© boardworks ltd 2004 1 of 43 statistics 3 eso bil mathematics

© Boardworks Ltd 2004 1 of 43

Statistics

3 ESO BIL Mathematics


D1

D1

D1

D1

D1.1 Basic concepts

Contents

D1 Statistics

D1.2 Collecting data

D1.3 Organizing data

D1.4 Writing a statistical report


What is the Statistics?

The science related to the collection, organization, interpretation and analysis of data is called Statistics.

The science related to the collection, organization, interpretation and analysis of data is called Statistics.

A statistic gives us information about a characteristic studied in a set of individuals called population.

In order to study the characteristic you can choose a subset of members of the population called sample.

Each element of the elements of the population of the sample is called individual.

The characteristics we want to study is called statistics variable.


Choosing the sample

When collecting data it is usually impractical to include every member of the group of population that is being investigated.

How big should a sample be?

The sample should be as large as possible.

If the sample size is too small, then the results will be unrepresentative.

A sample is therefore choose to represent the group that is being investigated.


Choosing the sample

Suppose, for example, that you wish to investigate the favourite sports of 11 to 15 year-olds.

Would it be reasonable to question a sample of people outside a football ground following a game?

Can you suggest a better sample?

You would have to make sure that you ask equal numbers of girls and boys and that the sample is spread out across all age groups in the range.


Different kinds of data

Qualitative data is data that is non-numerical. Qualitative data is data that is non-numerical.

For example,

Sometimes qualitative data can contain numbers.

For example,

favourite football team,

eye colour,

birth place.

favourite number,

last digit in your telephone number,

most used bus route.


Quantitative data

Discrete data can only take certain values. Discrete data can only take certain values.

Continuous data comes from measuring and can take any value within a given range.

Continuous data comes from measuring and can take any value within a given range.

Quantitative data is numerical. It can be discrete or continuous.

For example,

For example,

shoe sizes,

the number of children in a class, the number of sweets in a packet.

the weight of a banana, the time it takes for pupils to get to school, the height of 13 year-olds.


Discrete or continuous data


Summary of basic concepts

Ex.: Feeding habits of infant population (3 to 6 years old)

Population: set where you realize the study.

Childs from 3 to 6 years old

Sample: Subset of the population studied.

The set of children chosen to do the study.

Individual: Every element of the population or the sample.

Every single boy and girl.

Statistics variable: characteristic studied on the population.

The aspect stydied: consumer of vegetables, fruit, meet, fish, etc.

Types of variables:

Qualitative (qualities) Ex: What kind of fruit do you eat?

Quantitative (numerical): discrete o continuous

Ex: How many pieces a week? orWhich quantity in weight per week?


D1

D1

D1

D1

D1.1 Concepts


Contents


D1 Statistics



Deciding on the data

Data can be collected from a primary source or a secondary source.

Data can be collected from a primary source or a secondary source.

Data from a primary source is data that you have collected yourself, for example:

Data from a secondary source is data that you have collected from somewhere else including the Internet, reference books or newspapers.

From a survey or questionnaire of a group of people.

From an experiment involving observation, counting or measuring. In this case you use an observation sheet.


Sources of data


Designing a questionnaire

It is important to design a questionnaire so that:

People will co-operate and answer the questions honestly.

The answers to the questions can be analysed and presented.

The questions are not embarrassing or personal.

The questions, if possible, have a specific answer.



Make sure that questions are not embarrassing or personal.

For example, you need to think carefully about questions asking about age or income.

Do not ask : How old are you?

A better question is : Tick one box for your age group.

15-20 21-25 26-30 31 +


Suggest a better question

How much do you weigh?

This is too personal, also some people don’t know their weight.

Underweight Average weight Overweight

Would you consider yourself to be:

A better question would be:



For example :

People could answer :

Yes

No

Not much

Only the best bits

Once a day

Sometimes

If possible, write questions so that they have a specific answer.

Did you see the Olympics on TV ?




How much of the Olympics coverage did you watch?Tick one box only.

None

Less than 1 hour a day

Between 1 to 2 hours a day

More than 2 hours a day

Every eventuality has been accounted for and the person answering the question cannot give another choice.


How would you rate the leisure facilities available in your local area? Tick one box only.


A scale can be used when asking for an opinion.

For example,

Excellent UnsatisfactoryPoorSatisfactoryGood


Suggest a better question

The intervals given overlap. Also, if a person has read more than 6 books there is nowhere to tick.


How many books did you read last month?Tick one box.

