Data handling
What this chapter covers
- Write statistical questions (questions whose answers vary) and choose a method to collect the data — interview, survey, observation or experiment (12.1.1)
- Classify data as categorical or numerical (discrete or continuous) and organise ungrouped data in a frequency table using tally marks (12.1.2)
- Construct a bar chart, dual bar chart, pie chart, line graph, dot plot and stem-and-leaf plot, and say why the representation suits the data (12.1.3)
- Convert one data representation to another suitable representation and justify the choice (12.1.4)
- Read and interpret bar charts, pie charts, line graphs, dot plots, stem-and-leaf plots, histograms and frequency polygons, and make inferences and predictions (12.1.5)
- Explain why data must be represented ethically, and spot misleading scales and inaccurate charts (12.1.6)
Malaysia's population in 2014 was about 30.6 million: 15.8 million males and 14.8 million females. Numbers like these are data. Data handling means collecting data, organising it, showing it in a chart, and then reading the chart to find out what the data tells us. You meet it in newspapers, on television, in sport and in almost every job.
| English | Bahasa Melayu |
|---|---|
| statistical question | soalan statistik |
| categorical data, numerical data | data kategori, data numerik |
| frequency table | jadual kekerapan |
| bar chart, pie chart, line graph | carta palang, carta pai, graf garis |
| dot plot, stem-and-leaf plot | plot titik, plot batang-dan-daun |
| histogram, frequency polygon | histogram, poligon kekerapan |
| interpreting data | mentafsir data |
12.1Data collection, organisation and representation process, and interpretation of data representation
Statistical questions and collecting data
| Question | Statistical? | Why |
|---|---|---|
| "How tall are the students in Class 1 Amanah?" | Yes | The heights vary: 150 cm, 156 cm, 164 cm, … |
| "What is the favourite food of the students in Class 1 Amanah?" | Yes | The answers vary: nasi lemak, fried noodles, laksa, … |
| "How tall is Rosmee?" | No | It has only one answer (one person, one height). |
After writing the question, choose how to collect the data (textbook p. 266).
| Method | Example |
|---|---|
| Interview | Ask students one by one how they travel to school. |
| Survey (questionnaire) | Give a form asking for each student's favourite TV programme. |
| Observation | Count the cars that pass a junction every hour. |
| Experiment | Record the temperature of hot water every five minutes as it cools. |
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b) No — there is only one answer.
c) Observation (watch and count).
Classifying data and frequency tables
Numerical data is a quantity measured with numbers. It can be discrete (counted in whole units: number of family members) or continuous (measured on a scale: mass 53 kg, 56.2 kg, 66.5 kg).
Example 1 (textbook p. 267)
- The body temperature of each student: numerical (continuous — it is measured).
- The number of trees planted in each district: numerical (discrete — it is counted).
- The causes of road accidents: categorical.
Ungrouped data is raw data that has not been organised. We organise it in a frequency table, using tally marks in groups of five.
Example 2 (textbook p. 267)
The number of children in each of 20 families: 2, 0, 1, 1, 2, 1, 3, 0, 4, 3, 2, 4, 1, 0, 2, 1, 0, 2, 2, 3.
- List the possible values 0, 1, 2, 3, 4 in the first column.
- Go through the data once, making a tally mark for each value. The fifth mark crosses the first four.
- Count the tallies to get each frequency, and check that the total is 20.
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The sizes are categorical data.
Constructing data representations
A chart makes data easier to read. The best chart depends on the type of data and the purpose. Every chart needs a title, labelled axes (or labelled sectors) and a uniform scale.
| Representation | Best for |
|---|---|
| Bar chart | Comparing categories (dual bar chart: comparing two sets of data) |
| Pie chart | Showing each category as a part of the whole |
| Line graph | Showing changes over a period of time (trends) |
| Dot plot | Showing the distribution of a small set of numerical data; spotting clusters and extreme values |
| Stem-and-leaf plot | Showing the distribution of numerical data while keeping every original value |
(a) Bar chart. Bars of equal width, evenly spaced; the height of each bar is the frequency.
Example 3 (textbook p. 269)
Leisure activities of Form 1 Bakti: Reading 8, Watching television 9, Surfing the Internet 7, Exercising 6, Listening to music 4.
