Topic 10 Data Handling

7/27/2019 Topic 10 Data Handling

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INTRODUCTIONMost of the important decision making of modern society is based on statistics,graphs and probability. In politics, advertising and economics, samples areorganised, survey questions developed, answers sought, results tabulated andorganised and predictions displayed with averages and graphs to show

distributions, relationships and trends of the data collected before decisions aremade. What will be the next flavour of cakes manufactured? Where will the landfor the next supermarket be bought? Data handling has become an importantaspect of life for many people today.

Graphs and statistics are indispensable to comprehending the raw data on whichdecision making is based. A mass of data is incomprehensible. Averages supply aframework with which to describe what happens. Graphs supply a visual way ofpresenting the range of alternatives available and indicating where the density of

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DataHandling

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Use vocabulary related to data handling correctly as required by theYear 5 and Year 6 KBSR Mathematics Syllabus;

2. Apply the major mathematical skills and basic pedagogical contentknowledge related to data handling;

3. Use the vocabulary related to data organisation in graphs correctly;

4. Apply the major mathematical skills and basic pedagogical contentknowledge related to data organisation in graphs; and

5. Plan basic teaching and learning activities for data handling and data

organisation in graphs.


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TOPIC 10 DATA HANDLING178

interest lies. The forms of graph that are commonly used are bar graphs,histograms, picture graphs, line graphs and pie charts.

Statistics within the primary school is predominantly the study of procedures forcollecting, recording, organising and interpreting data. Data handling isintroduced in primary schools in the belief that it is crucial for children to beginstudy of the concepts and processes in statistics, graphs and probability as early aspossible. The difficulty lies in the lack of knowledge of what aspects of datahandling are suitable for primary children. Many primary school teachers havelittle preparation for teaching data handling and little experience of it being taughtto them. By reading and applying what is written in this topic, it is expected thatteachers will be able to:

(a) Show pupils that statistics and graphs are part of mathematical activities intheir daily lives;

(b) Show pupils the connections between statistics and graphs to basic numbersand space concepts; and

(c) Allow pupils to conduct simple statistical investigations and graphicalpresentations.

PEDAGOGICAL CONTENT KNOWLEDGE

Important information regarding the content and pedagogical aspects for teachingdata handling covers the following aspects:

(a) Statistical measures such as range, mode, median and mean;(b) Collecting, recording, organising and interpreting data;(c) Statistical procedures on organising data such as tables, charts and diagrams;

and

(d) Types of graphs used to visualise data.

10.1

ACTIVITY 10.1

Can you think of reasons why data handling exists in our lives? List

down the reasons before you could compare them with your partner.


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TOPIC 10 DATA HANDLING 179

Mean: 60.07 inchesMedian: 62.50 inches

Range: 42 inchesVariance: 117.681Standard deviation: 10.85 inchesMinimum: 36 inchesMaximum: 78 inchesFirst quartile: 51.63 inchesThird quartile: 67.38 inchesCount: 58 bears

Sum: 3438.1 inches

0

1

2

3 4 5 6 7 8

Frequency

Length in Inches

Black Bears

Figure 10.1:Histogram showing the statistics of Black Bears

10.1.1 Statistical Measures

Computational statistics is a large and complex branch of mathematics with

significance for the social as well as physical and biological sciences. However, in

primary schools, pupils will be exposed only to the simplest of descriptive

statistics. The statistical measures studied in Year 5 and Year 6 are range, mean,

mode and median.

(a) RangeIn a list of data, range is the difference between the greatest and the leastvalue. Consider the following results (out of 20) in a mathematics test fortwo groups of students (theBLUEand theRED):

TheBLUEscores: 6, 8, 10, 10, 5, 6, 11, 8, 11, 6, 7

TheREDscores: 7, 9, 12, 14, 7, 9, 9, 5, 16, 9, 13

ACTIVITY 10.2

Figure 10.1 above shows an example of how a histogram can be used tovisualize data on black bears. List down four other graphicalrepresentations and show how they differ from one another.


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The range for theBLUE group is 11 5 = 6, while the range for theRED

group is 16 5 = 11.

(b) MeanMean is the average of the scores. To calculate it, the scores are added andthe result is divided by the number of scores. In the example above, themean for theBLUEgroup is

6 + 8 + 10 + 10 + 5 + 6 + 11+ 8 + 11 + 6 + 7 = 88, 88 divided by 11 is 8.

