ACARA v9 CONTENT DESCRIPTION “select and construct appropriate representations, including tables, graphs, descriptive statistics, models and mathematical relationships, to organise and process data and information”
Builds on choosing a table or a graph to reveal a pattern. Here you process the data further: you fit a mathematical relationship that turns a trend into an equation you can calculate with, and you summarise repeated readings with descriptive statistics. The skill is matching the representation to the data and to the purpose you have in mind.
One dataset, many representations
A class investigates Newton’s second law. They load a trolley of fixed mass and pull it with a measured force, recording the acceleration each time. The raw readings start as a table of paired numbers. A table keeps every exact value, but a trend is hard to read off rows of digits. Plot acceleration against force and the points line up; fit a line and the relationship becomes an equation. No single representation is best for everything, so the first inquiry skill is selecting the one that fits the data and what you want it to show.
Acceleration of a fixed-mass trolley at rising force
Acceleration, in metres per second squared, for a 2 kilogram trolley pulled with rising force. Start with the table, then switch to the line graph to see the trend.
In the table you read paired numbers row by row. As a line graph the relationship is plain: the points climb in a straight line through the origin, so acceleration is directly proportional to force. Each rise of 2 N adds 1 unit of acceleration, which is the gradient. The line graph is the right representation when the purpose is showing how one quantity changes with another.
From a straight line to a mathematical relationship
When the points rise in a straight line through the origin, the data is more than a trend: it is directly proportional. For a fixed mass, doubling the force doubles the acceleration, so acceleration equals force divided by mass, written a equals F divided by m. That equation is the most compact representation of all. The gradient of the line is one over the mass, so the graph even measures the trolley for you. Best of all, the equation is a model you can calculate with: it predicts the acceleration at a force you never tested, as long as you stay in the range where the line holds.
Summarising repeated readings with descriptive statistics
A single mathematical relationship is not always what you need. When the same measurement is repeated, each run scatters a little, and you want a few numbers that stand for the set. These are descriptive statistics. The mean adds all the values and divides by how many there are, giving a typical value. The median is the middle value once they are sorted. The range is the largest minus the smallest, a measure of how widely the readings scatter. Each processes the data differently, so reporting the mean alongside the range tells a reader both the typical result and how reliable it is.
Reaction time across five repeated runs
Time for the same reaction to finish, in seconds, measured five times under identical conditions. View the table, then the bar chart, and read off the centre and the spread.
The five runs add to 110 seconds, so the mean is 110 divided by 5, which is 22 seconds. The median, the middle of the sorted runs, is also 22 seconds. The range is 24 minus 20, which is 4 seconds, so the readings scatter only a little around the centre. Reporting the mean of 22 seconds with a range of 4 seconds processes the raw runs into a typical result and a measure of its reliability.
Choosing the representation that fits the question
A table records the exact data, a graph reveals its shape, a fitted equation turns a trend into a rule you can calculate with, and descriptive statistics compress repeated readings to a few telling numbers. Each one organises and processes the data for a different purpose. Sorting which conclusions the representations actually support is how you avoid claiming more than the data can show.
Test conclusions against the representations
Use the force-acceleration data and the repeated reaction times above, with their fitted relationship a equals F divided by m and their descriptive statistics. Decide which conclusions the data actually supports.
Claim: The graphs, the mathematical relationship and the descriptive statistics above support the statement being judged.
For the fixed-mass trolley, doubling the force roughly doubles the acceleration.
A line graph through the origin is a good choice for showing the force-acceleration trend.
The reaction time mean of 22 seconds with a range of 4 seconds gives both a typical value and its spread.
The equation a equals F divided by m will hold for any force, however large, with no limit.
The range of 4 seconds means every single reaction run took exactly 22 seconds.
Decide whether each statement is evidence for the claim, or not.
Why this matters
A table records data, a graph reveals its shape, descriptive statistics compress repeated readings to a few telling numbers, and a mathematical relationship turns a trend into a rule you can calculate with. Choosing the representation that fits the purpose is a skill in itself. Engineers, physicists and data analysts all depend on selecting the representation that turns raw measurements into knowledge a reader can trust.
Quick self-check
1. A trolley of fixed mass is pushed with forces of 2, 4, 6 and 8 N, giving accelerations of 1, 2, 3 and 4 metres per second squared. These points plot as...
2. On the graph of acceleration against force for a single trolley, the gradient of the line is linked to the trolley’s...
3. A reaction is timed five times under the same conditions: 20, 21, 22, 23 and 24 seconds. The range of these readings is...
4. You want to predict the acceleration of the trolley for a force you never tested. The most useful representation is the...
5. Five timed runs of the same reaction scatter a little around their mean. The clearest representation for reporting that typical time and its spread is a...