With reference to this table, a statistician has proposed the following criteria for determining the "most geographically typical" of the...

Owning the Dataset

Let's start by understanding what we're working with in this table about Australian states:

The table shows 8 Australian states with 4 different measurements for each:

• Land Area (in square kilometers)
• Population
• Population Density (people per square kilometer)
• Percentage of Population in Capital City

Key insight: The question defines "typical" as being in positions 3-6 when the states are sorted by any metric. This means the middle-ranked states are considered "typical," while the highest (positions 7-8) and lowest (positions 1-2) are not typical.

This understanding is crucial because we don't need to remember exact values - we just need to determine each state's position when sorted.

Analyzing Statement 3: ACT is not typical in any category

Statement 3 Translation:

Original: "ACT is not typical in any category"
What we're looking for:

• ACT must NOT be in positions 3-6 when sorted by any metric
• ACT must always be in positions 1-2 or 7-8

In other words: ACT is always among the extremes (very high or very low) in all categories.

Why Start With Statement 3?

Let's begin with Statement 3 because it focuses on just one state and has the simplest verification process. It's much easier to check one state across four categories than comparing multiple states.

Let's sort each category and check ACT's position:

Sorting by Land Area:
When we sort from smallest to largest, ACT is the smallest territory - position 1.
→ Not typical ✓

Sorting by Population:
When sorted from smallest to largest, ACT is the second smallest - position 2.
→ Not typical ✓

Sorting by Population Density:
When sorted from lowest to highest, ACT has the highest density - position 8.
→ Not typical ✓

Sorting by % in Capital City:
When sorted from lowest to highest, ACT has the highest percentage - position 8.
→ Not typical ✓

Conclusion: ACT is never in positions 3-6, so Statement 3 is Yes.

Teaching moment: Notice how sorting immediately reveals each state's position without requiring us to write down all the values or create a complex tracking table. This is much faster than calculating and comparing each value manually.

Analyzing Statement 1: NSW is typical in more categories than any other state

Statement 1 Translation:

Original: "NSW is typical in more categories than any other state"
What we're looking for:

• Count how many categories NSW is typical in (positions 3-6)
• Check if any other state is typical in MORE categories than NSW

In other words: NSW must have the highest count of "typical" appearances.

Targeted Analysis of NSW

Let's check NSW by sorting each category:

Sorting by Land Area:
NSW is in position 4 → Typical ✓

Sorting by Population:
NSW has the highest population (position 8) → Not typical ✗

Sorting by Population Density:
NSW is in position 6 → Typical ✓

Sorting by % in Capital City:
NSW is in position 4 → Typical ✓

NSW Summary: Typical in 3 categories (Land Area, Population Density, % in Capital)

Now, for Statement 1 to be true, no other state can be typical in 3 or more categories. Let's check SA, which is mentioned in Statement 2:

SA's typicality:

• Land Area: Position 5 → Typical ✓
• Population: Position 4 → Typical ✓
• Population Density: Position 3 → Typical ✓
• % in Capital City: Position 7 → Not typical ✗

SA Summary: Typical in 3 categories (Land Area, Population, Population Density)

Since SA is also typical in 3 categories, NSW is not typical in more categories than any other state. Statement 1 is No.

Teaching moment: We strategically checked SA next since it's mentioned in Statement 2. This gives us information for both statements simultaneously, saving significant time.

Analyzing Statement 2: No state is more typical than SA

Statement 2 Translation:

Original: "No state is more typical than SA"
What we're looking for:

• We already know SA is typical in 3 categories
• Check if any state is typical in 4 categories (which would be more than SA)

In other words: SA must be tied for the most "typical" appearances.

Efficient Verification

We've already established:

• SA is typical in 3 categories
• NSW is typical in 3 categories
• ACT is typical in 0 categories

For Statement 2 to be false, we would need to find at least one state that is typical in all 4 categories. Let's scan our sorted lists to see if any state appears in positions 3-6 across all categories.

Checking for a state that appears in positions 3-6 in all categories:
Reviewing our sorted lists, we can see that no state appears in positions 3-6 for all four categories. Some states might be typical in 3 categories (like NSW and SA), but none are typical in all 4.

Conclusion: No state is more typical than SA, so Statement 2 is Yes.

Teaching moment: We didn't need to check every single state individually. Once we knew SA was typical in 3 categories, we just needed to determine if any state could be typical in all 4 categories. This targeted approach saved us from unnecessary work.

Final Answer Compilation

Let's compile our findings:

• Statement 1: No (NSW is typical in 3 categories, but so is SA)
• Statement 2: Yes (No state is typical in more than 3 categories)
• Statement 3: Yes (ACT is never typical)

The answer is B (No/Yes/Yes).

Learning Summary

Skills We Used

• Strategic Sorting: We used sorting to instantly identify positions rather than manually tracking values
• Targeted Analysis: We focused only on states mentioned in the statements (ACT, NSW, SA) rather than analyzing all 8 states
• Strategic Statement Ordering: We tackled Statement 3 first because it was easiest to verify and focused on just one state
• Parallel Processing: While checking NSW and SA's typicality, we gathered information for both Statements 1 and 2

Strategic Insights

• Start with the most specific statement: Statement 3 was focused on just one entity (ACT) and was easiest to verify
• Sort for position identification: Sorting immediately reveals positions without requiring extensive calculations
• Recognize when to stop: Once we found that SA and NSW were both typical in 3 categories, we knew Statement 1 was false
• Focus on statement-specific entities: We prioritized checking the states mentioned in statements rather than analyzing all states

Common Mistakes We Avoided

• Creating a comprehensive tracking table for all states across all categories
• Checking states in alphabetical order rather than in strategic order
• Processing statements in numerical order (1, 2, 3) rather than efficiency order
• Calculating exact values when we only needed to know positions

Remember: Table analysis questions reward strategic thinking over comprehensive analysis. Focus on gathering just the information you need to answer each statement, and use sorting to quickly identify positions rather than calculating every value.