👋 Welcome to Analysing Data! (Problem Solving Section 3)
Hi there! This chapter is all about digging into the data you've organised and processed to find out what it really means. It’s where you become a data detective!
This isn't just about calculation; it’s about visualising information, spotting patterns, and working backwards to understand the rules behind the numbers.
Why is this important? In Problem Solving (especially in Papers 1 and 3), you are often presented with complex data sets (tables, graphs, etc.). To solve the problem, you need the skills to quickly shift perspective (transform data) and logically explain why the numbers are behaving the way they are (explain trends).
3.1 Transform Data: Seeing the Same Thing Differently
Often, the way data is first presented might not be the most helpful for solving the problem. Transforming data means changing its form (e.g., from a table to a chart) or changing its spatial representation (e.g., rotating a tile) while keeping the core information intact.
3.1.1 Recognising Alternative Representations
The core skill here is identifying different formats that convey the exact same information. You must be able to move flexibly between these representations.
- Text to Table: Taking a written description of quantities and laying it out clearly in rows and columns.
- Table to Graph: Converting raw numbers into visual formats like bar charts, line graphs, or pie charts.
Example: Bar Chart vs. Pie Chart
If a bar chart shows that Class A scored 50% on a test, Class B scored 30%, and Class C scored 20%, a pie chart representing the same data will show three wedges with corresponding sizes (a half, just under a third, and a fifth). They look completely different, but they are equivalent representations.
Key Takeaway: Don't let the visual format trick you. Always look at the underlying values and proportions.
3.1.2 Identifying Features of a Model Based on Representation
When data represents a *model* (a simplified set of rules for a real-world scenario), different representations (like graphs or diagrams) highlight different features of that model.
A. Interpreting Features in Graphs (The Gradient)
A very common model feature represented on a graph is the rate of change, known as the gradient (or slope).
Imagine a graph showing the total cost of a taxi ride (Y-axis) against the distance travelled (X-axis).
- A Steep Gradient: Means the cost increases quickly relative to the distance. (The price per kilometre is high).
- A Shallow Gradient: Means the cost increases slowly relative to the distance. (The price per kilometre is low).
- A Break Point: If the line suddenly gets steeper, it means the rate (price per km) changed after a certain distance (a threshold value in the model).
Tip: Interpret the gradient appropriately in the context of the model. If the graph plots speed vs. time, the gradient is acceleration. If it plots cost vs. quantity, the gradient is the unit price.
B. Spatial Transformations (Rotations and Reflections)
Sometimes, transforming data involves identifying how shapes or objects relate to each other through movement.
If you are looking at a pattern or a tile design, you must be able to:
- Identify if two shapes are the same object, but one has been rotated (turned).
- Identify if two shapes are the same object, but one has been reflected (flipped).
Did you know? In geometry problems, objects that are "identical shapes in opposite corners of a tile" are often related by 180-degree rotation or a combined reflection/rotation.
Quick Review: Transforming Data
Transforming Data involves two skills:
1. Showing data in an equivalent format (e.g., Pie Chart = Bar Chart).
2. Extracting features from representations (e.g., reading the rate of change from the gradient of a model graph, or identifying a rotated shape).
3.2 Explain Trends in Data: Finding the 'Why'
Once you’ve successfully analysed the numbers and graphs, the next step is to understand the story they tell. This involves suggesting reasons for patterns and making sure your mathematical model accurately reflects reality.
3.2.1 Suggest Possible Explanations for Trends
A trend is a general direction in which data is moving (up, down, or stable). When asked to explain a trend, you must look for external factors that could influence the data.
Example: Ice Cream Sales
Data shows ice cream sales steadily increase from March to August, then suddenly decrease sharply in September.
- Trend Explanation (March–August): Sales rise due to warmer weather and school holidays.
- Explanation for the Change/Break Point (September): The sharp decrease is due to colder weather beginning and students returning to school, reducing opportunistic purchases.
Step-by-Step for Explaining Trends:
1. Describe the trend: State clearly what the data shows (e.g., "The data shows a linear increase until point X").
2. Identify the break/change: Note where the pattern shifts (e.g., "At point X, the trend reverses sharply").
3. Suggest the cause: Propose a plausible, real-world reason for both the trend and the change (e.g., "A new law was introduced at time X," or "Seasonality effects took over").
Common Mistake to Avoid: Don't just re-state the data. Saying "The sales went down because they sold less" is circular reasoning. You need to suggest *why* they sold less (the external factor).
3.2.2 Fit a Model to the Information Available
A mathematical model uses specific parameters (fixed or variable values) to describe a situation. When you "fit a model," you are usually given the overall structure of the equation and must deduce the missing values (the parameters) that make the model match the observed data.
Don't worry if this seems tricky at first—this often boils down to simple simultaneous equations or reversing a calculation.
Analogy: Finding the Recipe Ingredients
If you taste a cake (the result, or the output data) and you know the basic recipe structure (the model: flour + sugar + eggs), you are trying to deduce the exact amount of each ingredient (the parameters) that was used.
Syllabus Example: Deducing Price per Kilometre
The pricing model for a taxi company is:
Total Cost = Fixed Charge + (Price per Kilometre \(\times\) Distance)
We have two recorded journeys:
1. A 5 km journey cost \$11.
2. A 10 km journey cost \$16.
Goal: Find the fixed charge (F) and the price per kilometre (P).
Step 1: Set up the equations
(1) \(11 = F + 5P\)
(2) \(16 = F + 10P\)
Step 2: Deduce the difference
The difference between the two journeys is 5 km and \$5. This means the extra 5 km cost \$5.
Therefore, \(5P = 5\), so the Price per Kilometre (P) = \$1.
Step 3: Find the fixed charge
Substitute \(P=1\) back into Equation (1):
\(11 = F + 5(1)\)
\(11 = F + 5\)
\(F = 6\)
Result: The fixed charge is \$6 and the price per kilometre is \$1. You have successfully fitted the parameters to the model using the available information.
Key Takeaway: Fitting a model requires using known outcomes (data points) to calculate the hidden constant values (parameters) that define the rules of the situation.
Quick Analysis Toolkit: Summary
To successfully "Analyse Data," remember these two core actions:
1. Transform: Change the data format or spatial representation so it is easier to understand, but ensure the core information remains the same. (Look for equivalent charts or rotational symmetry.)
2. Explain: Suggest plausible reasons for the patterns (trends) you see, especially where the pattern changes. If you are given a model structure, use the data to calculate the specific numbers (parameters) that make that model work.
Keep practising interpreting graphs and tables in context—you’ve got this!