Inequalities Recap Inequalities are:Greater Than Equal to or Less Than Equal to or Greater Than.

InequalitiesInequalities

Recap

Inequalities are: < Less Than

> Greater Than

Equal to or Less Than

Equal to or Greater Than

Notation on the numbers line

0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1-10

x < 2Note: the open circle denotesx can be very close to, but not equal to 2

Note: the closed circle denotesx can be very close to, AND equal to 2


0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1-10

x 2

0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1-10

0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1-10

-6 < x 4

Means

Meansx -3 or x > 3

0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1-10


0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1-10

0 1 2 3 4 5 6 7 8 9-9 -8 -7 -6 -5 -4 -3 -2 -1-10

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 >3

4

-1, 0, 1, 2

InequalitiesInequalitiesNow try these:

-2, -1, 0, 1, 2, 3List all the integer values that satisfy these inequalities:1. -2 x 32. -4 < x < 03. -8 4n 154. -3 < 2n 125. -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

Types of DataTypes of Data

Discrete Data

Data that can only have a specific value (often whole numbers)

For example Number of people You cannot have ½ or ¼ ofa person.

Shoe size You might have a 6½ or a 7but not a size 6.23456

Continuous Data

Data that can have any value within a range

For example Time A 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 youwill at some point every height on the way

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 DataAverages from Grouped Data


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

2535

45

55

135

150

175

175

180

110

Mean estimate = 925/55 = 16.8 minutes

55f 925f x



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


(55+1)/2= 28


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 28th data value is in the 10 - 20 class


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


mp x fmidpoint(x)

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequencynumber of laps


3

8

1318

23

2833

38

6

72195360

39170066

381828fx 91f

Mean estimate = 1828/91 = 20.1 laps


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


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

91f

The 46th data value is in the 16 – 20 class



midpoint(x) mp x f

250 - 60

440 - 50

530 - 40

720 - 30

1010 - 20

270 - 10

frequencyminutes Late

Averages from Grouped DataAverages from Grouped DataWorksheet 1

mp x fmidpoint(x)

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequencynumber of laps


Averages from Grouped DataAverages from Grouped DataWorksheet 2

Inequalities Recap Inequalities are:Greater Than Equal to or Less Than Equal to or Greater Than.

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