Data squares: Level 3


This unit provides a way of looking at multivariate data from a group of individuals. Data squares hold several pieces of information about individuals, and by sorting and organising a set of data squares, things can be found out about the group. 

Achievement Objectives
S3-1: Conduct investigations using the statistical enquiry cycle: gathering, sorting, and displaying multivariate category and whole-number data and simple time-series data to answer questions; identifying patterns and trends in context, within and
Specific Learning Outcomes
  • collect information
  • sort information into categories
  • answer questions by sorting, organizing and arranging information
  • make sensible statements about the information with supporting evidence
Description of Mathematics

This unit focuses on sorting and organising data sets, i.e. collections of information from a group of individuals. Looking at the data, sorting and organising it first, with things of interest and questions arising from this. This is a different approach than starting with a question then collecting data to see if it is correct. The data squares allow students to consider issues needing more than one category to be considered at the same time.

Understanding the difference between individual data and group data is central to the unit. The goal is to have students make statements about the group in general. Increasing student’s ability to accurately describe aspects of a data set, including developing statistical vocabulary, is part of the unit. As students become comfortable with making statements and describing data, more precise vocabulary is to be encouraged.

Required Resource Materials
  • Data Set One Master
  • scissors
  • paper and pencils
  • Data Set Two Master
  • Data Set Three Master
  • Blank Data Squares
Key Vocabulary

 multivariate data, categories, data squares, data sets, evidence, multi-dimensional, prediction, average, representative, reaction score


Session One

Part One – Making Class Data Squares
Show the following data square to the class and explain what a data square is, i.e. a square piece of paper contains four pieces of information about one person.

The information on this data square comes from four questions:

  • Are you a boy or girl?
  • Can you whistle?
  • Are you the oldest, youngest or a middle child in your family? (Only children are classified as oldest)
  • Which hand do you write with to produce your neatest work?

data square


Ask the class to tell you something about this student.

  • Does anyone in the class fit this data square?
  • Do you know someone that fits this data square not in this class?
  • How many people could this data square be correct for?

Discuss the importance of knowing exactly what each piece of data is about, i.e. the importance of specific questions. Discuss how some students could answer the same question differently, e.g. "Are you right handed or left handed?" could give two different answers for the person who throws a ball with one hand and writes with the other. Questions need to be specific, with no ambiguous answers.

What would a data square about you look like?

Hand out a blank data square for each student to fill out. Once completed collect all data squares.

After this session the teacher needs to photocopy all the data squares onto a piece of paper, one set for each pair of students in the class. Photocopying onto coloured paper is suggested to make it easy to recognise the class’s data set. This data set will be used during the next session.

Part Two – Data Set One
Organise the students into pairs, hand out to each pair a set of data squares, Data Set One Master, and get them to cut out all the data squares. Once cut out, have the student sort and organise the data squares to find out things about this group of students.

Encourage the students to look for multi dimensional interesting things. This means looking for interesting things within different categories rather than simply counting the number in categories. For example, rather than seeing if there are more girls than boys or more whistlers than non whistlers, look to see if more boys than girls are left handed or if there is a link between place in family and the ability to whistle.

Arranging the data squares like below, is one way to help see things in the data.

arranging data squares


The teacher is to move around getting the students to explain and show what they have found out. The teacher is to encourage students to add detail to their observations. This could include thinking proportionally. For example, rather than "One more girl is right handed than boys", "A larger proportion of girls are right handed, 8 out of 12 girls in comparison to 7 out of 12 boys are right handed." More able students are to be encouraged to think proportionally when the number in comparing groups is not the same, e.g. 8 out of 20 is a smaller proportion than 7 out of 9.

The following question could be asked to encourage thinking:

  • Are there proportionally more whistling right-handers or whistling left-handers?
  • Is there anything interesting when comparing place in the family and whistling?
  • All the boys in this group who are the youngest can whistle, does this mean every boy who is the youngest in their family can whistle?

On a large piece of paper write up what the students discover or get each pair of students to write down what they found out about this group. Keep this information, as it can be used in the next session to compare with the class data set.

