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SLIDE 1
360 degrees = 360°
A circle is divided into 360 degrees.
An angle is a measure of the separation of two lines. Angles are measured using a protractor in degrees.
SLIDE 2
Use a protractor to measure the angles to the nearest degree.
SLIDE 3
SLIDE 4
The angle of a circle is 360°
State the angle of
?
A straight line ?
A right-angle ?
An equilateral triangle
SLIDE 5
?
A straight line ?
A right-angle ?
An equilateral triangle
180° 90° 60°
SLIDE 6
Find the angles
?
Regular hexagon ?
Regular pentagon ?
Regular octagon
SLIDE 7
?
Regular hexagon ?
Regular pentagon ?
Regular octagon
SLIDE 8
A pie chart represents information
There are 360 children. Each child opts for one sport.
90 children choose football
120 children choose tennis
45 children choose hockey
105 children choose basketball
The pie chart shows this information.
SLIDE 9
A farm has 360 animals
100 cows
90 pigs
45 horses
125 sheep
Draw a pie chart to show this information
SLIDE 10
A farm has 360 animals: 100 cows, 90 pigs, 45 horses, 125 sheep
SLIDE 11
At an election 1800 votes were cast. The results are shown in the pie chart. Measure the angles and estimate the number of votes cast for each of the four candidates?
SLIDE 12
At an election 1800 votes were cast. The results are shown in the pie chart. Measure the angles and estimate the number of votes cast for each of the four candidates?
Number of votes is 5 times the measured angle
Candidate Angle Fraction Votes
Sir Blue
525
Ms. Red
625
Mr. Green
350
Baron Yellow
300
SLIDE 13
Example
The angle measured in a pie chart is 100°. What is this as a percentage? Give your answer to 1 decimal place.
Exercise
Express the following angles in a pie chart as percentages. Where appropriate, give your answer to 1 decimal place.
? 72° ? 85° ? 240°
SLIDE 14
? 72°
? 85°
? 240°
SLIDE 15
In a survey people were asked which of four brands of smart phone they thought was the best. The results were displayed in a pie chart.
Measure the angles and estimate to the nearest 0.5% the percentage of people who chose each brand as best.
SLIDE 16
In a survey people were asked which of four brands of smart phone they thought was the best. The results were displayed in a pie chart. Measure the angles and estimate to the nearest 1% the percentage of people who chose each brand as best.
XXX
Alpha
Supreme
Excellence
SLIDE 17
Example
In a blind test 22% of people could not tell the difference between two brands of marmalade. Convert this to (a) a fraction and (b) an angle for a pie chart.
Exercise
Express each of the following percentages as (a) fractions and (b) angles in a pie chart.
? 33.3% ? 35% ? 37%
SLIDE 18
? 33.3%
? 35%
? 37%
SLIDE 19
The table shows the percentage of cattle breeds imported to a country.
Breed Percentage
Aberdeen Angus 27
Hereford 15
Highland 25
Jersey 8
Longhorn 25
Make a pie chart from the data.
SLIDE 20
Breed Percentage
Aberdeen Angus 27
Hereford 15
Highland 25
Jersey 8
Longhorn 25
SLIDE 21
A stem and leaf diagram is a method of organizing data.
Example
The length, in cm, of twenty different plant shoots were measured 3 weeks after germination.
Draw a stem and leaf diagram to show this information. Include a key.
SLIDE 22
The weight, in grams, of twenty different newly dug-up potatoes was measured.
Draw a stem and leaf diagram to show this information. Include a key.
SLIDE 23
Data
Stem and leaf diagram
SLIDE 24
Continuous data describes things that be measured. For instance, the length of a bed, or the weight of potatoes on a market stall.
Discrete data describes things that are being counted but not measured. For instance, the number of children in a school, or the number of cars parked on a street at a given time.
Question
Which option is correct?
A. Number of magazines on a table ? Continuous
Weight of the magazines on the table ? Discrete
B. Number of magazines on a table ? Discrete
Weight of the magazines on the table ? Continuous
C. Number of magazines on a table ? Continuous
Weight of the magazines on the table ? Continuous
D. Number of magazines on a table ? Discrete
Weight of the magazines on the table ? Discrete
SLIDE 25
Option B is correct
Number of magazines on a table ? Discrete
Weight of the magazines on the table ? Continuous
SLIDE 26
Discrete data is represented by a bar chart.
A manager kept a record of the number of fashion magazines bought in the shop on Monday.
Magazine BoF Cosmopolitan Elle Harper’s Vogue W
Number 8 10 14 12 19 6
Make a bar chart to represent this data.
SLIDE 27
Magazine BoF Cosmopolitan Elle Harper’s Vogue W
Number 8 10 14 12 19 6
SLIDE 28
Continuous data comes from measurements. Continuous data is represented by a histogram.
