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SLIDE 1
Revision
The composite bar chart shows information of profits from a food retailer over three years.
Find the percentage increase in profits from Internet home delivery between 2017 and 2019 and the percentage decrease in profits from Supermarkets between the same two years.
SLIDE 2
Internet home delivery profits in 2017 were £40m, and in 2019 £170m.
Supermarket profits in 2017 were , and in 2019 were .
SLIDE 1B
Revision
The composite bar chart shows information of profits from a food retailer over three years.
Find the percentage increase in profits from Internet home delivery between 2017 and 2019 and the percentage decrease in profits from Supermarkets between the same two years.
SLIDE 2B
Internet home delivery profits in 2017 were $40m, and in 2019 $170m.
Supermarket profits in 2017 were , and in 2019 were .
SLIDE 3
Revision
60 tourists were asked to name their favourite London attraction. The results of the survey were as follows.
Site Votes
Tower of London 24
Madam Tussauds 20
British Museum 10
National Gallery 6
Make a pie chart of this data.
SLIDE 4
Site Votes Angle
Tower of London 24
Madam Tussauds 20
British Museum 10
National Gallery 6
SLIDE 5
Revision
Calculate the mean, median and mode of the following 12 numbers.
31 26 19 30 22 28
22 25 31 32 35 22
SLIDE 6
31 26 19 30 22 28
22 25 31 32 35 22
The data in rank order
19 22 22 22 25 26
28 30 31 31 32 35
The mode is the most frequent number, 22
The median is the average of the two middle numbers,
Summary
Mode 22, Median 27, Mean 26.9
SLIDE 7
Using the calculator in Stat Mode
You need to know how to find the mean using a calculator.
Calculators differ in how they do this, and even the simplest modern calculator is far too complex an instrument for most uses. The more basic types of calculator have gone out of fashion, so you need to learn to do simple operations on the calculator you have.
Statistics calculations are performed in Statistics Mode, which is usually abbreviated to SD.
STEP 1
Enter stat mode
Press Mode
Select SD, here by pressing the button 2. That puts you in statistics mode.
You will see an SD in the display.
When you need the calculator for ordinary calculation, you need to exit SD mode by clearing and resetting ALL.
SLIDE 8
Using the calculator in Stat Mode
It is assumed you are in statistics mode (SD)
STEP 2
Clear the statistics memory
Press Shift CLR
Select Sc1, here by pressing the button 1.
Do not select 3, All. This will clear the SD mode, and return you to calculator mode.
Press ON, and you are ready to enter data.
STEP 9
We will enter the following data
31 26 19 30 22 28
22 25 31 32 35 22
STEP 3
Key in the value of the first data point, which is 31 and press the M+ button. This button is usually located on the right-hand side, above the AC button. Below it you may find the symbol DT, which in fact indicates that you are adding the value to the stat memory.
In the display the value indicates that you have entered the first data point. As you enter the other data this will increase.
Repeat the process of the other data points. There are twelve data points in all in our example.
SLIDE 10
Finding the total sum
You will find S-SUM and S-VAR buttons on you calculator. These are often located above the 1 and 2 buttons. Because they are above these buttons, you must press SHIFT to use them.
STEP 4
Press SHIFT 1, S-SUM
The sigma symbol indicates a sum.
31 26 19 30 22 28
22 25 31 32 35 22
Here this shows that the sum of all the values is
We can use this to calculate the mean. Since the number of data points is , the mean is
SLIDE 11
Finding the mean
We can calculate the mean as
but the calculator can also do this for us.
STEP 5
Press SHIFT 2, S-VAR
Select 1
It displays
SLIDE 12
Use the calculator to find the total sum and mean of the following time for ten competitors in the 100m
12.3 13.7 11.8 12.9 12.8 13.2 11.9 12.1 13.3 12.4
SLIDE 13
Use the calculator to find the total sum and mean of the following time for ten competitors in the 100m
12.3 13.7 11.8 12.9 12.8 13.2 11.9 12.1 13.3 12.4
Solution
SLIDE 14
The sigma notation
The Greek letter is used to indicate the process of making a sum.
in the Greek alphabet corresponds to “S”.
