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1 CUMULATIVE FREQUENCY AND OGIVES

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2 AS 10.4.1 (a) Collect, organise and interpret univariate numerical data in order to determine measures of dispersion, including quartiles, percentiles and the interquartile range AS 11.4.1 (a) Calculate and represent measures of central tendency and dispersion in univariate numerical data by drawing Ogives

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3 The word ogive is used to describe various smooth curved surfaces. S-shaped. Cumulative frequency curve.

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4 Cumulative Frequency Table In a frequency table you keep count of the number of times a data item occurs by keeping a tally. The number of times the item occurs is called the frequency of that item. In a frequency table you can also find a ‘running total’ of frequencies. This is called the cumulative frequency. It is useful to know the running total of the frequencies as this tells you the total number of data items at different stages in the data set.

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5 Cumulative Frequency Table showing the marks obtained by students in a test MarkFrequencyCumulative frequency This tells you that 1 11 1 students scored 1 mark 2 33+1=4 4 students scored marks of 2 or less 3 44+4=8 8 students scored marks of 3 or less 4 66+8=14 5 99+14=23 6 1111+23=34 7 1515+34=49 8 1818+49=67 9 1010+67=77 10 55+77=82 Total82 Check that the final total in the cumulative column is the same as the total number of students

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6 Activity 1 1.b) 34 learners c) 82 – 34 = 48 learners d) 77 learners 2.a) b) i) 23 learners ii) 3 learners iii) 25 learners Number of pets012345 frequency866321 Cumulative frequency81420232526

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7 A cumulative frequency table can be drawn up from: Ungrouped data (see page 3) Grouped discrete data (see page 4) Grouped continuous data (see page 5)

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8 Activity 2 – question 1 a)28 b)59 c)21 + 10 = 31 or 90 – 59 = 31 Height, h, in cm Freq Cum. Freq. 90< h ≤95 55 95< h ≤100 914 100< h ≤105 1731 105< h ≤110 2859 110< h ≤115 2180 115< h ≤120 1090

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9 Activity 2 – question 2 a)The way the interval is given 0< x ≤ 10 versus 1 – 10 b) c)15 learners d)16 + 11 = 27, or 140 – 113 = 27 e)Couldn’t % No of learners Cumul. Freq. 0< h ≤10 00 10< h ≤20 22 20< h ≤30 68 30< h ≤40 715 40< h ≤50 1429 50< h ≤60 2049 60< h ≤70 3584 70< h ≤80 29113 80< h ≤90 16129 90< h ≤100 11140 Total = 140

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10 We represent data given on a frequency table by drawing – A broken line graph – A pie chart – A bar graph – A histogram – A frequency polygon We represent data given on a cumulative frequency table by drawing a cumulative frequency graph or ogive

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11 Drawing a Cumulative Frequency Curve or Ogive Running total of frequencies S – shape Starts where frequency is 0.

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12 Activity 3 MarkFreq.Cum Freq Co- ords 000(0;0) 111(1;1) 234(2;4) 348(3;8) 4614(4;14) 5923(5;23) 61134(6;34) 71549(7;49) 81867(8;67) 91077(9;77) 10582(10;82) Tot=82

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13 Activity 4 – question 1 a)59 learners b)Yes c)90 – 59 = 31 learners

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14 Activity 4 – question 2 Time (in sec) FreqCum Freq Co-ords 35< x ≤ 40 00(40;0) 40< x ≤ 45 22(45;2) 45< x ≤ 50 79(50;9) 50< x ≤ 5 5 817(55;17) 55< x ≤ 60 825(60;25) 60< x ≤ 65 631(65;31) 65< x ≤ 70 536(70;36) 70< x ≤ 75 541(75;41) 75< x ≤ 80 445(80;45)

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15 Activity 4 – question 2 continued

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16 THE MEDIAN AND QUARTILES FROM A CUMULATIVE FREQUENCY TABLE Suppose we have the marks of 82 learners. We can divide the marks into four groups containing the same number of marks in the following way: 20 terms Q 1 score of the 21 st learner 20 terms M Average of the 41 st and 42 nd scores 20 terms Q 3 Score of the 62 nd learner 20 terms

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17 These values can be found in the cumulative frequency table by counting the data items: MarkFrequencyCumulative frequency 1 11 2 34 3 48 4 614 5 923 6 1134 7 1549 8 1867 9 1077 10 582 Total82 The 21 st student is here. Q 1 = 5 The 41 st and 42 nd students are here. Median = 7 The 62 nd student is here. Q 3 = 8

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18 The Median and Quartiles from an Ogive Q 3 is the 62 nd value Estimate of upper quartile is read here. Q 3 ≈8 Median is the 41½ th value Estimate of median is read here. M ≈ 7 Q 1 is the 21 st value Estimate of lower quartile is read here. Q 1 ≈ 5

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19 Percentiles Deciles : They divide the data set into 10 equal parts Percentiles : They divide the data set into 100 equal parts The median is the 50 th percentile. This means 50% of the data items are below the median Q 1 = 25 th percentile. This means 25% of the data items are below Q 1 Q 3 = 75 th percentile. This means 75% of the data items are below Q 3

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20 Percentiles should only be used with large sets of data. Example: The 16 th percentile of the data on the previous page is found like this: 16% of 82 = 13,12 On vertical axis find 13 then read across to curve and then down to horizontal axis 16th percentile 4 This means 16% of the class scored 4 marks or less.

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21 Activity 5 1.(a) 10% of 82 = 8,2 10 th percentile ≈ 3 90% of 82 = 73,8 90 th percentile ≈ 9 (b) 80% of the marks lie between 3 and 9. (c) 50% of the class got 7 or less out of 10 for the test. 2.(a) 50 th (b) 25 th (c) 75 th

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22 Activity 5 – question 3 MarksFrequency Cumulative frequency Points 1 – 1011(10;1) 11 – 2023(20;3) 21 – 301316(30;16) 31 – 402440(40;40) 41 – 503272(50;72) 51 – 601688(60;88) 61 – 701199(70;99) 71 – 801100(80;100)

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23 Activity 5 – question 3 continued c)Median is the 50½ th term. It lies in the interval 41 – 50. Median ≈ 45,5 f)Lower quartile is the 25½ th term. It lies in the interval 31 – 40. Q 1 ≈ 35,5 Upper quartile is the 75½ th term. It lies in the interval 51 – 60. Q 3 ≈ 55,5

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