acquire data sets for discrete and continuous numerical variables and calculate the range, median, mean and mode; make and justify decisions about which measures of central tendency provide useful insights into the nature of the distribution of data.
We are learning to acquire discrete and continuous numerical data and interpret measures of centre and spread.
Discrete numerical data come from counts with separated possible values, such as the number of messages received, while continuous numerical data come from measurements on a scale, such as reaction time or mass. Rounding a measurement for recording does not change the underlying variable from continuous to discrete.
The mean is the sum divided by the number of values and uses every observation, the median is the middle of the ordered data, and the mode is the most frequent value. The range is maximum minus minimum, so it describes total spread rather than centre.
No single summary tells the whole story. Extreme values can move the mean and range substantially while leaving the median almost unchanged, and data quality depends on consistent units, sensible precision, credible sources and checks for missing, duplicated or impossible values before digital calculations are interpreted.
Success criteria
I can distinguish a count from a measurement and record numerical data with suitable units and precision.
I can calculate mean, median, mode and range accurately from an ordered data set.
I can select and interpret summaries by considering outliers, variation and the data-collection context.
Key vocabulary
discrete numerical variable
A discrete numerical variable is produced by counting and has separated possible numerical values.
continuous numerical variable
A continuous numerical variable is measured and can take values anywhere within an interval.
median
The median is the middle value of ordered data, or the mean of the two middle values when the count is even.
range
The range is the maximum value minus the minimum value and measures the total spread.
Visual models and representations
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
Digital tools accelerate calculations but require transparent formulas and careful source or measurement checks.
A code-specific application model for distinguish discrete and continuous variables and data quality: discrete count of messages:…
4 worked numerical & application examples
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 1
Calculating centre and spread for five observations.
Use the ordered data set 4, 5, 5, 7 and 9, whose sum is 30 and whose count is 5.
Calculate the mean as 30 divided by 5, which equals 6.
Identify the middle value 5 as the median and the repeated value 5 as the mode.
Calculate the range as maximum 9 minus minimum 4, which equals 5.
Final answer: The data have mean 6, median 5, mode 5 and range 5.
Check: Multiplying the mean 6 by the count 5 returns the total 30, and the ordered positions and endpoints confirm the other summaries.
Example 2
Judging the effect of an extreme reaction time.
Five reaction times in seconds are 0.42, 0.45, 0.47, 0.49 and 1.20, and they total 3.03 seconds.
Calculate the mean as 3.03 divided by 5, which equals 0.606 seconds, and identify the median as 0.47 seconds.
Calculate the range as 1.20 minus 0.42, which equals 0.78 seconds, and recognise 1.20 seconds as an extreme value relative to the other four.
Use the median to describe the typical recorded reaction time and report the extreme value and range rather than hiding the variation.
Final answer: The median of 0.47 seconds is more representative of the main cluster than the mean of 0.606 seconds, while the 0.78-second range reveals the extreme result.
Check: Removing the 1.20-second value changes the mean to 0.4575 seconds but changes the median of the remaining four values only to 0.46 seconds.
Example 3
Application problem 1
Problem: For the data 4, 5, 5, 7, 9, 10, 16, calculate the mean, median and range, then explain which statistic is most affected by the value 16.
Plan: Represent the information first, then calculate, interpret and independently check the result.
Work: Mean = 56/7 = 8, median = 7, range = 12. The mean and range are strongly affected by the high value 16; the median is less affected.
Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Final answer: Mean = 56/7 = 8, median = 7, range = 12. The mean and range are strongly affected by the high value 16; the median is less affected.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Application problem 2
Problem: Plan how to acquire reliable continuous data about Year 7 reaction times, including units, precision and two quality checks before calculating summaries.
Plan: Specify how every measurement is made and how you will detect inconsistent, missing or impossible values.
Work: Use the same digital timer and procedure for every student, record reaction time in seconds to the same precision, repeat or flag impossible readings, check missing and duplicate entries, and then calculate and interpret centre and spread with the sample size stated.
Interpret: A standard measurement procedure and transparent cleaning protect the meaning of later numerical summaries.
Final answer: Use the same digital timer and procedure for every student, record reaction time in seconds to the same precision, repeat or flag impossible readings, check missing and duplicate entries, and then calculate and interpret centre and spread with the sample size stated.
Check: A standard measurement procedure and transparent cleaning protect the meaning of later numerical summaries.
Common misconceptions
Common mistake: The median can be found from the middle position before the data are ordered.
Correction: The values must be arranged in numerical order before the middle position is identified.
Common mistake: A measurement recorded to the nearest whole unit becomes a discrete variable.
Correction: The underlying quantity remains continuous even when its recorded value is rounded.
Common mistake: The mean is always the best description of a typical value.
Correction: An extreme value can distort the mean, so the median and the distribution must also be considered.
10 important problems to solve
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. Classify the number of siblings each student has as a discrete or continuous numerical variable.