- Draw the axes. Mark the vertical axis "Number of students" with a uniform scale from 0 (here 0 to 10).
- Label the horizontal axis "Activity" and draw one bar for each activity.
- Write the title.
Justification: a bar chart is suitable for comparing the number of students in the different activities.
Example 4: dual bar chart (textbook p. 270)
A dual bar chart puts two bars side by side for each student, with a key, so we can compare the two quizzes. For example, Ai Fen did better in Mathematics (45) than in Science (30).
(b) Pie chart. Sectors of a circle show each category as a part of the whole.
Example 5 (textbook pp. 270–271)
Cars sold by colour: Red 9, Yellow 12, White 10, Blue 5. Total = 9 + 12 + 10 + 5 = 36.
| Colour | Number of cars | Angle of sector |
|---|---|---|
| Red | 9 | 936 × 360° = 90° |
| Yellow | 12 | 1236 × 360° = 120° |
| White | 10 | 1036 × 360° = 100° |
| Blue | 5 | 536 × 360° = 50° |
| Total | 36 | 360° |
Draw a circle, draw a radius, measure each angle with a protractor, label the sectors and write the title.
Justification: a pie chart is suitable for comparing each colour with the total number of cars.
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Rasa Sayang 30 × 3° = 90°; Ikan Kekek 120°; Bangau Oh Bangau 60°; Geylang Si Paku Geylang 45°; Lompat Si Katak Lompat 30°; Dayung Sampan 15°.
Check: 90 + 120 + 60 + 45 + 30 + 15 = 360°.
(c) Line graph. Points joined by straight lines show how data changes over time. Time goes on the horizontal axis.
Example 6 (textbook pp. 271–272)
A patient's temperature from 1 a.m. to 10 a.m.: 37.8, 37.9, 38.2, 38.4, 38.2, 37.9, 37.9, 37.6, 37.6, 37.5 (°C).
The scale goes up in 0.2 °C steps. The zigzag on the vertical axis shows that the values from 0 to 37.4 are left out. The graph shows clearly that the temperature was highest (38.4 °C) at 4 a.m. and then fell.
Justification: a line graph is suitable for showing the changes in temperature over 10 hours.
(d) Dot plot. Each value is a dot above a number line. It shows where the data clusters, how it spreads, and any extreme value (an unusual value far from the rest).
Example 7 (textbook pp. 272–273)
The duration of dental treatment, in minutes, for 14 patients: 23, 24, 21, 24, 25, 24, 25, 24, 22, 17, 21, 23, 22, 23.
- Draw a number line covering the smallest value (17) to the largest (25).
- Put one dot above the value for each patient; stack repeated values.
- Write the title.
The times range from 17 to 25 minutes and cluster around 21 to 25 minutes. 17 minutes is an extreme value.
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Smallest 3, largest 11, so the number line runs from 3 to 11.
(e) Stem-and-leaf plot. Each value is split by place value: the leaf is the last digit and the stem is the digit or digits in front. Every original value is kept, so we can still calculate with the data.
Example 8 (textbook pp. 273–274)
Mathematics marks of 20 students: 60, 56, 69, 32, 63, 58, 71, 86, 52, 64, 50, 67, 82, 63, 75, 50, 69, 78, 77, 59.
- Write the stems (tens digits) from 3 to 8 down the left, including 4 even though no mark is in the 40s.
- Write each units digit as a leaf beside its stem, in the order the data comes.
- Rearrange the leaves in ascending order. Repeated marks repeat their leaf (50 and 50 give 0 0).
- Write the key and the title.
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3 | 0 1 1 2 3 4 6 8 9
4 | 2 4 4 7 7
5 | 1 1 3 7 8 8
6 | 0 5
Key: 2 | 2 means 22 years. (Check: 2 + 9 + 5 + 6 + 2 = 24 leaves.)
Converting one representation to another
A representation can be changed into another one that suits the purpose better. Always justify the change.
Example 9: bar chart to line graph (idea from textbook pp. 275–276)
The data is collected over six years, so a line graph is more suitable: it shows the profit trend over time (here rising from RM30 million to RM70 million, with a small drop in 2012).
| From | To | Justification |
|---|---|---|
| Bar chart of data over time | Line graph | Shows changes over a period of time |
| Stem-and-leaf plot | Dot plot | Both show the distribution of numerical data and keep the original values |
| Pie chart (percentages) | Bar chart | Shows and compares the actual number in each category |
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Check: 7 + 5 + 9 + 4 = 25.
