Statistics - Mathematicssmcmaths.webs.com/Form-4/Statistics Notes Form 4 Part 1.pdf · Mr. Jonathan...

Mr. Glen Farrugia Stella Maris College Mr. Jonathan Camenzuli Form 4 Mathematics

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Statistics Data is continuously being collected everyday, by means of surveys,

questionnaires and through online technologies. Our main aim in Statistics is to

make sense of this raw data which is collected.

One way to do so is to find the average of the data. By doing so we are able to

compare one set of data to another set by comparing just two values – their

averages.

There are several ways of expressing an average, but the most commonly used

averages are the mean, mode, median and range.

The mean of a set of data is given when all the values in

the set are added, and then divided by the number of

values in that set. That is:

Example 1

The ages of 11 players of Juventus F.C. are:

37, 26, 32, 21, 31, 28, 24, 34, 28, 21, 31, 34

Find the mean age of the team.

Sum of all ages = 37 + 26 + 32 + 21 + 31 + 28 + 24 + 34 + 28 + 21 + 31 + 34 = 347

Total number in team = 11

Mean age =

(up to 1 decimal place)

Mean


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Example 2

14 students were asked how many hours they spend on Facebook everyday. Their responses were the following:

2.5, 5, 3, 3, 3, 1, 1.5, 1.5, 2.5, 1, 0.5, 1, 3, 1

Find the mean time (in hours) spent on social media by these students.

Example 3

The total (added) score of all students in a Maths test was 975. The average mark of the class was 75. How many students are there in this class?

Example 4

The mean weight of a group of 6 boys is 54kg. Two boys leave the group and the mean

weight of the remaining 4 boys is 58kg. What is the mean weight of the two boys that left

the group?


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The mode of a set of data is the value that occurs the

most. That is, it is the value with the highest frequency.

The mode is very easy to find, and it can also be applied

to non-numerical data. For example, you can find the

modal birthday month of the class.

Example 1

13 families were asked how many cars they own as a family. The following were their

responses:

5, 2, 2, 1, 3, 3, 1, 2, 4, 2 , 2, 1, 3

What is the modal number of cars owned by these families?

Example 2

A group of pensioners were asked how many times they visited Gozo last year. Their

esponses were as follows:

4, 2, 2, 5, 4, 4, 3, 2, 1, 1, 4, 5, 5, 2

What is the mode of this set of data?

Example 3

The following data represents the weather of the first 12 days of March 2015:

rain, sun, cloud, sun, rain, fog, thunder, rain, fog, sun, thunder, sun

What is the modal weather in these 12 days?

Mode


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The median is the middle number of an ordered set of

numbers.

To find the median follow these steps:

We can also use the following formula, to find the position of the median:

Example 1

Find the median of the following set of numbers:

1, 5, 9, 15, 21

Therefore, the median is 9

OR

position of median =

rd position

9 is in the 3rd position. Therefore 9 is the median.

Median

where n is the number of values in

the set of data.

1. Arrange the numbers in order from smallest to largest.

2. Find the middle number


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Special Case

Find the median of the following set of numbers:

22, 25, 29, 31, 36, 40

Therefore, in this example,

Median =

Note that in this case,

position of median =

th position

In other words the median lies between the 3rd and the 4th positions.

Example 2

A group of students were asked how many cousins they have. The following were their

responses:

4, 5, 4, 8, 1, 0, 1, 5, 9, 4, 11, 13, 2, 3, 2

Find the median number of cousins of this group of students.

If there are two median numbers, we simply

find the mean of these two numbers.


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Example 3

A group of teenagers were asked how many books they read last year. Their responses were

as follows:

11, 5, 4, 8, 2, 2, 3, 1, 5, 6, 20, 16, 9, 1

Find the median number of books read by these teenagers.

The range is not an average. It shows the spread of the

data. It can be used to compare two or more sets of

similar data.

It can also be used to comment on the consistency of the

data.

