Collecting and Representing Data

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

Statistics is about answering questions with data. At Stage 6 you plan your own small investigation, collect the right kind of data, record it neatly and choose a diagram that shows it clearly. The next page (Mode, Median, Mean and Range) shows how to describe and interpret the data you collect.

Key words

1. Planning an investigation

A statistical investigation follows a cycle. You start with a question, and at the end you check whether your prediction was right. Usually you investigate a set of related questions — one main question split into smaller questions that you can answer with data.

1. Ask the main question and the related questions. Example — main question: “Do older students sleep less on school nights?” Related questions: How long do Stage 3 students sleep? How long do Stage 6 students sleep? Is there a big difference between students in the same stage?
2. Predict: “I think Stage 6 students sleep about 1 hour less than Stage 3 students.”
3. Plan the data: What will you collect (hours of sleep — continuous data)? From whom (for example 20 students from each stage)? How (a questionnaire for parents, or a sleep diary for one week)?
4. Collect and record the data in a tally chart or table.
5. Represent the data in a suitable diagram.
6. Interpret: describe what the data shows, answer each question, and check your prediction.

Good planning tips: ask enough people (a few answers can be misleading); ask the same question in the same way to everyone; for measurements, decide the unit and how accurate to be (e.g. to the nearest centimetre); and make sure the people you ask fit the question (ask Stage 6 students about Stage 6, not your family).

2. Three types of data

3. Tally charts and frequency tables

A tally chart is the quickest way to record data as you collect it: draw one stroke for each answer and draw every fifth stroke across the four before it, so the marks come in groups of five that are easy to count. The frequency is the total for each row. A frequency table shows the frequencies as numbers.

Always check the total: 8 + 5 + 11 + 6 = 30, the number of students asked. For continuous data, use groups (class intervals) that do not overlap, for example 130 ≤ h < 135, 135 ≤ h < 140… A height of exactly 135 cm goes in the group 135 ≤ h < 140, because “135 ≤ h” includes 135 but “h < 135” does not.

4. Venn diagrams and Carroll diagrams (sorting)

Both diagrams sort items using two properties. In a Venn diagram, each circle is one property. Items with both properties go in the overlap; items with neither go outside the circles but inside the rectangle.

A Carroll diagram is a table. Each property has a column (or row) for “yes” and one for “not”. Every item goes in exactly one cell. Here are the same numbers:

5. Bar charts

Use a bar chart for categorical data or discrete data. Each bar shows a frequency. The bars are the same width and have gaps between them. A good bar chart has a title, labels on both axes and a scale with equal steps that starts at 0.

Reading the scale: first work out what one small step is worth. Here the labelled lines go up in 2s, so halfway between two lines is 1 more.

6. Waffle diagrams and pie charts (parts of a whole)

Use these when you want to show how a whole is split into parts — proportions. A waffle diagram is a 10 × 10 grid of 100 squares, so each square is 1%. Count the squares of one colour to get the percentage.

A pie chart is a circle split into sectors. The whole circle is all the data. Half the circle is half the data; a quarter (a right angle at the centre) is a quarter of the data. To find a number of people, find that fraction of the total.

7. Frequency diagrams for continuous data

For grouped continuous data (heights, times, masses) we draw a frequency diagram. It looks like a bar chart, but the bars touch because there are no gaps between the groups — every value from 130 cm to 155 cm belongs in some group. The horizontal axis is a number scale, like a ruler.

8. Line graphs

Use a line graph to show how something changes over time, when the measurement is continuous (temperature, height of a plant, distance travelled). Plot each reading as a cross, then join the crosses with straight lines. Time always goes on the horizontal axis.

You can read values between the points: at 13:00 the temperature was about 29 °C (halfway between 28 °C and 30 °C). This only makes sense because time and temperature are continuous. Never draw a line graph for categorical data such as favourite fruits — the line between “apple” and “banana” would mean nothing.

9. Scatter graphs

A scatter graph shows whether there is a relationship between two sets of measurements taken from the same people or things. Each cross is one student: across for height, up for hand span. The points are not joined.

• Crosses going up from left to right: as one value increases, the other tends to increase (e.g. height and hand span).
• Crosses going down from left to right: as one value increases, the other tends to decrease (e.g. age of a car and its price).
• Crosses spread everywhere: there is no relationship (e.g. height and the number of your house).

10. Dot plots

A dot plot shows each piece of discrete (or small numerical) data as one dot above a number line. It is quick to draw and you can still see every value, so it is good for small data sets.

11. Choosing the best representation

Worked examples

Worked example 1 — Types of data [2 marks] Write whether each set of data is categorical, discrete or continuous: (a) the mass of each student’s school bag; (b) the number of goals each team scored; (c) each student’s favourite subject; (d) the time each student takes to walk to school.