Interpreting data representations
To interpret a chart: read the values carefully from the scale, calculate (total, difference, mean, percentage), then make an inference (a conclusion about the data) or a prediction (what will probably happen next).
Example 10: line graph (textbook pp. 277–278)
- Mass in 2010 = 1.9 thousand tonnes = 1 900 tonnes.
- 2011 and 2014 both show 2.1 thousand tonnes: the masses are the same.
- Total = 1 900 + 2 100 + 2 900 + 2 600 + 2 100 + 1 600 = 13 200 tonnes; mean = 13 2006 = 2 200 tonnes.
- Inference: the mass of waste decreases each year after 2012.
- Prediction: the trend continues downwards, so 2016 is about 1 100 tonnes (the textbook's answer).
Example 11: dot plot (based on textbook p. 278; numbers adapted)
The lifespans of 24 dry cells are tested. Cells lasting less than 10 hours are rejected.
- Maximum lifespan = 20 hours; minimum = 8 hours.
- Rejected (less than 10 hours): the dots at 8 and 9 = 1 + 2 = 3 cells. Percentage = 324 × 100% = 12.5%.
- 50% of 24 = 12 cells. Counting from the right: 3 + 2 + 4 + 3 = 12 dots from 17 to 20, so 50% of the cells last at least 17 hours.
Histograms show grouped data (data in class intervals). The bars touch; each bar's width is an interval and its height is the frequency. A histogram does not show the actual values, but it can show a large data set.
Example 12: histogram (textbook p. 279)
- Girls with heights 155–159 cm: 10.
- Girls 160 cm and above (eligible for volleyball): only the 160–164 bar, 3 girls.
- Inference: most girls are 145 cm to 159 cm tall. (Check: 6 + 13 + 18 + 10 + 3 = 50 girls.)
Example 13: frequency polygon (textbook p. 280)
- Total participants = 10 + 24 + 30 + 26 + 12 + 8 = 110 (the end points at 0 add nothing).
- 16–20 minutes: 24 participants.
- Inference: most participants took 16 to 30 minutes.
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b) Exceeding 27.5 mm: 27.7, 27.8, 27.8, 28.2, 28.3, 28.6 = 6 axles (27.5 itself does not exceed 27.5).
630 × 100% = 20%
Representing data ethically
Example 14: a misleading scale (textbook p. 282)
- Adam scored 14 goals and Ravi 10. Twice Ravi's goals = 20, so Adam did not score twice as many.
- On the first chart Adam's bar (14 − 6 = 8 units) looks twice Ravi's (10 − 6 = 4 units) because the axis starts at 6. This is misleading. The vertical scale should start at 0, as on the second chart.
Example 15: inaccurate data (textbook p. 282)
Total = 22 + 36 + 28 + 10 = 96%. The pie chart is not accurate because the percentages must add up to 100%.
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The manager might use graph P, where the line looks flatter. The data is the same, so the choice of scale can mislead the reader.
Summary
| Idea | Key fact | Example |
|---|---|---|
| Statistical question | Answered by collecting data that varies | "How tall are the students in 1 Amanah?" |
| Collecting data | Interview, survey, observation, experiment | Counting cars at a junction: observation |
| Categorical data | A characteristic, not measured | Blood group, colour of cars |
| Numerical data | Discrete (counted) or continuous (measured) | Number of siblings; mass 56.2 kg |
| Frequency table | Tally in fives, then count; check the total | Children in 20 families: 4, 5, 6, 3, 2 |
| Bar chart | Compare categories; equal widths, equal gaps | Leisure activities |
| Pie chart | Part of the whole; angle = frequency ÷ total × 360° | 9 of 36 cars → 90° |
| Line graph | Changes over time; trend and prediction | Temperature every hour |
| Dot plot | Distribution, clusters, extreme values | Dental treatment times |
| Stem-and-leaf plot | Keeps every value; leaves sorted; needs a key | 3 | 2 means 32 |
| Histogram, frequency polygon | Grouped data; polygon joins midpoints of bar tops | Heights 140–144, 145–149, … |
| Ethical representation | Scale starts at 0 and is uniform; data accurate | Pie chart percentages total 100% |