107i - Acquiring Data Sets

Lesson 107 — Acquiring Data Sets for Discrete and Continuous Variables

Strand: Statistics | Descriptor: AC9M7ST01 | Duration: 45 minutes

Equipment: tape measures, rulers, a stopwatch or phone timers, scales if available.

Learning Intentions

• To distinguish discrete from continuous numerical variables.
• To collect a data set accurately and record it clearly.

Success Criteria

I can:

1. Classify a numerical variable as discrete or continuous.
2. Explain why a variable is one and not the other.
3. Collect data accurately, recording units and precision.
4. Organise raw data into a frequency table.

Warmup

(6 minutes — sort the variables, pairs)

Sort these into two groups, using any rule you choose. Be ready to state your rule.

Number of siblings; height; shoe size; time to run m; number of pets; mass; number of goals scored; arm span; number of pages in a book; temperature at noon.

Expected sorts: many students separate by “things you count” versus “things you measure”. Collect the rules; name the mathematical distinction:

• Discrete — from counting; only certain separate values are possible (whole numbers, or steps like half shoe sizes).
• Continuous — from measuring; any value in a range is possible, limited only by the instrument.

Answers: Discrete — siblings, shoe size, pets, goals, pages. Continuous — height, running time, mass, arm span, temperature.

Activities

Activity 1 — Explicit Instruction: Discrete versus Continuous (12 min)

The test question: between any two possible values, is another value possible?

Variable	Between and ?	Type
Number of siblings	No — siblings is impossible	discrete
Height in metres	Yes — m, m, m…	continuous

Two cases worth arguing about — raise them deliberately:

1. Shoe size. Sizes go , , — steps, not a continuum. Discrete, even though it is not whole numbers. Discrete does not mean “whole number”; it means “separate values with gaps”.
2. Money. Prices go up in cents. Strictly discrete — but with steps so small that it is often treated as continuous. Naming the convention matters more than winning the argument.

Precision belongs to continuous data. A height “is” m only to the nearest centimetre; the true value lies in a range (Lesson 61’s tolerance work). Discrete counts are exact — siblings is exactly .

We do — classify and justify:

1. Number of students in a class.
2. Time taken to complete a puzzle.
3. Number of text messages sent yesterday.
4. Mass of a school bag.
5. Number of correct answers on a -question test.
6. Length of a pencil.

(Answers: discrete; continuous; discrete; continuous; discrete; continuous. Justifications must use the between-values test.)

Activity 2 — Collecting Real Data (16 min)

Groups of four. Two variables — one discrete, one continuous — collected from the whole class.

Variable 1 (discrete): number of letters in each student’s first name.

Variable 2 (continuous): arm span, in centimetres to the nearest centimetre.

Collection protocol — taught explicitly before starting:

1. Define the variable precisely. “Arm span” means fingertip to fingertip, arms horizontal. Without a definition, different groups measure different things.
2. State the precision. Nearest centimetre. Everyone rounds the same way.
3. Record raw data first — a simple list, in the order collected. Do not sort or tally while measuring.
4. Then tally into a frequency table.

Frequency table for the discrete variable:

Letters	Tally	Frequency
3
4
…

Grouped frequency table for the continuous variable — because every arm span is likely different, individual values are useless:

Arm span (cm)	Tally	Frequency
–
–
–
–
–

The grouping point — teach it explicitly. Continuous data usually needs class intervals, because raw values rarely repeat. The intervals must not overlap and must leave no gaps: – then – works; – then – does not (where does go?).

Circulating prompts:

Prompt	Purpose
What exactly are you measuring, and where do you start the tape?	Definition precision.
Someone measures — what do you record?	Precision convention, agreed in advance.
Why not one row per arm span in the table?	Motivates grouping.
Would cm intervals be better than ?	Interval choice is a judgement — too many rows or too few.

Activity 3 — Inquiry: how Good is Our Data? (11 min)

Pairs, then class discussion.

Look at the class’s arm span data.

1. Name two sources of error in how it was collected.
2. Would a second measurement of the same person give exactly the same number? Why?
3. If we measured the whole school, would the picture look different? How?
4. Our data represents Year 7 students at this school. What can it not tell us?

Socratic scaffolding:

Prompt	Purpose
What could vary between measurers?	Tape tension, arm height, rounding, whether shoes were on.
So is cm exact?	No — it is cm at best (Lesson 61’s tolerance).
Q3: what would a whole-school data set add?	Older students are taller — a wider spread and a higher centre.
Q4: what is our data’s population?	Year 7 at this school — not all Year 7s, not all people.
Looking back	A data set answers questions about the group it came from, and no further.