Example
Find
Solution
SLIDE 15
Find when
?
?
SLIDE 16
?
?
SLIDE 17
The mean or average is also called the arithmetical mean.
The arithmetical mean of two numbers, x and y, is 8. The arithmetical mean of the same two numbers with a third, z, is 10. What is the value of z?
SLIDE 18
The arithmetical mean of two numbers, x and y, is 8. The arithmetical mean of the same two numbers with a third, z, is 10. What is the value of z?
Solution
SLIDE 19
Remark
Examiners like to set number puzzles based on the mean and median, so the student must be familiar with these.
Question
are all integers. If is the median, which of the following could be a value of x?
6 8 10 12 14
SLIDE 20
are all integers. If is the median, which of the following could be a value of x?
6 8 10 12 14
Solution
We work out each expression for the given values
Of these options, only when is the middle (that is, median) value. Answer: .
SLIDE 21
The median of 47 consecutive numbers is 35. What is the least integer in the list?
SLIDE 22
The median of 47 consecutive numbers is 35. What is the least integer in the list?
Solution
The median is the middle value when the numbers are arranged in ascending order. Taking out this middle number, we have numbers. Half of these lie below the median, and half above; that is are below 35. Hence the least number is .
SLIDE 23
The arithmetic mean of six numbers is 7. Find the new arithmetic mean if 3 is subtracted from each of four of the numbers.
SLIDE 24
The total of the six numbers is . When 3 is subtracted from four of these numbers, we are subtracting , and the new total becomes . The new arithmetic mean is .
SLIDE 25
Data that is discrete can be counted by integers.
Continuous data is data that can be measured by real numbers.
Which option is correct?
A Number of people at football matches ? Continuous
Height of mountains ? Discrete
B Number of people at football matches ? Discrete
Height of mountains ? Continuous
C Number of people at football matches ? Continuous
Height of mountains ? Continuous
D Number of people at football matches ? Discrete
Height of mountains ? Discrete
SLIDE 26
Option B is correct
Number of people at football matches ? Discrete
Height of mountains ? Continuous
What is the difference between a bar chart and a histogram?
SLIDE 27
In a bar chart the columns are separated by gaps. In a histogram, there are no gaps, and measurements are put at the boundaries of the columns.
Which option is correct?
A Continuous ? Bar chart
Discrete ? Bar chart
B Continuous ? Histogram
Discrete ? Bar chart
C Continuous ? Bar chart
Discrete ? Histogram
D Continuous ? Histogram
Discrete ? Histogram
SLIDE 28
Option B is correct
Continuous ? Histogram
Discrete ? Bar chart
SLIDE 29
A sample of 150 adult male Atlantic Herring was taken, and the size of the fish measured.
Length / mm Frequency Length / mm Frequency
4
37
13
19
28
6
43
Construct a histogram representing this data.
SLIDE 30
Length / mm Frequency Length / mm Frequency
4
37
13
19
28
6
43
SLIDE 31
Frequency polygon
To make a frequency polygon we mark off the mid-point of each interval with a dot at the top of each bar in the histogram and join the dots. The start and end points cannot fall outside the range of the data.
Complete the above frequency polygon. The first three points and two lines have been done for you.
One usually does not display the original rectangles of the histogram. Remove the rectangles from your final version.
SLIDE 32
SLIDE 33
Mean of a grouped frequency table
When the data is organized into intervals it is called a grouped frequency table.
We need to find the mean of a grouped frequency table.
Interval / mm Mid-interval value, x / mm Frequency
5 4
15 13
28
43
37
19
6
Complete the above table with the other mid-interval values.
SLIDE 34
Interval / mm Mid-interval value, x / mm Frequency, f x f
5 4 20
15 13 195
25 28
35 43
45 37
55 19
65 6
We find the product of the mid-interval value and its frequency, this is denoted . For example, .
Complete the above table. The sum of all the frequencies is denoted . Here it is , as there are 150 data-points. Find also and complete the missing entry.
SLIDE 35
Interval / mm x f x f
5 4 20
15 13 195
25 28 700
35 43 1505
45 37 1665
55 19 1045
65 6 390
The mean is given by .
Find the mean of the above data.
