Reading data displays and sampling methods
Reading and interpreting numerical data displays, then choosing and evaluating sampling methods for collecting data fairly
About three lessons of 45 to 60 minutes
Before you can trust ANY statistic, you have to ask: who, and how many, were actually asked?
A data display, a table, a stem-and-leaf plot, a simple frequency count, only tells you the truth about the data it was built from. Reading one accurately is a skill in its own right: how many values in total, which value is most common, how many fall above a threshold.
But the display is only ever as trustworthy as the SAMPLE behind it. Surveying every single Year 12 student who happens to be at the canteen at lunchtime is not the same as surveying a genuine random sample of the whole school, even if both produce a neat-looking table afterward. This unit builds both skills: reading a data display accurately, and judging whether the sample behind any statistic was actually collected fairly.
- A frequency table of a class's shoe sizes or pocket moneyreading total count, the most common value, and values above a threshold
- A street survey of 'randomly selected' shopperswho is actually walking past at 2pm on a Tuesday is not a random sample of everyone
- An online poll only people who follow a certain account can seea self-selected sample of people who already agree tends to skew results
- Surveying every student on the school roll versus a sample of 50a census asks everyone; a sample asks a representative subset
What students will be able to do
Students will read and interpret a numerical data display, describing its shape, centre and spread, investigate techniques for collecting data including census and sampling and explain their practical implications, and analyse how the choice of sampling method can affect a survey's results and be used to support a particular point of view.
- I can find the total count, the most common value, and how many values are above or below a threshold from a data display.
- I can describe a data set's shape, centre and spread, including any outliers.
- I can explain the difference between a census and a sample, and describe random, convenience and systematic sampling.
- I can judge whether a described sample is likely to be representative of the whole population, and suggest a fairer method.
- I can explain how the CHOICE of sampling method, not just the sample size, can be used to make a survey support a particular point of view.
Standards this unit teaches
- AC9M7ST02Australian Curriculum v9 (ACARA)Numerical data displays
Create different types of numerical data displays including stem-and-leaf plots; describe and compare the distribution of data, commenting on the shape, centre and spread including outliers.
- AC9M8ST01Australian Curriculum v9 (ACARA)Data collection methods
Investigate techniques for data collection including census, sampling, experiment and observation, and explain the practicalities and implications of obtaining data through these techniques.
- AC9M9ST02Australian Curriculum v9 (ACARA)Sampling and representation
Analyse how different sampling methods can affect the results of surveys and how choice of representation can be used to support a particular point of view.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Data display
- an organised way of presenting a data set, such as a frequency table or a stem-and-leaf plot
- Frequency
- how many times a particular value occurs in a data set
- Outlier
- a value much higher or lower than the rest of the data set
- Census
- collecting data from EVERY member of a population, rather than a sample
- Sample
- a subset of a population selected to represent the whole population
- Representative sample
- a sample that genuinely reflects the characteristics of the whole population it was drawn from
- Convenience sample
- a sample made up of whoever is easiest to reach, often biased and not representative
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Reading and interpreting numerical data displays
ConcreteA numerical data display organises raw data into values paired with how often each occurs (its frequency). Reading one accurately means being able to answer three kinds of question from it directly: the total count, the most common (or least common) value, and how many values satisfy some condition, like being above a threshold.
A survey of 20 students' weekly pocket money (rounded to the nearest $5) gives: $5 (2 students), $10 (5 students), $15 (8 students), $20 (3 students), $25 (2 students). The total is 2+5+8+3+2 = 20, matching the number surveyed. The most common (modal) amount is $15, with the highest frequency, 8. The number of students receiving MORE than $15 is 3 + 2 = 5.
A data display shows weekly pocket money: $5 (2 students), $10 (5 students), $15 (8 students), $20 (3 students), $25 (2 students). Find the total number of students, the most common amount, and how many students receive more than $15.
- Total students: 2 + 5 + 8 + 3 + 2 = 20.
- Most common (modal) amount: $15, which has the highest frequency, 8.
- Students receiving more than $15: the $20 and $25 groups, 3 + 2 = 5.
Answer: 20 students in total, the most common amount is $15, and 5 students receive more than $15.
- How do you find the total number of data values from a frequency display without listing every single value out?
- Why is the mode simply the value with the tallest bar, rather than the value that appears first in the table?
