is a graphical representation of a five-figure summary.

20. 12 49 66 97 32 0 10 20 30 40 50 60 70 80 90 100 Example: Draw a box plot for this five-figure summary, which represents candidates marks in an exam out of 100 L = Q1 = Q2 = Q3 = H = Marks out of 100 ● 25% of the candidates got between 12 and 32 (the lower whisker) ● 50% of the candidates got between 32 and 66 (in the box) ● 25% of the candidates got between 66 and 97 (the upper whisker) P152/153: Ex 8

21. 0 10 20 30 40 50 60 70 80 90 100 Semi-interquartile range = (Q3 – Q1) (SIQR) 2 Q1 Q3 Comparing Distributions When comparing two or more distributions it is (VERY) useful to consider the following … ● the typical score (mean, median or mode) ● the spread of marks (the range can be used, but more often the interquartile range or semi-interquartile range is used Interquartile range = Q3 – Q1 Marks out of 100

22. Results of two exams These boxplots compare the results of two exams, one in January and one in June. Note … that the January results have a median of 38 and a semi-interquartile range of 14; the June results have a median of 51 and a semi-interquartile range of 23. On average the June results are better than January’s (since the median is higher) but … scores tended to be more variable (a larger semi-interquartile range). Note … the longer the box … the greater the interquartile range … and hence the variability.

23. Mr Tennent’s example Boxplots showing spread of marks in two successive tests. Test 2 Test 1 Which would you hope to be test 1 and which test 2? Has the class improved? (give reasons for your answer)

24. Q1 H L Q3 Q2 A suitable scale Boxplots A boxplot is a graphical representation of a five-figure summary.

25. The Five-Figure Summary When a list of numbers is put in order it can be summarised by quoting five figures: H Highest number L Lowest number Q2 Median of the full list (middle score) Q1 Lower quartile – the median of the lower half Q3 Upper quartile – the median of the upper half

26. Example: Draw a box plot for this five-figure summary, which represents candidates marks in an exam out of 100 L = Q1 = Q2 = Q3 = H = 12 49 66 97 32 0 10 20 30 40 50 60 70 80 90 100 Marks out of 100 ● 25% of the candidates got between 12 and 32 (the lower whisker) ● 50% of the candidates got between 32 and 66 (in the box) ● 25% of the candidates got between 66 and 97 (the upper whisker)

27. Example Make a five-figure-summary for the following data ... 6 3 7 8 11 8 6 10 9 8 5 3 5 6 6 7 8 8 8 9 10 11 Q3 Q2 Q1 L = Q1 = Q2 = Q3 = H = 3 8 9 11 6