0-2 3-5 6-8 8+

How many books did you read last month?

0-2 2-4 4-6


Designing an observation sheet

An observation sheet can be used to record data that comes from counting, observing or measuring.

It can also be used to record responses to specific questions.

For example, to investigate a claim that the amount of TV watched has an impact on weight we can use the following:

age gender height (cm) weight (kg) hours of TV watched per week



For example, in our integrated unit we take the following data to test our physical condition:



For example, in our integrated unit we take the following data to test our physical condition:

BOUCHARD INDEX

Years weight height B.I.= W/H

1º Evaluation

2º Evaluation

RUFFIER DICKSON INDEX

P1 P2 P3 R.I.

1º Evaluation

2º Evaluation

EUROFIT TEST

COURSE NAVETTE LONG JUMP ABDOMINALS SPEED FLEXIBILITY

1º Evaluation

2º Evaluation


D1

D1

D1

D1


Contents


D1 Statistics

D1.1 Concepts



Using a tally chart

When collecting data that involves counting something we often use a tally chart.

For example, this tally chart can be used to record people’s favourite snacks.

favourite snack tally frequency

crisps

fruit

nuts

sweets

The tally marks are recorded, as responses are collected,and the frequencies are then filled in.

13

6

3

8


Using a tally chart


Grouping discrete data

We take the temperatures during one month:

With these data we can make the frequency table:

24 33 30 32 27 34

24 28 33 34 35 33

27 32 35 33 34 34

25 25 33 35 33 34

27 27 35 34 33 35

Number of data 24

Temperatures tally frequency

24

25

27

33

2

2

4

7

34 6

35 4



We take the temperatures during one month:

Can you find the mistake?

24 33 30 32 27 34

24 28 33 34 35 33

27 32 35 33 34 34

25 25 33 35 33 34

27 27 35 34 33 35

Number of data 24


24

25

27

33

2

2

4

7

34 6

35 4



To avoid these kind of mistakes we add another row to calculate the total frequency which must be the same as the number of data.


24

25

27

33

2

2

4

7

34 535 4

Total frequency 24


Using a frequency table

Once data has been collected it is often organized into a frequency table.

For example, this frequency table shows the favourite take-away meals of a group of pupils:

Favourite take-away

Pizza

Fish and chips

Burgers

Indian

Frequency

11

7

8

5

Chinese 8



A group of 20 people were asked how much change they were carrying in their wallets. These were their responses:

34p£1.7283p £6.36

£4.07£2.97£3.53 6p

£9.5434p£1.68 50p

82p£7.54£1.09 £2.81

£2.4346p£1.70 £1.29

Each amount of money is different and the values cover a large range.

This type of data is usually grouped into equal class intervals.


Choosing appropriate class intervals

When choosing class intervals it is important that they include every value without overlapping and are of equal size.

For the following data:

34p£1.7283p £6.36

£4.07£2.97£3.53 6p

£9.5434p£1.68 50p

82p£7.54£1.09 £2.81

£2.4346p£1.70 £1.29

We can use class sizes of £1:

£0.01 - £1.00, £1.01 - £2.00, £2.01 - £3.00, £3.01 - £4.00,

£4.01 - £5.00, Over £5. This is an open class interval.


Over 5.00

4.01 - 5.00

3.01 - 4.00

2.01 - 3.00

1.01 - 2.00

0.01 - 1.00

FrequencyAmount of money (£)

3

1

1

3

5

7


34p£1.7283p £6.36

£4.07£2.97£3.53 6p

£9.5434p£1.68 50p

82p£7.54£1.09 £2.81

£2.4346p£1.70 £1.29

Complete the following frequency table for this data:



The size of the class intervals depends on the range of the data and the number of intervals required.

Explain why class sizes of £5 would be inappropriate.

Could we use a class size of 20p?

For the following data:

34p£1.7283p £6.36

£4.07£2.97£3.53 6p

£9.5434p£1.68 50p

82p£7.54£1.09 £2.81

£2.4346p£1.70 £1.29


Grouping continuous data

Continuous data is usually grouped into equal class intervals.

What is wrong with the class intervals in this grouped frequency table showing lengths?

30 ≤ length

20 ≤ length ≤ 30

10 ≤ length ≤ 20

0 ≤ length ≤ 10

FrequencyLength (cm)

This is an open class interval.

30 ≤ length

20 ≤ length < 30

10 ≤ length < 20

0 ≤ length < 10

FrequencyLength (cm)

The class intervals are written using the symbols ≤ and <.