The range for a set of data is given as the highest value minus the lowest value

in the set:

Example 1

The price of petrol in the last six months was as following:

€1.38, €1.40, €1.40, €1.44, €1.38, €1.35

Find the range in petrol prices.

Range = Highest value – Lowest values = 1.44 – 1.35 = 0.09

Range

Range = Highest value – Lowest value


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Example 2

The number of teenagers attending a youth group on 10 consecutive Fridays was as follows:

20, 18, 13, 22, 26, 20, 15, 13, 22, 12

Find the range of the attendance during these 10 weeks.

Example 3

The manager of Liverpool F.C. has to choose between Moreno and Clyne on who plays first

in the next match. In the previous 11 league matches, there performance scores were as

follows:

Moreno: 14, -19, 12, -1, 36, 13, 12, 30, 14, 2, 20

Clyne: 27, 30, -12, -9, 27, 23, -10, 15, 19, 5, 11

a. Calculate the mean performance score for each player separately.

b. Find the range of each player separately.

c. If you were the manager, whom would you choose to play first in the next match?

Explain why.


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Example 4

I have 5 numbers.

Their mean is 4.

Their median is 3.

Their mode is 3.

Find 3 different sets of five positive whole numbers that satisfy these conditions.

Support Exercise Pg. 265 Exercise 17A Nos. 1-10


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When a lot of information has been gathered, it is often

convenient to put it together in a frequency table.

From this table we can then find the values of the three

averages and the range.

Example 1

The table below shows the total number of goals scored in the Champion’s

League so far.

Number of goals by team 5 6 7 8 9 10 11 12 13

Frequency 7 3 4 2 1 2 1 0 1

For the number of goals scored, calculate:

a) the mode b) the median c) the mean

a) The modal number of goals scored is the number with the largest

frequency, which is 7.

Therefore, the modal number of goals scored is 5.

b) Remember that the median is found by working out where the middle of

the set of numbers is located.

In order to find the median we need to first add a new row in the table,

where in each cell, we add the frequencies of the preceding cells. We

call this new row the cumulative frequency.


Frequency 7 3 4 2 1 2 1 0 1

Cumulative Frequency 7 7+3 =10

10+4 =14

14+2 =16

16+1 =17

17+2 =19

19+1 =20

20+0 =20

20+1 =21

Frequency

Tables


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Next, we calculate the middle position:

From the cumulative frequency row, we can now find which group

contains the 11th item. Note that up to the 10th item are in the second

group (i.e. those teams which scored a total of 6 goals), therefore, the

11th item must be in the third group. Hence, the median number of goals

scored is 7.

c) To calculate the mean number of goals scored, we add an extra row in

which we multiply the number of goals scored by the frequency. We

then add the whole row to find the total number of goals and divide by

the total frequency (the number of teams included).


Frequency 7 3 4 2 1 2 1 0 1

Total number of goals scored

5 x 7 = 35

6 x 3 = 18

7 x 4 = 28

8 x 2 = 16

9 x 1 = 9

10 x 2 = 20

11 x 1 = 11

12 x 0 = 0

13 x 1 =13

Hence, the mean number of goals scored is


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Example 2

A survey of the shoe size of all boys in one year of a school gave these results:

Shoe size 4 5 6 7 8 9 10

Number of students (Frequency)

12 30 34 35 23 80 3

From this frequency table, find the a) mode b) median c) mean.


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Example 3

Roll the dice for 20 times. Record the resulting number each time:

Draw the corresponding Frequency Table:


Raw Data


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Example 4

A school did a survey on how many times in a week students arrived late at school. These

are the findings:

Number of times late 0 1 2 3 4 5


481 34 23 15 3 4


Support Exercise Pg. 267 Exercise 17B Nos. 1-6


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Sometimes the infrmation we are given is

grouped in some way. Normally grouped

data is continuous data, which is data

that can have any value within a range of

values (eg. height, mass and time). In

these situations, the averages can only be

estimated, as we do not have all the

information.

Discrete data is data that consists of separate numbers, for example, goals

scored, marks in a test, number of people and shoe sizes.