SLIDE 36
Interval / mm x f x f
5 4 20
15 13 195
25 28 700
35 43 1505
45 37 1665
55 19 1045
65 6 390
SLIDE 37
Calculator to find the mean of a grouped frequency table
We need also to be able to find the mean using the calculator.
Interval / mm x f
5 4
15 13
25 28
35 43
45 37
55 19
65 6
Put the calculator in stat mode and clear the stat memory.
Your calculator may do things differently.
In the good old days, all you had to do was input and press M+. While your calculator might still do it that way, it is more likely to do things in a complex and counter-intuitive way. What follows is a popular method of inputting a grouped frequency table into a basic calculator.
SLIDE 38
Interval / mm x f
5 4
15 13
25 28
35 43
45 37
55 19
65 6
Input sequence
The in the display indicated that you have inputted the value 5 four times. You repeat this until you reach the last row.
When you reach the last row, you should see , because there are 150 data points in this table.
SLIDE 39
Use the same buttons as before to find n, and
SLIDE 40
Interval / mm x f
5 4
15 13
25 28
35 43
45 37
55 19
65 6
We obtain with the calculator the same results as before.
SLIDE 41
Interval / mm f
2
8
17
58
21
11
7
1
Use your calculator to find , and for the above data. Confirm that found by the calculator is .
SLIDE 42
Interval / mm x f xf
2.5 2 5
7.5 8 60
12.5 17 212.5
17.5 58 1015
22.5 21 472.5
27.5 11 302.5
32.5 7 227.5
37.5 1 37.5
SLIDE 43
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.
SLIDE 44
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
SLIDE 45
Using this data
? make a grouped frequency table
? construct a histogram and a frequency polygon
? find the mean
SLIDE 46
Interval 40-50 50-60 60-70 70-80 80-90 90-100 100-110
x 45 55 65 75 85 95 105
f 2 8 9 15 9 5 2
90 440 585 1125 765 475 210
SLIDE 47
The range is the largest value less the smallest value. The median is the middle value. In this set of data, there are 50 data points, so the median is the average of the 25th and 26th values.
About the median
For an odd number of data points there is a definite answer to the question: what is the median value? Consider the following five data points from a complete set of data: 1 1 3 5 7. There is a middle number, so the median is 3. When there is an even number of data points, there is no definite median. In this case one simply has to make a rule (a convention) as to what the median shall be. The rule in this text is that the median is the average of the two middle numbers. In this way a set of data can have a fraction as a median. Some statisticians in other textbooks allow either the lower or the upper number to be the median, and some examiners allow all three to be a valid answer.
Question
Find the range and the median of the above data.
SLIDE 48
Range
The last value is 108. The first value is 43.
The range is
Median
There are 50 data points in the set, so the median is the average of the 25th and 26th values. These are 74 and 76 respectively. The median is 75.
SLIDE 49
Quartiles
We can also divide the data into quarters. The first quartile is also called the 25th percentile and is denoted Q1.
Here there are 50 data points.
The data is discrete (individual numbers), so can be divided exactly into two blocks of 25 numbers. This makes the median into the average of the two middle numbers.
Each block of 25 numbers to the left and right of the median has a distinct middle value. For Q1 it is the 13th value, which divides the 25 numbers into blocks of 12 numbers to the left and right: . Thus, Q1 is 62.
Find the median third quartile and the inter-quartile range of the above data.
SLIDE 50
The two middle numbers are the 25th and 26th numbers. Here 74 and 76 respectively, so .
Q1 is the 13th value, and Q3 is the 38th value
SLIDE 51
The median is also denoted Q2.
Police measured the speed of 25 cars travelling in a school zone. The results in mph were as follows.
21 27 22 19 27 45 25 25 31 41
37 18 30 25 24 19 26 23 20 28
31 25 29 38 23
Make a stem and leaf diagram of the data.
Find the range, median , first and third quartiles and the inter-quartile range of the data.
SLIDE 52
21 27 22 19 27 45 25 25 31 41
37 18 30 25 24 19 26 23 20 28
31 25 29 38 23
1 8 9 9
2 0 1
3 4 5 5
6 7 7 8
3
1 7 8
4 1 5
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