1. Inequalities Recap Inequalities are:<Less Than >Greater Than Equal to or Less Than Equal to or Greater Than

2. Notation on the numbers line 0 1234567 8 9 -9-8 -7 -6 -5 -4-3-2 -10 x < 2 Note: the open circle denotes x can be very close to, but not equal to 2 Note: the closed circle denotes x can be very close to, AND equal to 2 Inequalities 0 1234567 8 9 -9-8 -7 -6 -5 -4-3-2 -10 x 2 0 1234567 8 9 -9-8 -7 -6 -5 -4-3-2 -10 0 1234567 8 9 -9-8 -7 -6 -5 -4-3-2 -10 - 6 < x 4 Means x - 3 or x > 3

3. 0 1234567 8 9 -9-8 -7 -6 -5 -4-3-2 -10 Inequalities 0 1234567 8 9 -9-8 -7 -6 -5 -4-3-2 -10 0 1234567 8 9 -9-8 -7 -6 -5 -4-3-2 -10 List the integer values for - 3 < x 1 - 2, - 1, 0, 1 List the integer values for 7 x > - 1 0, 1,2,3,4,5,6,7 List the integer values for 6 3x > - 4 3 2 x > - 1, 0, 1, 2

4. Inequalities Now try these: - 2, - 1, 0, 1, 2, 3 List all the integer values that satisfy these inequalities: 1. - 2 x 3 2. - 4 < x < 0 3. - 8 4n 15 4. - 3 < 2n 12 5. - 5 2n-1 < 6 - 3, - 2, - 1 - 2, - 1, 0, 1, 2, 3 - 1, 0, 1, 2, 3, 4, 5, 6 - 2, - 1, 0, 1, 2, 3

5. Types of Data Discrete Data Data that can only have a specific value (often whole numbers) For exampleNumber of peopleYou cannot have ½ or ¼ of a person. Shoe size You might have a 6½ or a 7 but not a size 6.23456 Continuous Data Data that can have any value within a range For exampleTimeA person running a 100m race could finish at any time between10 seconds and 30 seconds with no restrictions Height As you grow from a baby to an adult you will at some point every height on the way

6. 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. 250 < x ≤ 60 440 < x ≤ 50 530 < x ≤ 40 720 < x ≤ 30 1010 < x ≤ 20 270 < x ≤ 10 frequencyminutes late (x) Data is grouped into 6 class intervals of width 10. Averages from Grouped Data

7. 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. midpoint(x) mp x f 250 < x ≤ 60 440 < x ≤ 50 530 < x ≤ 40 720 < x ≤ 30 1010 < x ≤ 20 270 < x ≤ 10 frequencyminutes 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. 5 15 25 35 45 55 135 150 175 180 110 Mean estimate = 925/55 = 16.8 minutes Averages from Grouped Data

8. 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. The Modal Class 250 < x ≤ 60 440 < x ≤ 50 530 < x ≤ 40 720 < x ≤ 30 1010 < x ≤ 20 270 < x ≤ 10 frequencyminutes late The modal class is simply the class interval of highest frequency. Modal class = 0 - 10 Averages from Grouped Data

9. ( 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. The Median Class Interval The Median Class Interval is the class interval containing the median. 250 < x ≤ 60 440 < x ≤ 50 530 < x ≤ 40 720 < x ≤ 30 1010 < x ≤ 20 270 < x ≤ 10 frequencyminutes late The 28 th data value is in the 10 - 20 class Averages from Grouped Data

10. 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. 136 - 40 231 – 35 2526 – 30 1721 – 25 2016 – 20 1511 – 15 96 – 10 21 - 5 frequency (x)number of laps Averages from Grouped Data

11. mp x f midpoint(x) 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. 3 8 13 18 23 28 33 38 6 72 195 360 391 700 66 38 Mean estimate = 1828/91 = 20.1 laps Averages from Grouped Data

12. Modal Class 26 - 30 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 Averages from Grouped Data

13. 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. (91+1)/2 = 46 The 46 th data value is in the 16 – 20 class Averages from Grouped Data

14. 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. midpoint(x) mp x f 250 - 60 440 - 50 530 - 40 720 - 30 1010 - 20 270 - 10 frequencyminutes Late Averages from Grouped Data Worksheet 1

15. mp x f midpoint(x) 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. Averages from Grouped Data Worksheet 2