Answer: (a) continuous, (b) discrete, (c) categorical, (d) continuous. 2 marks for all four, 1 mark for three correct. Common mistake: calling every set of numbers “discrete”. Ask: was it counted or measured?

Worked example 2 — Completing a frequency table [2 marks] 40 students chose their favourite colour. Red 9, blue 13, green ?, yellow 6. (a) How many chose green? (b) Which colour was the mode (the most popular)?

1. Add the known frequencies: 9 + 13 + 6 = 28.
2. Subtract from the total: 40 − 28 = 12 chose green.
3. Compare: blue 13 is the largest frequency.

Answer: (a) 12 (b) blue. Check: 9 + 13 + 12 + 6 = 40 ✓

Worked example 3 — Carroll diagram [1 mark] A Carroll diagram has columns “has a right angle” / “has no right angle” and rows “quadrilateral” / “not a quadrilateral”. Write where these shapes go: square, right-angled triangle, rhombus (not a square), equilateral triangle.

Method: take each shape and answer both questions. Square: 4 sides (quadrilateral) and right angles → top-left. A rhombus that is not a square has no right angles → top-right. Each shape goes in exactly one cell.

Worked example 4 — Waffle diagram [2 marks] Use the waffle diagram of how 100 students travel to school (section 6). (a) What percentage of students walk or cycle? (b) In the whole school 600 students travel in the same proportions. How many travel by car?

1. (a) walk 15% + bike 10% = 25%.
2. (b) car = 30%. 10% of 600 = 60, so 30% of 600 = 3 × 60 = 180.

Answer: (a) 25% (b) 180 students. Common mistake: answering “30” for part (b). 30 is the percentage (30 out of 100), not the number of students out of 600.

Worked example 5 — Choosing a graph [2 marks] Amira measured the height of a bean plant every Monday for 8 weeks. Ali measured the heights of 30 bean plants on one day and grouped them. Which type of graph should each of them draw? Explain.

1. Amira: one plant, measured over time → line graph, because it shows how the height changes from week to week and she can read heights between the Mondays.
2. Ali: many measurements of continuous data, grouped → frequency diagram with touching bars, because height is continuous so the groups have no gaps.

Exam technique: in “explain” questions, name the graph and give a reason linked to the data (“because the data is continuous” / “because it shows change over time”).

Worked example 6 — Planning an investigation [2 marks] Youssef wants to find out: “Do students who read more books get higher spelling scores?” (a) Write a prediction. (b) What data should he collect, and which graph should he draw?

1. (a) Prediction: “Students who read more books each month will get higher spelling scores.”
2. (b) For each student he needs two pieces of data: the number of books read in a month (discrete) and the spelling test score (discrete). He should ask a good number of students, for example the whole class.
3. Two measurements for each student, looking for a relationship → scatter graph.

Answer: prediction as above; collect books read and test score for each student; a scatter graph. If the crosses go up from left to right, his prediction is supported.

Practice

All pages in this course

Start here

• Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

• Counting On and Back and Negative Numbers
• Sequences, Letters and Position-to-Term Rules
• Order of Operations, Brackets and Laws of Arithmetic
• Multiplying Whole Numbers (up to 10 000 × 2-digit)
• Dividing Whole Numbers (up to 1000 ÷ 2-digit)
• Factors, Multiples and Tests of Divisibility
• Square Numbers and Cube Numbers

Number: place value, decimals, fractions, percentages and ratio

• Place Value and Decimals to Thousandths
• Rounding Decimals
• Fractions: Division, Operators and Simplest Form
• Adding and Subtracting Fractions
• Multiplying and Dividing Fractions by Whole Numbers
• Fractions, Decimals and Percentages: Equivalence and Ordering
• Adding, Subtracting, Multiplying and Dividing Decimals
• Ratio and Direct Proportion

Geometry and measure

• Time: Decimals and Mixed Units
• Quadrilaterals: Properties and Classification
• Circles: Parts and Constructing Circles
• Angles: Measuring, Drawing and Angles in a Triangle
• Area of Rectangles and Right-angled Triangles
• 3D Shapes, Compound Shapes and Nets
• Capacity, Volume and Surface Area
• Coordinates in Four Quadrants
• Rotational Symmetry, Translation, Reflection and Rotation

Statistics and probability

• Collecting and Representing Data
• Mode, Median, Mean and Range, and Interpreting Data
• Probability

Practice tests

• Checkpoint 6 Maths — Practice Test 1
• Checkpoint 6 Maths — Practice Test 2
• Checkpoint 6 Maths — Practice Test 3