In both cases, when using a grouped table to estimate the mean, we first find

the midpoint of each interval by adding the two end-values and then dividing

by two, as shown in the following example.

Example 1

Students were asked how much pocket money they received weekly. The data

was then grouped in the following way:

Pocket money, p (€) 0 < p ≤ 1 1 < p ≤ 2 2 < p ≤ 3 3 < p ≤ 4 4 < p ≤ 5


2 5 5 9 15

a) Write down the modal class.

b) Find the range in which the median value lies.

c) Calculate an estimate of the mean weekly poket money.

a) The modal class is that with the largest frequency, that is, the range

4 < p ≤ 5.

Frequency Tables

With Grouped Data


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b) Like for the grouped data, we need to add an extra row for the

cumulative frequency.

Pocket money, p (€) 0 < p ≤ 1 1 < p ≤ 2 2 < p ≤ 3 3 < p ≤ 4 4 < p ≤ 5


2 5 5 9 15

Cumulative Frequency 2 2+5 = 7

7+5 = 12

12+9 = 21

21+15 = 36

Next, we calculate the middle position:

This means that the median lies between the 18th and 19th positions.

From the cumulative frequency, we can now find which group contains

the 18th and 19th item. The third group contains up to the 12th item,

while the fourth group contains up to the 21st item. Therefore the 18th

and 19th positions lie in the fourth group. Hence the median lies in the

range 3 < p ≤ 4.

c) To estimate the mean, we assume that each student in each class

receives the ‘midpoint’ amount. Thus, we must add an extra row in the

table for the midpoint.

Pocket money, p (€) 0 < p ≤ 1 1 < p ≤ 2 2 < p ≤ 3 3 < p ≤ 4 4 < p ≤ 5 Total

Frequency, f 2 5 5 9 15 36

Midpoint, m

f x m 2 x 0.5

= 1 5 x 1.5 = 7.5

5 x 2.5 = 12.5

9 x 3.5 = 31.5

15 x 4.5 = 67.5

120


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We then procedd as usual, that is the estimated mean is given by:

(rounded to the nearest cent)

Example 2

The number in millions of Facebook users in the world in 2011 was as follows:

Age, a Frequency (millions)

13 ≤ a ≤ 15 56

16 ≤ a ≤ 17 66

18 ≤ a ≤ 24 248

25 ≤ a ≤ 34 210

35 ≤ a ≤ 44 109

45 ≤ a ≤ 54 61

55 ≤ a ≤ 64 29

65 ≤ a ≤ 80 19

a) How many people were using FB in 2011?

b) What is the modal age class of FB users in 2011?

c) What is the age class in which the median age of FB users falls?

d) What is the estimated mean age of FB users in 2011?

You cannot find the median or range from a grouped

table since you do not know the actual values.


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Example 3

In a health survey, people were asked how many litres of water they drink everyday. Their

responses were grouped as follows:

Litres of water (lt.) 0 - 0.5 0.5 - 1 1 – 1.5 1.5 - 2 2 – 2.5

Frequency 21 57 65 52 12

a) How many people participated in the survey?

b) What is the modal class of litres of water consumed?

c) In which range of litres does the median value lie?

d) What is the estimated mean of litres of water consumed?


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Example 4

In a survey, parish priests (kappillani) were asked how much money they spend in organising

their village feast. Their responses were as follows:

Expenditure (€ 1,000) 5 < c ≤ 10 10 < c ≤ 15 15 < c ≤ 20 20 < c ≤ 25 25 < c ≤ 30

Frequency 5 9 22 27 26

a) How many parish priests participated in the survey?

b) What is the modal class of expenditure in village feasts?

c) In which expenditure range does the mean value lie?

d) What is the estimated mean of expenditure in village feasts?

Support Exercise Pg. 275 Exercise 17D Nos. 1-5

Statistics - Mathematicssmcmaths.webs.com/Form-4/Statistics Notes Form 4 Part 1.pdf · Mr. Jonathan...

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