A histogram uses rectangles that touch each other.
A quality inspector kept a record of the weights of seventy grade A tomatoes.
Weight / g Number
5
13
28
17
7
Make a histogram to represent this data.
SLIDE 29
Weight / g Number
5
13
28
17
7
A histogram has no gaps because the data is continuous. The frequencies (number) are given for intervals, which are represented as lying on the boundaries of the rectangles.
SLIDE 30
An examination board kept a record of the scores of 200 students taking an exam.
Score (max 100) Number Score (max 100) Number
5
64
2
46
9
21
12
7
32
2
Make a histogram to represent this data.
SLIDE 31
Score (max 100) Number Score (max 100) Number
5
64
2
46
9
21
12
7
32
2
SLIDE 32
Where data is collected in categories, it is called a grouped frequency table. The numbers in each category is the frequency.
For example, the data in the preceding question had the following grouped frequency table.
Score (max 100) Frequency Score (max 100) Frequency
5
64
2
46
9
21
12
7
32
2
SLIDE 33
The times in seconds spent by 50 customers waiting to be served in a store were recorded.
43 57 51 78 74 54 92 85 79 61
61 69 64 101 65 94 68 88 72 72
72 108 53 76 76 76 77 78 52 81
83 56 87 62 88 89 78 92 66 96
99 64 73 51 72 56 83 79 85 46
? Make a stem and leaf diagram of the data.
? Make a grouped frequency table from the stem and leaf diagram
? Construct a histogram for the data
SLIDE 34
43 57 51 78 74 54 92 85 79 61
61 69 64 101 65 94 68 88 72 72
72 108 53 76 76 76 77 78 52 81
83 56 87 62 88 89 78 92 66 96
99 64 73 51 72 56 83 79 85 46
?
? Interval 40—50 50—60 60—70 70—80 80—90 90—100 100—110
Frequency 2 8 9 15 9 5 2
?
SLIDE 35
The mode is the item that comes up most frequently in a list.
Group each of the following by category and find the mode.
? Essay grades
A D C C A B C E C
? Vegetables
Potato Carrot Carrot Cabbage Sprout Carrot
Leak Carrot Potato Spinach Carrot Parsnip
Potato Carrot Cabbage Bean Bean Carrot Bean
? Time in seconds
20.5 20.3 19.9 20.0 20.1 20.2 20.3 21.0 20.6
20.2 20.4 20.2 20.1 20.3 20.2 20.2 20.4 20.2
SLIDE 36
? Essay grades
A D C C A B C E C
A B C D E
2 1 4 1 1
Mode = C
? Vegetables
Potato Carrot Carrot Cabbage Sprout Carrot
Leak Carrot Potato Spinach Carrot Parsnip
Potato Carrot Cabbage Bean Bean Carrot Bean
Potato Carrot Cabbage Sprout Leak Spinach Parsnip Bean
3 7 2 1 1 1 1 3
Mode = Carrot
? Time in seconds
20.5 20.3 19.9 20.0 20.1 20.2 20.3 21.0 20.6
20.2 20.4 20.2 20.1 20.3 20.2 20.2 20.4 20.2
19.9 20.0 20.1 20.2 20.3 20.4 20.5 20.6
1 2 2 6 3 2 1 1
Mode = 20.2
SLIDE 37
There can be no mode and there can be more than one mode
Which of the following have (a) one mode (b) no mode or (c) more than one mode?
? Orange Yellow Green Blue Magenta Violet Red
? 23 22 19 25 19 22 23 25 18
? Apollo Venus Zeus Hades Persephone Hera Zeus
Zeus Hermes Hermes Artemis Poseidon Zeus Zeus
? Peter Paul Andrew Jude Paul Matthew Mark John
John Paul John Matthias Philip Bartholomew Judas
SLIDE 38
? Orange Yellow Green Blue Magenta Violet Red
No mode
? 23 22 19 25 19 22 23 25 18
Three modes: 22, 23 and 25
? Apollo Venus Zeus Hades Persephone Hera Zeus
Zeus Hermes Hermes Artemis Poseidon Zeus Zeus
One mode: Zeus
? Peter Paul Andrew Jude Paul Matthew Mark John
John Paul John Matthias Philip Bartholomew Judas
Two modes: Paul and John
SLIDE 39
In a set of ordered data, the range is the highest value less the lowest value.
If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of the two middle values.
Arrange the following sets of data in order. For each find the range, mode and median.