While the mean for theREDgroup is

7 + 9 + 12 + 14 + 7 + 9 + 9 + 5 + 16 + 9 + 13 = 110, 110 divided by 11 is 10.

(c) Mode

Mode is the most commonly occurring score. In the example above, the

mode for theBLUEgroup is 6, while the mode for theREDgroup is 9.

(d) MedianMedian is the middle score when the scores are arranged in ascending order.

In the above example, there are 11 scores altogether, therefore the median is

the sixth score when the scores are arranged in ascending order.

BLUE: 5, 6, 6, 6, 7, 8, 8, 10, 10, 11, 11

RED: 5, 7, 7, 9, 9, 9, 9, 12, 13, 14, 16

Hence, the median for the BLUE group is 8 and the median for theRED

group is 9.

Note: If there is an even number of scores (say 10), then the median is

halfway between the half score and the next score (example: half way

between the 5th

and the 6th

score in ascending order). For example, for scores

5, 9, 3, 8, 6, 4, 6, 3

The arrangement of the scores in ascending order is

3, 3, 4, 5, 6, 6, 8, 9

And the fourth score is 5 and the fifth score is 6.

This means that the median is 5 + 6 = 11 divided by 2, and that will be 5.5.


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10.1.2 Collecting, Recording, Organising andInterpreting Data

Data handling can be a valuable aid in decision making. A commonly used formatto investigate problems (Thompson et al; 1976) is stated in the following 5 steps:

(a) Recognise and clearly formulate a problem;(b) Collect relevant data;(c) Organise the data appropriately;(d) Analyse and interpret the data; and(e) Relate the statistics obtained from the data to the original problem.The five step format in using data to make decisions can be illustrated with theexample adapted from Thompson et al (1976).

(a) A group of children wished to send a representative to a softball throwingcontest. Three children volunteered. Each volunteer was asked to make fivethrows which were measured with a trundle wheel to the nearest metre. Theresults were:

Table 10.1: Result of softball throwing contest

Volunteers Their 5 throws ( to the nearest metre)

Shahar 28, 23, 22, 24, 27

Bala 24, 23, 27, 24, 27

Tony 23, 27, 29, 18, 26

(b) To help comprehend these results, the children tallied them into a frequencytable and graphed them onto bar graphs. They then calculated the mean,median and range for each volunteer. The tables and the bar graphs areshown below:

Table 10.2:The frequency

Distance of

throw (m)

18 19 20 21 22 23 24 25 26 27 28 29

Shahar 1 1 1 1 1

Bala 1 2 2

Tony 1 1 1 1 1


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Bar Graph: Shahar

Length of throw (m)

Bar Graphs: Bala

Length of throw (m)

Bar Graphs: Tony

Length of throw (m)

18 19 20 21 22 23 24 25 26 27 28 29

Fr

equen

cy

1

2

3

0

18 19 20 21 22 23 24 25 26 27 28 29

Frequen

cy

1

2

3

0

18 19 20 21 22 23 24 25 26 27 28 29

Frequenc

y

1

2

3

0

Figure 10.2:Statistical measures

Next, the three statistical measures, mean, median an range are calculated andtabulated in the table below.


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Table 10.3: Three Statistical Measures

Mean Median Range

Shahar 24.8 24 6

Bala 25.0 24 4

Tony 24.6 26 11

Based on the frequency table, bar graphs and the statistical measuresconstructed, ask your students the following questions.

(c) Who would be the best representative? Why?Who is the most consistent? Why?

Who has the longest throw?

(d) What should be our criteria for selecting the best representative?Who has the best typical throw?How do we define typical?Is consistency important?Should we have measured more or less than five throws?Should bad throws be excluded?Is anything important lost in rounding to the nearest metre?

(e) Would it make it easier if we tallied the throws into sections, say 15-19, 20-

24, 25-29 etc.?

10.1.3 Methods of Organising Data

The appropriate methods of organising data that seem suitable for the primary

years are interpreting and constructing simple tables, charts and diagrams that are

commonly used in everyday life to display information. The basis of this

component is the organisation of raw data into collections. This means

determining the extent of the possible outcomes, forming these into categories and

ACTIVITY 10.3

Write your answers for these two questions and compare them with yourpartner next to you.