2. Data collection: census versus sampling, and choosing a fair method
PictorialA census collects data from every member of a population; a sample collects data from a smaller, chosen subset, used to estimate what the whole population would show. A sample is only useful if it is REPRESENTATIVE, meaning it genuinely reflects the population, which depends heavily on HOW it was chosen: randomly, conveniently, or systematically.
A school wants to know what proportion of ALL students, Years 7 to 12, support moving the canteen. They survey every Year 12 student who eats at the canteen at lunchtime. This is a convenience sample: only one year level, only students who already use the canteen, only at one time of day. It very likely overstates support, since non-canteen users and other year levels are never asked. A fairer method would be a genuinely random sample drawn from the WHOLE school roll, across every year level.
A school surveys every Year 12 student who eats at the canteen at lunchtime, to find out what proportion of ALL students support moving the canteen. Is this sample likely to be representative? Explain, and suggest a better method.
- Identify who was actually asked: only Year 12 students, only ones who already use the canteen, only at lunchtime.
- Compare that to who the survey is meant to represent: ALL students, Years 7 to 12, regardless of whether they use the canteen.
- Conclude the sample is a convenience sample, and is very likely biased toward pro-canteen opinions.
Answer: No, the sample is not representative: it only includes canteen-using Year 12 students, a convenience sample likely biased toward support for the canteen. A better method is a random sample drawn from the whole school roll, across every year level.
- What THREE separate ways does the canteen survey's sample fail to represent the whole school?
- Why is a census (asking every single student) not always practical, even though it would remove sampling bias entirely?
3. How sampling method can be used to support a point of view
AbstractThe SAME question can produce very different results depending on how the sample was chosen, and this can be used deliberately, not just accidentally, to make a survey support a particular conclusion. Comparing sampling methods, not just sample sizes, is essential before trusting a reported statistic.
Two surveys ask the same question, whether parents support a new school uniform. Survey A reports '80% support', based on 25 parents surveyed only at the school gate during pickup, a small, self-selecting convenience sample of parents who happened to be there and chose to respond. Survey B reports '52% support', based on a genuinely random sample of 300 families drawn from the whole school roll. Survey B is far more trustworthy: it is both a much larger sample AND a properly random one, while Survey A's method (who was asked, and how they were chosen) makes it easy for someone to have picked a time and place likely to produce a favourable result.
Survey A reports 80% support for a new uniform (25 parents surveyed at the school gate during pickup). Survey B reports 52% support (a random sample of 300 families from the whole school roll). Which is more trustworthy, and why?
- Compare sample size: Survey B (300) is far larger than Survey A (25).
- Compare sampling method: Survey A is a convenience sample (whoever was at the gate); Survey B is a genuinely random sample of the whole population.
- Conclude that Survey B's combination of a larger, properly random sample makes it far more trustworthy.
Answer: Survey B is more trustworthy: it uses a much larger, genuinely random sample of the whole school, while Survey A's small convenience sample (only parents at the gate at one particular time) could easily be chosen, deliberately or not, to skew the result.
- Besides the sample size, what specifically makes Survey B's SAMPLING METHOD more trustworthy than Survey A's?
- How could someone deliberately choose a sampling method to make a survey support a predetermined conclusion?
Common misconceptions and how to address them
MisconceptionA bigger sample is always representative, regardless of HOW it was chosen.
Why it happens: Sample size is the easiest thing to compare, so it can overshadow the (more important) question of whether the sample was chosen randomly or conveniently.
How to address it: A large but biased sample (e.g. 1,000 people who all follow the same social media account) can still be far less representative than a smaller, genuinely random one. Check the sampling METHOD, not just the size, before trusting a result.
MisconceptionThe mode of a data display is whichever value is listed first or last in the table, not the value with the highest frequency.
Why it happens: Students scan the table's order rather than comparing the frequency numbers themselves.
How to address it: The mode is found by comparing every frequency and picking the value attached to the LARGEST one, regardless of where it sits in the table.
MisconceptionA census and a very large sample are the same thing.
Why it happens: Both involve 'a lot of data', which can blur the distinction between 'everyone' and 'most people'.
How to address it: A census collects data from EVERY single member of the population, with none left out; a sample, no matter how large, is still a chosen subset, and always carries some risk of not perfectly representing the whole population.
MisconceptionSampling bias only happens by accident, so a survey's method does not need to be checked once the results are already reported.
Why it happens: It feels more natural to assume researchers made an honest mistake than to consider a sampling method might have been chosen deliberately to produce a favourable result.