? Speed outside the school in miles per hour
30 17 20 18 33 19 25 27 20 21 17 23 24 16 24 26 20 39 19
? Year of birth of residents in a home
1932 1925 1941 1937 1944 1929 1930 1931 1932 1926 1937 1940 1923 1936 1943 1941 1946 1935
SLIDE 40
? Speed outside the school in miles per hour
16 17 17 18 19 19 20 20 20 21 23 24 24 25 26 27 30 33 39
? Year of birth of residents in a home
1923 1925 1926 1929 1930 1931 1932 1932 1935 1936 1937 1937 1940 1941 1941 1943 1944 1946
SLIDE 41
Continuous data describes things that be measured. For instance, the length of a bed, or the weight of potatoes on a market stall.
Discrete data describes things that are being counted but not measured. For instance, the number of children in a school, or the number of cars parked on a street at a given time.
Which option is correct?
A Number of children in a park ? continuous
Volume of fuel in a tank ? discrete
Continuous data ? bar chart
Discrete data ? histogram
B Number of children in a park ? discrete
Volume of fuel in a tank ? continuous
Continuous data ? bar chart
Discrete data ? histogram
C Number of children in a park ? continuous
Volume of fuel in a tank ? discrete
Continuous data ? histogram
Discrete data ? bar chart
D Number of children in a park ? discrete
Volume of fuel in a tank ? continuous
Continuous data ? histogram
Discrete data ? bar chart
SLIDE 42
Continuous data describes things that be measured. For instance, the length of a bed, or the weight of potatoes on a market stall.
Discrete data describes things that are being counted but not measured. For instance, the number of children in a school, or the number of cars parked on a street at a given time.
The correct option is D
D Number of children in a park ? discrete
Volume of fuel in a tank ? continuous
Continuous data ? histogram
Discrete data ? bar chart
SLIDE 43
Continuous data describes things that be measured. For instance, the length of a bed, or the weight of potatoes on a market stall.
Continuous data has an average. The average is also called the mean. To find the mean, you add up all the values and divide by the total number of values.
Example
31 32 32 33 37 42 43 47 51 56
The number of values is also called the total frequency.
Question
16.5 16.7 17.1 18.3 18.4 18.8 19.0 19.3 19.6 20.1
Find the mean
SLIDE 44
16.5 16.7 17.1 18.3 18.4 18.8 19.0 19.3 19.6 20.1
SLIDE 45
Mean, median and mode are all different measures of the central value of a set of data.
Continuous data has a mean. Data that can be ordered has a median.
Which of the following sets of data have a mean? Find the mean, median and mode where appropriate.
? Fastest eight 100 m times
Bolt 9.58 Gay 9.69 Blake 9.72 Powell 9.72
Gatlin 9.74 Coleman 9.76 Carter 9.78 Green 9.79
? Cars parked on a street
Volkswagen, BMW, Porsche, Honda, Ford, BMW, BMW, Volvo, Porsche, Nissan, Porsche, Ford, Audi, Volvo, Honda
? First ten prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
? Mass of children in a class / kg
31.3 32.7 29.3 27.3 28.4 29.7 31.5 26.0 22.9
33.4 40.1 38.7 30.6 33.5 29.9 41.3 36.5 34.0
SLIDE 46
? Fastest eight 100 m times
Bolt 9.58 Gay 9.69 Blake 9.72 Powell 9.72
Gatlin 9.74 Coleman 9.76 Carter 9.78 Green 9.79
Continuous data ? has a mean
No mode, median = ,
mean =
? Cars parked on a street
Volkswagen, BMW, Porsche, Honda, Ford, BMW, BMW, Volvo, Porsche, Nissan, Porsche, Ford, Audi, Volvo, Honda
Data in categories ? no mean, no median
Mode = BMW and Porsche
? First ten prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Not continuous ? no mean
Median = , no mode.
? Mass of children in a class / kg
26.0 22.9 27.3 28.4 29.3 29.7 29.9 30.6 31.3
31.5 32.7 33.4 33.5 34.0 36.5 38.7 40.1 41.3
Continuous data ? has a mean
mean = median =
SLIDE 47
The first ten prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
These numbers are placed in order from smallest to largest in the above list.
? Find the average of these ten numbers
? Explain why the first ten prime numbers has an average but does not have a mean
? What is the difference between an average and a mean?
SLIDE 48
The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
? Average
? These numbers are not measurements. They are counting numbers, also called natural numbers. Only data that can be measured continuously has a mean. Natural numbers are not continuous data.
? Given any collection of numbers we can always calculate the average, but we will only call that the mean if in fact the data originates in measurements and is, therefore, continuous.
SLIDE 49
It is customary to use average and mean interchangeably, and to also call the average of any numbers (continuous or not) the mean of those numbers. Strictly, a mean is calculated on continuous data.
The context of the question makes the use clear. If the context is information and data, then the data only has a mean if they are continuous measures.