Session Two

During this session, students will be sorting and arranging data squares about themselves, i.e. the students own data squares. Before the class data set is handed out, remind the students about what they found out about Data Set One in Session One and how they organised the data square to see things.
Briefly discuss what they expect to find out about their class.

  • What do you expect to find out about the class?
  • Will the things we found out from Data Set One, be different or similar to our class?

Hand out a set of class data squares to each pair of students. The pairs are to cut out the data squares, sorting and arranging them to look for things of interest. The teacher is to move around getting students to explain and show what they have found out.

Conclude the session by considering the statements the students made at the beginning of the session and sharing other things of interest.

Session Three

Show the following data square to the class, telling them it is information from a student in a class like ours, then ask them the following questions:

  • What do you think the letter and numbers mean?
  • Why are letters and numbers used instead of words?
  • What specific questions could give the answers: B, 6, 10 or 13?

data square


Explain that the four questions from this data square are:

  • Are you a boy or girl? - B
  • What year level are you at school? – 6
  • How many years old are you? – 10
  • What is your reaction score for catching a ruler? – 13

The reaction score is the average length a ruler falls, before being caught, when it is dropped four times. To work out the reaction score, one student holds a ruler vertically above the test student’s first finger and thumb; the bottom of the ruler is in line with the top of the thumb. The ruler is released and the test student closes their finger and thumb as quickly as they can to catch the ruler. The number of centimetres the ruler falls through the finger and thumb is the score. This is repeated four times, with the scores averaged to give the reaction score. For example, if the ruler fell 12 cm first time, 15 cm second time, 11 cm third time and 14 cm fourth time, the average is 12 + 15 + 11 + 14 = 52, 52 ÷ 4 = 13, therefore, the reaction score is 13.

reaction startreaction end



Hand out a set of data squares, Data Set Two, to each pair of students. The pairs are to cut out the data squares, sorting and arranging them to look for things of interest. The teacher is to move around getting each pair of students to explain and show what they have found out.

Data Set Three – Optional
A third data set has been included for teachers wishing to repeat the activity in this session. The data for this set was obtained from
Data Set Three is a data set of 24 students. The data is: top – male/female, left – arm span in cm, right – height in cm, bottom – age in years.

Session Four

Today the students, in pairs, will design and compile their own data square set. Each pair of students needs to design three questions to ask 24 other students in the class. The first question will be "Are you a boy or a girl?" with three new questions added.

Discuss and brain storm suitable questions

Sample questions:

  • How many centimetres tall are you?
  • How many centimetres is your right hand?

    Specific instructions will be needed with questions like this, so it is clear where to start and finish measuring.

  • How fast can you run 100m?
  • What is your favourite . . . ?

    A list of possible favourites to select from is best with questions like this.

  • What time did you go to bed last night?

    When organising the data from questions like this, categories may be needed, e.g. before 8 pm, 8 to 9 pm, 9 to 10 pm, and later than 10 pm.

Before starting to collect data each pair of students needs to write three statements about what they expect to find out about the class.

Each pair of students needs to collect information and make 24 data squares from students in the class.

blank data square


At this point, teachers may wish to discuss the likely difference in results between randomly selecting 24 students from the class and hand picking 24 friends. A quick example is a good way to illustrate this point at this level of the curriculum. The point to get across is that hand picking students to answer a question can give a misleading impression of the class, if it is assumed that it is representative of the whole class. For example, the teacher selects five rugby-loving boys in the class and asks them to name their favourite sport. All the boys are likely to say rugby, with the resulting statement make, "Everyone answered rugby, so the favourite sport in the class is rugby" or "Everyone in this class loves rugby".

Once the data squares are completed, students are to sort and arrange them to look for things of interest. The pairs of students are to prepare a brief report of the things they have found out.

Session Five

This session has the students compiling a new set of data squares based on questions they develop themselves. The questions could be comparing information from the data sets presented during this unit, looking at new information or asking a different group of student the same questions. Students can use the Blank Data Squares

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