1. What are statistical measures?

2. Why is it necessary for children to know how to collect, record,organise and interpret data?


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organising the data under these categories. The techniques that may have to be

used in this process are combinatorial counting (to determine all the possible

outcomes) and tallying (to organise the data under the categories). Let us begin

this section by introducing to you the tables.(a) Tables

(i) The simple tableAn example of this simple table is the table of contents on a cereal packet.It consists of words and figures in two columns (refer to Figure 10.3).Oats Meal Cereal: Average contents per serving:

Figure 10.3:Table of contents on a cereal packet

(ii) The regular tableThe regular table is the matrix style table where there are more thantwo columns (more than column of data). The everyday example is thebus timetable. It is useful when comparing, for example, results from

one year to another or between different people. Another commonexample of this table is in advertisements where prices at differentshops are compared (refer to Table 10.4).

Table 10.4:Materials Collected by the Children in 6 Orkid

Bakar Muthu Chong Mary Rokiah

Bottle tops 5 8 7 6 2

Cotton reels 9 3 5 2 8

Egg Cartons 5 7 2 9 3Plastic spoons 3 5 8 3 7

(b) ChartsCharts are less regular in terms of rows and columns. They attempt todisplay information more visually, to relate the display to what actuallyoccurs. As such, we have the road maps and bus routes of transport and thetime lines of history.

Vitamin C 25 mg

Iron 27 mgNiacin 11 mgRiboflavin 38 mg


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(i) The strip mapThis may be the bus route of an area or the time line of a history topic.A line is drawn and on this line are marked references to major

features (refer to Figure 10.4).

Bus Route:

Ipoh Tapah Bidor Sungkai

The Rule of King Willhem:

Coronation The

Great

War

Birth of

Prince

Henry

Birth of

Prince

Derek

Death of

the Duke

Figure 10.4:Bus route of an area or time line of a history topic

(ii) The branch mapThis is a combination of strip maps, involving branching as in a tree.The most straight forward examples are the road maps or genealogydiagrams (family tree of parents, grandparents etc.). The skill offollowing directions from a map is an important life skill that ourchildren must master. An example of a family tree is shown below.

Kamal Baharuddin and Fauziah Hamid

Kassim Fauziah m Ahmad Karim m Rokiah Kamsiah

KamarulSiti Yusuf

Figure 10.5:Kamal Baharuddins family tree


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(a) DiagramsThese are visual ways to represent membership in different sets and subsets.A Venn diagram and a Carroll diagram could be considered the most

favourable diagrams used to show the relationship between the members ofa given group of objects.

(i) Venn Diagram: An example of a Venn diagram for flowers in termsof red and scented.

Figure 10.6:Venn Diagram

(ii) Carroll Diagram: An example of a Carroll diagram for flowers interms of red and scented.

Red Not Red

Scented Red and scented

flowers

Not red and scented

flowers

Not Scented Red and not scented

flowers

Not red and not scented

flowers

Figure 10.7:Carroll diagram

10.1.4 Types of Graphs

The importance of graphs in primary schools arises from two simple ideas

(a) A picture is worth a thousand words; and(b) Mathematics is a study of relationships.

Neither red nor scented

RedFlowers Scentedflowers

Red and scented


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Graphs are not in the syllabus to give light relief to the numerical activities. Theirpurpose is to improve communication and understanding, especially for childrenof lower ability. However, we can all gain insight to complicated statistical

information if it is displayed in a graphical manner. Obviously, knowing how todraw graphs and to draw inferences from them are valuable skills to acquire.

Bar graphs, picture graphs, line graphs, circle graphs and scatter graphs, can all beused to visualise data. These various forms of graphs are commonly seen in reallife in magazines, newspapers, textbooks and advertisements. The objective inusing a graph is to visually present information in a form which enables it to beassimilated at a glance as compared to a list of numbers.

Graphs are yet further examples of representing information in such a way thatpatterns are evident or worthwhile seeking. If particular patterns emerge, time and

time again we can conclude that, indeed, some generalisation can be made aboutthe circumstances we are representing. Hypothesis can be formulated and testedand a visual display made of the results. Concepts are more clearly understood asa consequence and fundamental principals are consolidated.