The vocabulary to record:

• Population — the whole group we want to know about.
• Sample — the part we actually measured.
• Our class is a sample; conclusions about all Year 7 students would be an extrapolation, exactly like Lesson 83’s graph warning.

Checks for Understanding

(5 minutes — exit ticket)

1. Classify with a reason: (a) number of pets (b) time to swim m (c) shoe size (d) mass of an apple.
2. Why is arm span data usually grouped into intervals but name-length data is not?
3. What is wrong with the intervals –, –, –?
4. Name two things that must be agreed before a class collects measurements.
5. Reasoning. Our class’s arm span data — what population does it describe, and what can it not tell us?

Answers: 1. (a) discrete — counted, no values between (b) continuous — any value in a range (c) discrete — separate half-size steps (d) continuous — measured; 2. Arm spans rarely repeat, so a table of individual values would have one row each; name lengths take few distinct values and repeat naturally; 3. They overlap — and each belong to two intervals; 4. The precise definition of the variable, and the precision (rounding) to use; 5. Year 7 students at this school; it cannot tell us about other year levels, other schools, or adults.

Common Misconceptions

Misconception	How to pre-empt it
Discrete means “whole number”.	Shoe sizes are discrete but not whole — the gaps are what matter.
Continuous means “large” or “decimal”.	The between-values test is the criterion, not the appearance.
Recording a measurement as exact.	Precision stated in advance; tolerance discussed in the inquiry.
Overlapping class intervals.	Exit Q3; the no-gaps-no-overlaps rule.
Tallying while measuring, losing the raw data.	The protocol’s step 3 — raw list first.
Generalising a class sample to everyone.	Population versus sample, named and recorded.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which of these is continuous: number of cars in a car park, the time a car is parked, or the number of parking bays?

Answer

The time — any value in a range is possible. The other two are counts.

E2 (AMC Junior style). A data set of heights ranges from cm to cm. Suggest sensible class intervals and say how many classes result.

Answer

Range cm. Intervals of cm from : –, –, …, – — eight classes. (Rule of thumb: five to ten classes for a set this size.)

E3 (Challenge). A student records shoe sizes as and argues the variable is continuous because it has decimals. Refute the argument.

Answer

Decimals do not make a variable continuous — the test is whether values between the possible ones can occur. Size does not exist, so there are gaps: the variable is discrete despite the decimals.

E4 (Challenge). Two groups measure the same ten students’ heights and get slightly different data. Explain how both can be correct, and what should be reported.

Answer

Measurement has limited precision, so each value is a rounded estimate within a tolerance. Both sets can be correct to the nearest centimetre. Reports should state the precision used and, ideally, the measuring protocol — otherwise the difference looks like an error rather than ordinary variation.

Homework

1. Classify each as discrete or continuous, with a reason: (a) number of songs on a playlist (b) length of a song in seconds (c) number of siblings (d) temperature at midnight (e) number of goals in a match (f) volume of water in a bottle.
2. Explain the difference between discrete and continuous data using the between-values test and one example of each.
3. What is wrong with these class intervals: –, –, –? Rewrite them correctly.
4. Collect a small data set at home: the number of letters in the first names of ten people you know. Record the raw list, then make a frequency table.
5. Collect a continuous data set: the length in centimetres (to the nearest centimetre) of ten objects in your home. Record the raw list, then group into sensible intervals.
6. For your Q5 data, state (a) the variable’s precise definition (b) the precision used (c) two possible sources of error.
7. A survey asks “how many hours of sleep did you get?” Is the answer discrete or continuous? Discuss why it might be recorded as though it were the other.
8. Reasoning. Explain the difference between a population and a sample, using your Q4 data as the example.
9. Challenge. A data set has values from to . Design a grouped frequency table with (a) five classes (b) eight classes. Which would you choose, and why?

Answers: Q1 — (a) discrete (b) continuous (c) discrete (d) continuous (e) discrete (f) continuous. Q3 — the boundaries overlap; use –, –, – (or , etc.). Q7 — sleep time is continuous, but people report it to the nearest half hour, so it arrives looking discrete — a recording convention, not a change in the variable. Q8 — the population is everyone whose name length we care about; the sample is the ten people actually asked; conclusions strictly describe the sample. Q9 — range : (a) classes of : –, –, –, –, –, – (six needed, so use width for five) (b) classes of : – through – — eight classes. Eight gives more detail without becoming unwieldy; five may hide the shape.