How to address it: Before trusting any reported statistic, ask who was actually surveyed, how they were chosen, and when and where, since sampling method can be chosen (deliberately or not) to favour a particular outcome.
Guided practice (with answers)
1. A data display shows: 3 students scored 60-69, 9 scored 70-79, 6 scored 80-89, 2 scored 90-100. How many students in total?
Answer: 20, because 3 + 9 + 6 + 2 = 20.
2. Using the same display, which score range is the mode?
Answer: 70-79, because it has the highest frequency, 9.
3. Using the same display, how many students scored 80 or above?
Answer: 8, because 6 + 2 = 8 (the 80-89 and 90-100 groups).
4. A council wants residents' views on a new park. They only survey people walking in the existing park on a Saturday morning. What is the problem with this sample?
Answer: It only includes people who already use parks and are free on Saturday mornings, a convenience sample that excludes non-park-users and people busy at that time, so it likely overstates support for more park spending.
5. Is a census or a sample more practical for finding the average height of every student in a school of 2,000 students?
Answer: A sample, because measuring all 2,000 students (a census) would be very time-consuming; a well-chosen random sample can estimate the average height reliably with far less effort.
6. A company only surveys customers who left a 5-star review. What proportion of customers does this sample likely overrepresent?
Answer: Very satisfied customers, since only people motivated to leave a positive review were counted, a self-selected sample that overstates overall satisfaction.
Independent practice worksheets
Practise reading data displays and reasoning about sampling methods with computed, never-wrong answer keys.
Differentiation
- For reading data displays, colour-code or circle the frequency column before answering any question, so the frequencies (not the values) are compared directly.
- Give a simple sentence frame for judging a sample: 'This sample only includes ___, but the population is ___, so it is likely to ___.'
- Provide a reference card listing the three named sampling types (random, convenience, systematic) with one concrete example of each.
- Work the canteen and uniform examples as whole-class discussions before asking students to critique a sample independently.
- Design a genuinely random sampling method for a real school-based question (e.g. every 10th name on an alphabetical roll, a systematic sample).
- Investigate why online polls and social media surveys are almost always convenience samples, even when they collect thousands of responses.
- Compare two hypothetical surveys with the same sample size but different sampling methods, and predict which is more trustworthy and why.
- Research (or design) a real-world example of a survey whose sampling method was later criticised, and explain what a fairer method would have looked like.
Assessment: exit ticket
A three-question exit ticket sampling reading a data display, judging a sample, and comparing sampling methods.
1. A data display shows: 4 cars sold in week 1, 7 in week 2, 10 in week 3, 5 in week 4. Find the total and the mode.
Answer: Total = 26 (4+7+10+5); mode = week 3 (10 cars, the highest frequency).
2. A gym surveys only people who show up before 6am to ask if opening hours should change. What is the problem with this sample?
Answer: It only includes early-morning gym-goers, a convenience sample that excludes everyone with different schedules, so results are unlikely to represent all members.
3. Survey X samples 40 people outside one specific shop; Survey Y takes a random sample of 400 people across the whole town. Which is more trustworthy and why?
Answer: Survey Y, because it uses both a larger sample AND a genuinely random one, while Survey X's small, single-location sample is a convenience sample unlikely to represent the whole town.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 reading data displays (section 1), Lesson 2 census versus sampling and representativeness (section 2), Lesson 3 sampling method and bias plus the exit ticket (section 3).
- This unit assumes comfort with mean, median, mode and range (the Year 7 statistics unit). Revisit that first if summarising a data set's centre is not yet automatic.
- Language to keep repeating: the mode is found by comparing FREQUENCIES, not table order; a census asks everyone, a sample asks a chosen subset; check the sampling METHOD, not just the sample size, before trusting a result.
- The bar-model figure in section 1 uses the site's comparison bar chart to make frequency comparison visual; when presenting live, ask the class to identify the tallest bar (the mode) before revealing the answer.
- Curriculum note: AC9M7ST02 (numerical data displays) sits at Year 7; AC9M8ST01 (data collection and sampling) at Year 8; AC9M9ST02 (sampling method and point of view) at Year 9. This unit deliberately bundles all three since they form one connected 'where did this statistic actually come from' skill.
- This unit's focus is deliberately on SAMPLING METHOD as a source of bias, distinct from misleading graph AXES or scale (covered in the separate statistical-critique unit), so the two do not overlap.
- Present mode and print both work: discuss the canteen and uniform scenarios live with the class in Present mode, then print the worksheets for independent practice.