Check answer
Answer: The number of siblings is a discrete numerical variable.
Hint: Decide whether the value is counted or measured on a scale.
Why: Siblings are counted in separated whole-number outcomes.
2. Classify each student's arm span measured in centimetres as a discrete or continuous numerical variable.
Check answer
Answer: Arm span is a continuous numerical variable, even if it is rounded to the nearest centimetre.
Hint: Ask whether a more precise instrument could produce a value such as 151.4 centimetres.
Why: Length is measured and can take values between recorded whole centimetres.
3. Find the median and range of 3, 8, 5, 11 and 8.
Check answer
Answer: After ordering the data as 3, 5, 8, 8, 11, the median is 8 and the range is 8.
Hint: Order the five values before selecting the middle and subtracting the endpoints.
Why: The middle ordered value gives the median and 11 minus 3 gives the range.
4. Calculate the mean, median, mode and range of 2, 4, 4, 6, 9 and 11.
Check answer
Answer: The mean is 6, the median is 5, the mode is 4 and the range is 9.
Hint: Use separate lines for total and count, middle pair, frequency and endpoints.
Why: The sum 36 divided by 6 gives 6, the middle pair averages to 5, 4 repeats, and 11 minus 2 gives 9.
5. The waiting times 3, 4, 4, 5 and 19 minutes include one extreme value; compare the mean and median and choose the better typical value.
Check answer
Answer: The mean is 7 minutes and the median is 4 minutes, so the median better represents the cluster from 3 to 5 minutes while 19 should be reported as an extreme value.
Hint: Calculate both centres and compare each with the four closely grouped times.
Why: The high value pulls the mean away from most observations but does not move the middle ordered value as far.
6. A spreadsheet contains masses recorded as 42 grams, 0.045 kilograms, 47 grams, a blank cell and 44 grams; identify two data-cleaning actions required before calculating a mean.
Check answer
Answer: Convert all masses to one unit and investigate or mark the missing value rather than treating the blank as zero.
Hint: Check whether every entry uses the same scale and whether every participant has a valid measurement.
Why: Consistent units and explicit handling of missing data are necessary for a meaningful calculation.
7. A class's daily message counts have mean 12 and range 20; explain what each statistic communicates and one fact it does not reveal.
Check answer
Answer: The mean communicates an average of 12 messages and the range communicates a 20-message gap between maximum and minimum, but neither reveals the complete distribution or where values cluster.
Hint: Describe one location feature, one endpoint difference and one pattern that both summaries omit.
Why: Centre and spread answer different questions and do not replace the original distribution.
8. Construct two five-value data sets that both have mean 6 and range 8 but have different medians, and verify the conditions.
Check answer
Answer: One valid pair is 2, 4, 6, 8, 10 and 2, 3, 5, 10, 10; both total 30 for mean 6, both have range 10 minus 2 equals 8, and their medians are 6 and 5 respectively.
Hint: Fix the endpoints at 2 and 10, make all five values total 30, and change the third ordered value.
Why: Equal mean and range do not force equal middle positions, as the verified totals, endpoints and medians show.
9. Plan how to acquire reliable continuous data about Year 7 reaction times, including units, precision and two quality checks before calculating summaries.
Check answer
Answer: Use the same digital timer and procedure for every student, record reaction time in seconds to the same precision, repeat or flag impossible readings, check missing and duplicate entries, and then calculate and interpret centre and spread with the sample size stated.
Hint: Specify how every measurement is made and how you will detect inconsistent, missing or impossible values.
Why: A standard measurement procedure and transparent cleaning protect the meaning of later numerical summaries.
10. For the data 4, 5, 5, 7, 9, 10, 16, calculate the mean, median and range, then explain which statistic is most affected by the value 16.
Check answer
Answer: Mean = 56/7 = 8, median = 7, range = 12. The mean and range are strongly affected by the high value 16; the median is less affected.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Curriculum coverage and elaborations
Content description: acquire data sets for discrete and continuous numerical variables and calculate the range, median, mean and mode; make and justify decisions about which measures of central tendency provide useful insights into the nature of the distribution of data.
International curriculum mapping
The Australian Curriculum code above is exact. Victorian Year 7, NSW Stage 4, US Grade 7, England Key Stage 3, New Zealand Level 4 and comparable international curricula contain broadly related learning, but code-to-code equivalence varies by jurisdiction.
🎥 Optional Video Lesson
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Math Antics — Compare common measures of centre when summarising numerical data.
As you watch: Which measure changes most when one unusually large value is added?
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Acquire data sets for discrete and continuous numerical variables and...
Mapped skill: acquire data sets for discrete and continuous numerical variables and calculate the range, median, mean and mode; make and justify decisions about which measures of central tendency provide useful insights into the nature of the distribution of data
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Topic reference: AC9M7ST01 — Acquiring Numerical Data and Measures of Centre