1. 14 8 9 4 0 Example 1. The number of goals scored by a team in 20 games are given below : 3, 2, 4, 2, 2, 3, 2, 2, 0, 5, 1, 1, 2, 3, 0, 2, 1, 4, 1, 0 Mean from a Frequency Table f.xFrequency, fGoals x Calculating the Mean: If there are large amounts of data, it is easier if it is displayed in a frequency table. 0 1 2 3 4 5 3 4 7 3 2 1 5 ∑f= 20∑fx= 40 Mean = ∑fx ∑f = 40 20 = 2 Mode

2. 14 19 17 7 3 Example 1. The number of goals scored by a team in 20 games are given below : 3, 2, 4, 2, 2, 3, 2, 2, 0, 5, 1, 1, 2, 3, 0, 2, 1, 4, 1, 0 Median from a Frequency Table C. F.Frequency, fGoals x Calculating the median If there are large amounts of data, it is easier if it is displayed in a frequency table. 0 1 2 3 4 5 3 4 7 3 2 1 20 ∑f= 20 Mode ( 20)/2 = 10 ( 20)/2 + 1 = 11 The 10 th value is 2 The 11 th value is 2 ∴ MEDIAN = ( 2+2 ) / 2 = 4/2 = 2

3. Large quantities of data can be much more easily viewed and managed if placed in groups in a frequency table. Grouped data does not enable exact values for the mean, median and mode to be calculated. Alternate methods of analyising the data have to be employed. Example 1. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. Grouped Data 250 - 59 440 - 49 530 - 39 720 - 29 1010 - 19 270 - 9 frequencyminutes late Data is grouped into 6 class intervals of width 10.

4. Example 1. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. Grouped Data midpoint(c.c.) F × c.c. 250 - 59 440 - 49 530 - 39 720 - 29 1010 - 19 270 - 9 Frequency,fminutes Late Estimating the Mean: An estimate for the mean can be obtained by assuming that each of the raw data values takes the midpoint value of the interval in which it has been placed. 4.5 14.5 24.5 34.5 44.5 54.5 121.5 145 171.5 172.5 178 109 Mean estimate = 897.5/55 ≈ 16.32 minutes

5. Example 1. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. Grouped Data The Modal Class 250 - 59 440 - 49 530 - 39 720 - 29 1010 - 19 270 - 9 frequencyminutes late The modal class is simply the class interval of highest frequency. Modal class = 0 - 9

6. For the following set of data find : بالنسبة لمجموعة البيانات التالية أوجد : worksheet 2) الوسيط 3) المنوال 4) المدى 1) المتوسط الحسابي 6, 8, 5, 11, 3, 1, 7, 9, 3 1) The mean 2) The median 3) The mode 4) The range بالنسبة لمجموعة البيانات التالية أوجد : For the following set of data find : 9, 3, 8, 7, 1, 9, 11, 4, 3, 2 2) الوسيط 3) المنوال 4) المدى 1) المتوسط الحسابي 1) The mean 2) The median 3) The mode 4) The range

7. The ages of a random sample of 30 persons are given in the table : Age ( x ) ( f )X. f( c f ) 402 417 429 436 445 451 total أعمار عينة عشوائية من 30 شخص كما بالجدول : أوجد : worksheet 2) الوسيط للأعمار 3) المنوال 4) مدى الأعمار 1) المتوسط الحسابي للأعمار Find : 1) The mean age 2) The median 3) The mode 4) The range of the ages

8. The following frequency distribution represents the lengths of 20 persons intervals ( f )c.c.c.c.× f 150 - 1542 155 - 1598 160 - 1645 165 - 1694 170 - 1791 total التوزيع التكراري يمثل أطوال 20 شخص أوجد : worksheet 2) اكتب الفئة المنوالية وقدر المنوال 3) أوجد مدى الأطوال 1) المتوسط الحسابي للأطوال Find : 1) The mean of the lengths 2) The model class and estimate the mode 3) The range of the lengths

9. The grades of 25 students are given below : Intervals الفئات ( f )( c f ) total درجات 25 طالب كما يلي : worksheet 2) ارسم المضلع التكراري التراكمي 3) استخدم الرسم لتقدر قيمة الوسيط 1) ضع الدرجات في جدول تكراري ذو فئات 1) Put the grades in a frequency table with intervals 2) Draw the cumulative frequency polygon 3) Use the graph to estimate the median 42, 63, 47, 77, 46, 71, 68, 83, 91, 55, 67, 66, 63, 57, 50, 69, 73, 82, 77, 58, 66, 79, 88, 97, 86

10. ( 55+1)/2 = 28 Example 1. During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. Grouped Data The Median Class Interval The Median Class Interval is the class interval containing the median. 250 - 59 440 - 49 530 - 39 720 - 29 1010 - 19 270 - 9 frequencyminutes late The 28 th data value is in the 10 - 19 class

11. Data is grouped into 8 class intervals of width 4. Example 2. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps. (b) Determine the modal class. (c) Determine the class interval containing the median. Grouped Data 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps

12. c.c. X f midpoint(c.c) 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequencynumber of laps Example 2. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps. (b) Determine the modal class. (c) Determine the class interval containing the median. Grouped Data 3 8 13 18 23 28 33 38 6 72 195 360 391 700 66 38 Mean estimate = 1828/91 = 20.1 laps

13. Modal Class 26 - 30 Grouped Data Example 2. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps. (b) Determine the modal class. (c) Determine the class interval containing the median. 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps

14. 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps Example 2. A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps. (b) Determine the modal class. (c) Determine the class interval containing the median. Grouped Data (91+1)/2 = 46 The 46 th data value is in the 16 – 20 class, median ≈ 18 c. F. 2 11 26 46 63 88 90 91