(i) Bar graphsBar graphs facilitate comparisons of quantities. Bar graphs can be vertical aswell as horizontal (columns as well as rows). They can also be in the form ofblocks, or bar lines. The following are examples of bar graphs (Figure 10.8):

Cats

Do s

Fish

Birds

0 5 10 15 20 25 (a) Horizontal Bar Graph: Types of pets children have


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Bus Car Bicycle Motorcycle

5

10

15

20

25

(b) Vertical Bar Graph: Types of vehicles children use to go to schoolFigure 10.8 (a) & (b):Bar graphs

(ii) Picture GraphsPicture graphs can also facilitate comparisons of quantities just like bargraphs. They can easily be updated. Picture graphs are also calledpictographs and isotypes. An example of a picture graph is shown below.

KEY: represents RM 100

Class A

Class B

Class C

Class D

Figure 10.9: Picture Graph Money accumulated for classroom projects


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(iii) Line GraphsLine graphs can be used for comparisons and for expressing allocations ofresources, but they seem particularly useful for communicating trends. Here

is an example of a line graph.

10oC

20oC

30oC

40oC

Mon Tue Wed Thurs Fri

Figure 10.10:Line graph maximum temperatures during the week

(iv) Circle GraphsCircle graphs (also known as pie charts) are used to picture the totality of aquantity and to indicate how portions of the totality are allocated. Here is acircle graph indicating how one college student spent his budget.

Room and Board

College Costs

Miscellaneous

Clothing

Entertainment

Figure 10.11:Circle graph: Kamaruddins budget

(v) Scatter graphsScatter graphs are similar to line graphs which show the relationship betweentwo different sets of data. The scatter graph is made for data which is not insequence (in terms of the horizontal axis) and is unsuitable for a line graph.Here is a scatter graph which shows that mass is related to height.


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50 kg 100 kg 150 kg 200 kg

50 cm

100 cm

150 cm

200 cm

Figure 10.12:Scatter graph weight and height of students

MAJOR MATHEMATICAL SKILLS FOR

DATA HANDLING IN YEAR 5 ANDYEAR 6

Our students will learn the topic of data handling effectively if we plan thelessons systematically. A well organised conceptual development of statisticalmeasures, collecting, recording, organising and interpreting of data will be veryhelpful for our students to understand these concepts better. It is recommended toinstruct this topic within a problem solving environment and in a less stressfulmanner. Remember to provide opportunities for our students to differentiate thedifferent types of graphs and when they are best used.

10.2

1. Describe briefly the three methods of organising data.

2. Explain the five types of graphs with the help of visualrepresentations.

SELF-CHECK 10.1


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The major mathematical skills to be mastered by pupils studying the topic of datahandling in Year 5 and Year 6 are as follows:

(a) Average(i) Describe the meaning of average;

(ii) State the average of two, three, four or five quantities;

(iii) Calculate the average using a formula; and

(iv) Solve problems in real life situations.

(b) Data Collection

(i) Collect data;

(ii) Process data; and

(iii) Analyse data.

(c) Pictograph

(i) Identify pictograph which represents one or more than one unit;

(ii) Extract information from a pictograph; and

(iii) Construct a pictograph.

(d) Bar Charts(i) Identify characteristics of a bar chart;

(ii) Extract information from a bar chart; and

(ii) Construct a bar chart.

TEACHING AND LEARNING ACTIVITIES

This section begins by describing the teaching and learning activities for you toconduct a lesson on data handling. Let us do Activity 10.4 first. Enjoy!

10.3


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10.3.1 Average

Example of an Answer Sheet:

Learning Outcomes:

To state the average of two, three, four or five quantities To calculate the average using a formulaMaterials:

Task Cards Answer SheetsProcedure:

1. Divide the class into groups of five students and give each studentan Answer Sheet.

2. Ask the students to write their name on the Answer Sheet.

3. Shuffle the Five Task Cards and place them face down in a stack atthe centre.

4. Instruct each player to begin by drawing a card from the stack.

5. Instruct the player to write all the answers to the questions in thecard drawn on the Answer Sheet.

6. After a period of time (to be determined by the teacher), the pupilsin the group exchange cards with the pupil on their left in aclockwise direction.