The following question appeared in an exam.
Question
Look at the numbers
9 11 10
? Show that the mean of the three numbers is 10
? Explain why the median of the three numbers is 10
? Four numbers have a mean of 10 and a median of 10, but none of the numbers is 10. What could the four numbers be? Give an example.
SLIDE 50
Look at the numbers
9 11 10
? Show that the mean of the three numbers is 10
? Explain why the median of the three numbers is 10
In rank order the numbers are 9, 10, 11 and the middle number is 10. Therefore, median = 10.
? Four numbers have a mean of 10 and a median of 10, but none of the numbers is 10. What could the four numbers be? Give an example.
The total must be 40 and the two middle numbers must average to 10, and hence total 20. For example,
5 9 11 15
SLIDE 51
The mean of three numbers is 5. When I add one more number the mean is 6. What number have I added?
Remark
This is not really a problem about information or data, but a number puzzle. The term mean is used here because it is customary to use it. We think average is better.
SLIDE 52
The mean of three numbers is 5. When I add one more number the mean is 6. What number have I added?
Solution
With three numbers the average is 5, so the total of the three numbers is .
With four numbers the average is 6, so the total of the four numbers is .
Therefore, the number I have added is .
SLIDE 53
Five consecutive numbers have a mean of 21. What are the numbers?
SLIDE 54
Five consecutive numbers have a mean of 21. What are the numbers?
Solution
Consecutive numbers are numbers that follow one after the other, like 1, 2, 3, 4, 5, …
Since 21 is the average (or “mean”), the middle number is 21. (The total is .) Since the numbers are consecutive, they are
19 20 21 22 23
SLIDE 55
I am thinking of four numbers. Two of the numbers are 18 and 29. The median of the four numbers is 25. The average of the four numbers is 26. Find the other two numbers.
SLIDE 56
I am thinking of four numbers. Two of the numbers are 18 and 29. The median of the four numbers is 25. The average of the four numbers is 26. Find the other two numbers.
Solution
The average of the four numbers is 26, so the total is .
One of the numbers is 18 and . So, the sum of the other two numbers is .
If 29 were the largest of the four numbers, then the total of the two middle numbers would be 57 and the median would be . But the median is 25, so 29 is not the largest of the four numbers.
Then since the median is 25, the second number must be 21 because the average of 21 and 29 is 25. The sum of the first three numbers is , so the final number is . The four numbers are
18 21 29 36
Tutor note
This argument uses proof by contradiction to show that 29 cannot be the largest of the four numbers.
SLIDE 57
A scatter diagram shows a relationship in information.
The following table provides information about the marks of ten students in two tests
Student A B C D E F G H I J
Test 1 60 36 20 72 86 90 34 80 44 66
Test 2 44 36 18 64 66 78 23 58 46 58
Plot these values onto graph paper
SLIDE 58
The line of best fit is a straight-line that goes “closest to the mostest”. At this stage you draw it by eye.
Draw a line of best fit onto the graph. Do not extend the line beyond the first or last data-point.
SLIDE 59
Another student (not one of the ten) scored 50 marks in the test 1. Estimate the score of this student in test 2?
SLIDE 60
A score of 50 in test 1 correlates to a score of 41 in test 2. The estimate of the student’s score in test 2 is 41.
Correlation is positive if the line of best fit is upward sloping. Correlation is negative if the line of best fit is downward sloping.
? What kind of correlation exists between the score in test 1 and the score in test 2?
There is said to be no correlation if the data-points are so scattered that no sensible line of best fit can be drawn.
? Sketch a graph of ten data-points where there is no correlation.
SLIDE 61
? ?
The line is upward sloping, so the correlation is positive An example of a collection of ten data-points that are so scattered that we cannot say there is any correlation.
SLIDE 62
The table shows the altitude and temperature at ten locations in Nepal on a day in summer.
Note: altitude is the height of a location above sea-level.
Location A B C D E F G H I J
Altitude
/ m 40 120 540 920 1440 1700 1960 2400 2640 2920
Temperature
/ °C 34 33 29 27 28 22 22 16 19 17
? Plot a scatter diagram for the data
? Draw the line of best fit
? What kind of correlation does this data exhibit?
? Interpret the correlation as a relationship between quantities measured?
? Estimate the temperature at a location at altitude of 2000m
SLIDE 63
Location A B C D E F G H I J
Altitude
/ m 40 120 540 920 1440 1700 1960 2400 2640 2920
Temperature
/ °C 34 33 29 27 28 22 22 16 19 17
The data exhibits negative correlation. This is interpreted by the following relationship: as the altitude (of a location in Nepal) goes up, the temperature (at that location) goes down. We estimate the temperature at an altitude of 2000 m above sea-level to be 21°C.
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