Discrete vs continuous Count it or measure it?

Discrete Continuous You can count it. The units of measurement cannot be split up. E.g shoe size- you cannot have a size 5.2. If you were to put it on a number scale, only certain numbers have a meaning. You can measure it. The whole units can be split up. You can put it on a number scale. Each value on the scale has a meaning e.g. 1.05kg

How many cows are there in that field? Discrete Continuous

How fast can your car go? Discrete Continuous

How much water is there in this bottle? Discrete Continuous

How many CDs do you own? Discrete Continuous

What is your shoe size? Discrete Continuous

How much does a whale weigh? Discrete Continuous

How much does a Mars Bar cost? Discrete Continuous

What is the temperature of this fire? Discrete Continuous

How long does the bus to Birchington take? Discrete Continuous

Discrete Qualitative Continuous What is the breed of that dog? Discrete Qualitative Continuous

Types of Average- Silent Starter – Write down accurate definitions of Mean, Mode, Median and Range Mean - Add up the numbers and divide by how many there are Mode - the value that occurs most often Median - the middle value (you must put them in order first) Range - the difference between the largest and smallest values

6 6 6 8 4 WATCH CAREFULLY AND WRITE DOWN THE NUMBERS THAT APPEAR AND CALCUATE THE MMMR

3 5 2 6 4 7 1

What is your method for finding the median? Calculate the MMMR of the following data: 3,5,6,5,4,3,3,5,6,2,4,3,1,2,1,3,4,5,5,1,1,5,6,3,4,2,1,2,1,3,4,5,6,4,3,1,4,2,6,6 1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6   Mean = 140/40 = 3.5 Median = 4 Mode = 3 Range = 6-1= 5 2

Here is some data on class sizes in a school. Frequency Tables It is often convenient to record data in a frequency table. This is particularly useful when there is a large amount of data. The mean, mode, median and range can be calculated directly from the table. Here is some data on class sizes in a school. 27, 27, 28, 28, 28, 28, 29, 29, 29, 30, 30, 30, 30, 30, 31, 31, 32 Data placed in a frequency table Class Size 27 28 29 30 31 32 f 2 4 3 5 1

Objective (L7) Calculate the Mode, Median, Mean and Range of grouped data.

Or if entered directly into a calculator. What does the symbol for ‘the sum of’ look like? Original data in list form: 27, 27, 28, 28, 28, 28, 29, 29, 29, 30, 30, 30, 30, 30, 31, 31, 32 Frequency Tables Calculating the mean from the frequency table. Class Size x 27 28 29 30 31 32 f 2 4 3 5 1 fx 54 112 87 150 62 32 Or if entered directly into a calculator. (2 x 27 + 4 x 28 + 3 x 29 + 5 x 30 + 2 x 31 + 1 x 32) 17 Mean = = 497/17 = 29.2

Original data in list form: 27, 27, 28, 28, 28, 28, 29, 29, 29, 30, 30, 30, 30, 30, 31, 31, 32 Frequency Tables Finding the mode, median and range from the frequency table. Class Size 27 28 29 30 31 32 f 2 4 3 5 1 Median = 29 (The middle data value) [(17 + 1)/2 = 9th data value.] Mode = 30 (Highest frequency) Range = 32 – 27 = 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 midpoint mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes 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. 5 135 15 150 25 175 35 175 45 180 55 110 Mean estimate = 925/55 = 16.8 minutes

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 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late Data is grouped into 6 class intervals of width 10.

The modal class is simply the class interval of highest frequency. 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 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late The modal class is simply the class interval of highest frequency. Modal class = 0 - 10

The 28th data value is in the 10 - 20 class 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. 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late (55+1)/2 = 28 The 28th data value is in the 10 - 20 class

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 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency (x) number of laps Data is grouped into 8 class intervals of width 4.

Grouped Data Mean estimate = 1828/91 = 20.1 laps Example 2. mp x f midpoint(x) 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency 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 3 6 8 72 13 195 18 360 23 391 28 700 33 66 38 38 Mean estimate = 1828/91 = 20.1 laps

Grouped Data Example 2. Modal Class 26 - 30 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. 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency (x) number of laps Modal Class 26 - 30

Grouped Data Example 2. (91+1)/2 = 46 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 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 The 46th data value is in the 16 – 20 class  (91+1)/2 = 46

Your Turn!

Averages From Grouped Data 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. Averages From Grouped Data midpoint(x) mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes 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. 15 25 35 45 55 135 150 175 180 110 Mean estimate = 925/55 = 16.8 minutes Modal Class (55 +1 )/2 = 28 Median Class Interval

Grouped Data Example 1. mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 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 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes Late

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. mp x f midpoint(x) 1 36 - 40 2 31 – 35 25 26 – 30 17 21 – 25 20 16 – 20 15 11 – 15 9 6 – 10 1 - 5 frequency number of laps