7. Pupils repeat steps (5 and 6) until everyone has answered thequestions in all the cards.

8. The pupil with the most number of correct answers, wins.

9. Teacher summarises the lesson on the meaning of average.

ACTIVITY 10.4


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Example of a Task Card:

Task Card A1. Calculate the average of 264 and 246.

Average = _______________

2. Calculate the average of RM273, RM264 and RM 252.

Average = RM ___________

3. Find the average of 4.2 km, 5.1 km, 4900 m and 5 km.

Average = ___________ km

Name :___________________ Class :__________

Card A Card B Card C1._____ 1._____ 1._____

2._____ 2._____ 2._____

3._____ 3._____ 3._____

Card D Card E

1._____ 1._____

2._____ 2._____

3._____ 3._____

ACTIVITY 10.5

Work with your friend in class to prepare four other Task Cards.

There should be three questions in each card.

Make sure your cards are based on the learning outcomes oActivity 10.4.


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10.3.2 Organising and Interpreting Data

Learning Outcomes:

To recognise frequency, mode, range, average, minimum andmaximum value from a bar graph; and

To find the frequency, mode, range, average, minimum andmaximum value from a given bar graph.

Materials:

30 different Flash Cards; and

Clean writing papers.Procedure:

1. Divide the class into groups of three students and give each group aclean writing paper.

2. Ask the students to write their names on the clean paper given.

3. Shuffle the Flash Cards and place them face down in a stack at thecentre.

4. Instruct Player A to begin by drawing a card from the stack. Heshows the card to Player B.

5. Instruct Player B to read the answers within the stipulated time(decided by the teacher).

6. Instruct Player C to write the points below Player Bs name. Eachcorrect answer is awarded one point (a maximum of 6 points foreach Flash Card).

7. Players repeat steps (4 and 5) until all 10 cards have been drawn by

Player A.8. Repeat steps (3 through 6) until all the players have the opportunity

to read all 10 Flash Cards shown to them.

9. The winner is the group of students that has the most number opoints.

10. Teacher summarises the lesson on how to find the frequency,mode, range, average, minimum and maximum value from a givenbar graph.

ACTIVITY 10.6


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Example of a Flash Card:

Flash Card 1

Mon

50

100

150

Tue Wed ThursDays

Mass of fish caught in kg

1. What is the most common amount of fish caught?

Answer: __________ kg

2. What is the mass of fish caught on Monday?

Answer: __________ kg

3. Find the average mass of fish caught in the four days.

Answer: __________ kg

4. What is the minimum mass of fish caught?

Answer: __________ kg

5. What is the maximum mass of fish caught?

Answer: __________ kg

6. Find the range between the maximum and the minimum mass of fish

caught.

Answer: __________ kg

ACTIVITY 10.7

Work with three friends of yours in class to prepare twenty-nineotherFlash Cards. There should be six questions in each Flash Card. Makesure your cards are based on the learning outcomes of Activity 10.6.


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10.3.3 Pie Chart

ACTIVITY 10.8

Learning Outcomes:

To recognise frequency, mode, range, average, minimum andmaximum value from a pie chart; and

To find the frequency, mode, range, average, minimum andmaximum value from a given pie chart.

Materials:

Task Sheets; Clean writing papers; and Colour pencils.Procedure:

1. Divide the class into groups of four to six students. Give eachgroup a different colour pencil and a clean writing paper.

2. The teacher sets up five stations in the classroom and places a TaskSheet at each station.

3. The teacher instructs students to solve the questions in the TaskSheet at each station.

4. Each group will spend 10 minutes at each station.

5. At the end of 10 minutes, the groups will have to move on to thenext station in the clockwise direction.

6. At the end of 50 minutes, the teacher collects the answer papers.

7. The group with the highest score (highest number of correctanswers) is the winner.

8. Teacher summarises the lesson on how to find the frequency,mode, range, average, minimum and maximum value from a givenpie chart.


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Example of a Task Sheet:

STATION 1

The pie chart below shows the colours of 1,000 marbles owned by Gopal.

Blue

White

19%

Black

5%

Red

25%

Green

19%

1. What is the percentage of blue marbles?

Answer:___________

2. What is the most common colour of the marbles?

Answer:___________

3. Calculate the range.Answer:___________

4. Find the average percentage of the different colours of marbles owned by

Gopal.