Grouping continuous data

Continuous data is usually grouped into equal class intervals.

What is wrong with the class intervals in this grouped frequency table showing weights?

Weight (g) Frequency

0 < weight < 10

10 < weight < 20

20 < weight < 30

30 < weight

Weight (g) Frequency

0 ≤ weight < 10

10 ≤ weight < 20

20 ≤ weight < 30

30 ≤ weight


Using two-way tables

A two-way table can be used to organize two sets of data.

For example, pupils from Years 7, 8 and 9 were asked what they usually did during their lunch break. This two-way table shows the results:

Year 7

Year 8

Year 9

Eat school dinners

35

29

38

Eat a packed lunch

42

34

32

Eat at home

19

22

18


Integrated unit

In your integrated unit you need first to collect the data of the observations sheet and them organizing them using a frequency table.

RUFFIER DICKSON INDEX

Bouchard index Frequency

TOTAL:


D1

D1

D1

D1

D1.4 Processing data

Contents



D1 Statistics

D1.1 Concepts


D2

D2

D2

D2

D2

Finding the mode

Contents

Processing data

Calculating the mean

Calculating statistics

Finding the median

Finding the range


Finding the mode

A dice was thrown ten times. These are the results:

What was the modal score?

3 is the modal score because it appears most often.


Finding the mode

The mode or modal value in a set of data is the data value that appears the most often.

For example, the number of goals scored by the local football team in the last ten games is:

The modal score is 2, and 1.

Is it possible to have more than one modal value?

Is it possible to have no modal value?

Yes

Yes

2, 1, 2, 0, 0, 2, 3, 1, 2, 1.2, 1, 2, 1, 0, 2, 3, 1, 2, 1.


Finding the mode from a frequency table

The mode is the only average that can be used for categorical or non-numerical data.

For example, 30 pupils are asked how they usually travel to school. The results are shown in a frequency table.

What is the modal method of travel?

Method of travel Frequency

Bicycle 6

On foot 8

Car 2

Bus 6

Train 3

8Most children travel on foot.

Travelling on foot is therefore the modal method of travel.


Finding the mode from a frequency table

This frequency table shows the frequency of different length words in a given paragraph of text.

What was the modal word length?

For this data there are two modal word lengths: 2 and 4.

We need to look for the word lengths that occur most frequently.

Word length

Frequency

1

3

2

16

3

12

4

16

5

7

6

3

7

11

8

6

9

2

10

116 16


Finding the mode from a bar chart

This bar chart shows the scores in a science test:

What was the modal score?

6 is the modal score because it has the highest bar.

0

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9 10

Nu

mb

er o

f p

up

ils

Marks out of ten


78

2618

55

23

chocolate

fruit

vegetables

sweets

other

Finding the mode from a pie chart

This pie chart shows the favourite food of a sample of people:

What was the modal food

type?

The biggest sector of the pie chart is for chocolate, so this is the modal food type.


Finding the modal class for continuous data

This grouped frequency table shows the times 50 girls and 50 boys took to complete one lap around a race track.

Frequency

Time (minutes:seconds) Boys Girls

2:00 ≤ t < 2:15 3 1

2:15 ≤ t < 2:30 7 6

2:30 ≤ t < 2:45 11 10

2:45 ≤ t < 3:00 13 9

3:15 ≤ t < 3:30 8 12

3:30 ≤ t < 3:45 7 10

3:45 ≤ t < 4:00 1 2

What is the modal class for the girls?

What is the modal class for the boys?

What is the modal class for the pupils regardless of whether they are a boy or a girl?



The mean is the most commonly used average.

To calculate the mean of a set of values we add together the values and divide by the total number of values.

Mean =Sum of values

Number of values

For example, the mean of 3, 6, 7, 9 and 9 is

3 + 6 + 7 + 9 + 9

5=

34

5= 6.8


Calculating the mean from a frequency table

The following frequency table shows the scores obtained when a dice is thrown 50 times.

What is the mean score?

Score

Frequency

1 2 3 4 5 6

8 11 6 9 9 7

Score × Frequency

Total

8 22 18 36 45 42

50

171

The mean score =17150

= 3.42


Calculating the mean from a frequency table

To calculate the mean of a set of values we add together the values and divide by the total number of values.

Mean =Sum of Score x Frequency

Sum of frequencies

In the previous example, the mean is

8 + 22 + 18 + 36 + 45 + 42

8 +11 + 6 + 9 + 9 + 7=

171

50= 3.42


Problems involving the mean

A pupil scores 78%, 75% and 82% in three tests. What must she score in the fourth test to get an overall mean of 80%?