Answer:___________

ACTIVITY 10.9

Work with two of your friends to prepare four other Task Sheets for theother stations. There should be fourquestions in each sheet.

Make sure your sheets are based on the learning outcomes of Activity10.8.


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10.3.4 Problem Solving

ACTIVITY 1

ACTIVITY 10.10

ACTIVITY 10.10

Learning Outcomes:

To solve problems involving average; and To solve problems involving graphs.Materials:

Activity Cards;

Clean writing papers; and Colour pencils.Procedure:

1. Divide the class into groups of four pupils and give each group adifferent colour pencil and a clean writing paper.

2. Shuffle a set of 12 Activity Cards and place them face down in astack at the centre.

3. Teacher signals to the students to begin solving the questions in the

first Activity Card drawn.

4. Once they have completed the first Card, they may continue withthe next Activity Card.

5. At the end of 10 minutes, the groups will stop and hand theiranswer papers to the teacher.

6. The group with the highest score is the winner.

7. Teacher summarises the lesson on how to solve problems in realcontexts involving averages and graphs.


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Example of an Activity Card:

Activity Card 1

1. The total score of Ali, Babu and Chin in a mathematics test is 260. The

average score of Ali and Chin is 85. Find Babus score.

2. The average mass of four pupils is 22.9kg. Ali joins the group and theaverage mass of the pupils is now 23.6 kg. What is Alis mass in kg?

Questions 3 and 4 are based on the bar graph below.

Amount of money saved by four students

Suzy

50

100

150

Samy Sarah Samsul Girl

Money (RM)

3. What is the percentage of money saved by Sarah?

4. What is the difference between the amount of money saved by Samyand Samsul?

ACTIVITY 10.11

Prepare 11 other Activity Cards for the group. There should be fourquestions in each card.

Make sure your cards are based on the learning outcomes of Activity10.10.


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Most of the important decision-making carried out in modern society is basedon statistics, graphsand probabilities.

Graphs and statistics are indispensable to comprehending the raw data onwhich decision-making is based.

Statistics within the primary school is predominantly the study of proceduresfor the collection, recording, organisation and interpretation of data.

Many primary school teachers have little preparation for teaching datahandling and little experience of it being taught to them.

In a list of data, range is the difference between the greatest and the leastvalue. Mean is the average of the scores. Mode is the most commonlyoccurring score. Median is the middle score when the scores have beenarranged in an ascending order.

A commonly used format to investigate problems in data handling are thefollowing 5 steps:

Recognise and clearly formulate a problem;

Collect relevant data;

Organise the data appropriately;

Analyse and interpret the data; and

Relate the statistics obtained from the data to the original problem.

The appropriate methods of organising data that seem suitable for the primaryyears are interpreting and constructing simple tables, charts and diagramsthat are commonly used in everyday life to display information.

Bar graphs, picture graphs, line graphs, circle graphsand scatter graphs,can all be used to picture data. These various forms of graphs are commonlyseen in the real world in magazines, newspapers, textbooks andadvertisements.


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Chart

Diagram

Graph

Mean

Median

Mode

Probability

Range

Raw data

Statistics

Table

Anne Toh. (2007). Resos pembelajaran masteri: Mathematics year 5. PetalingJaya: Pearson Malaysia.

Bahagian Pendidikan Guru. (1998). Konsep dan aktiviti pengajaran danpembelajaran matematik: Ukuran. Kuala Lumpur: Dewan Bahasa danPustaka.

Burrows, D., & Cooper, T. (1987). Statistics, graphs and probability in the primaryschool (trial materials). Queensland, Australia: Carseldine Campus.

Nur Alia Abd. Rahman & Nandhini. (2008).Siri intensif: Mathematics KBSR year5. Kuala Lumpur: Penerbitan Fargoes.

Nur Alia Abd Rahman & Nandhini. (2008).Siri intensif : Mathematics KBSR year6. Kuala Lumpur: Penerbitan Fargoes.

Ng, S.F. (2002).Mathematics in action workbook 2B (Part 1). Singapore: Pearson

Education Asia.

Clarke, P. et al. (2002). Maths spotlight activity sheets 1. Oxford: HeinemannEducational Publishers.

Topic 10 Data Handling

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