To get a mean of 80% the four marks must add up to

4 × 80% = 320%

The three marks that the pupils has so far add up to

78% + 75% + 82% = 235%

The mark needed in the fourth test is

320% – 235% = 85%


Finding the median

The median is the middle value of a set of numbers arranged in order. For example,

Find the median of

10, 7, 9, 12, 7, 8, 6,

Write the values in order:

6, 7, 7, 8, 9, 10, 12.

The median is the middle value.


Finding the median

When there is an even number of values, there will be two values in the middle.

In this case, we have to find the mean of the two middle values.

For example,

Find the median of 56, 42, 47, 51, 65 and 43.

The values in order are:

There are two middle values, 47 and 51.

42, 43, 47, 51, 56, 65.


Finding the median

To find the number that is half-way between 47 and 51 we can add the two numbers together and divide by 2.

47 + 51

2=

98

2= 49

Alternatively, find the difference between 47 and 51 and add half this difference to the lower number.

51 – 47 = 4

½ of 4 = 2

2 + 47 = 49

The median of 42, 43, 47, 51, 56 and 65 is 49.


Rogue values

The median is often used when there is a rogue value – that is, a value that is much smaller or larger than the rest.

The mean of the data set is 168. This is not representative of the set because it is lower than almost all the data values.

What is the rogue value in the following data set:192, 183, 201, 177, 193, 197, 4, 186, 179?

The median of this data set is:

4, 177, 179, 183, 186, 192, 193, 197, 201.

The median of the data set is not affected by the rogue value, 4.


Mean or median?

Would it be better to use the median or the mean to represent the following data sets?

median

mean

mean

median

mean

median

34.2, 36.8, 29.7, 356, 42.5, 37.1?

0.4, 0.5, 0.3, 0.8, 0.7, 1.0?

892, 954, 1026, 908, 871, 930?

3.12, 3.15, 3.23, 9.34, 3.16, 3.20?

97.85, 95.43, 102.45, 98.02, 97.92, 99.38?

87634, 9321, 78265, 83493, 91574, 90046?


What does it mean if the range is large?

What does it mean if the range is small?

Finding the range

The range of a set of data is a measure of how the data is spread across the distribution.

To find the range we subtract the lowest value in the set from the highest value.

Range = highest value – lowest value

When the range is large it tells us that the values vary widely in size.

When the range is small it tells us that the values are similar in size.


Find the mean, the median and the range


Find the median


D2D2.5 Calculating statistics

D2

D2

D2

D2

Contents

D2.2 Calculating the mean

D2.1 Finding the mode

D2 Processing data

D2.4 Finding the range

D2.3 Finding the median


Remember the three averages and range

M O D EM O D ECOOMMON

M E A NM E A NAADDD I V I D E

M E D I A NM E D I A N

MIDDDLE R A N G ER A N G E

LAARGEST

SMALLEEST


The three averages and range

There are three different types of average:

MODE

most common

MEAN

sum of valuesnumber of values

MEDIAN

middle value

The range is not an average, but tells you how the data is spread out:

RANGE

largest value – smallest value


The three averages

Each type of average has its purpose and sometimes one is preferable to an other.

The mode is easy to find and it eliminates some of the effects of extreme values. It is the only type of average that can be used for categorical (non-numerical) data.

The median is also fairly easy to find and has the advantage of being hardly affected by rogue values or skewed data.

The mean is the most difficult to calculate but takes into account all the values in the data set.


Find the mean, median and range


Find the missing value


Calculating statistics

Look at the values on these five cards:

2 4 5 8 11

Choose three cards so that:

The mean is bigger than the median.

The median is bigger than the mean.

The mean and the median are the same.


Stem-and-leaf diagrams

Sometimes data is arranged in a stem-and-leaf diagram.

For example, this stem-and-leaf diagram shows the marks scored by 21 pupils in a maths test.

0

1

2

3

4

6

4

0

0

0

7

5

1

2

0

9

5

3

2

8

5

2

6

5

6

8

stem = tensleaves = units

Find the median, mode and range for the data.

5

There are 21 data values so the median will be the 11th value, that is ___ .25

The mode is ___ .32

The range is 40 – 6, which is ___ .34

2 2 2


Stem-and-leaf diagrams

© boardworks ltd 2004 1 of 43 statistics 3 eso bil mathematics

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