For each of 20 composers, the table lists the year of birth, the year of death, and the number of...

OWNING THE DATASET

Let’s start by understanding this composers table with the intention of “owning the dataset” completely.

We have a table showing 20 composers with their:

• Birth and death years
• Number of performances of their works

The total number of performances across all composers is 24,722 (approximately 25,000).

Looking at just one example row helps us understand the relationships:

• Mozart (1756-1791): 3,035 performances

Key insight: The performance numbers vary widely between composers (we’ll see this ranges from under 100 to over 3,000), suggesting an uneven distribution that will be crucial for our efficient solving approach.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: “The composers born in the 1700s account for more than one-third of the total performances.”
What we’re looking for:

• Identify which composers were born between 1700-1799
• Calculate their total performances
• Compare if this total exceeds \(\frac{1}{3}\) of all performances (approx. \(25,000 ÷ 3 ≈ 8,300\))

In other words: Do the 1700s composers represent more than 33.3% of all performances?

Let’s use our first powerful technique - sorting by birth year to make this question much easier to solve.

When we sort by “Year of birth,” all composers born in the 1700s appear together in sequence. This gives us immediate visual access to the relevant composers without having to scan the entire table.

Scanning our sorted data, we can see only three composers were born in the 1700s:

• Haydn (born 1732): approximately 1,000 performances
• Mozart (born 1756): approximately 3,000 performances
• Schubert (born 1797): approximately 1,000 performances

Instead of calculating exact totals, we can use rounded numbers for faster mental math:
1,000 + 3,000 + 1,000 = approximately 5,000 total performances

For the one-third threshold:
Total performances: approximately 25,000
One-third: \(25,000 ÷ 3 = \mathrm{approximately}\ 8,300\)

Our comparison:
5,000 is clearly less than 8,300

Statement 1 is No.

Teaching callout: Notice how sorting instantly showed us which composers were born in the 1700s without having to check birth years for all 20 composers. This visual identification after sorting saved us significant effort.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: “The three composers with the most performances account for more than half of the total performances.”
What we’re looking for:

• Identify the top 3 composers by number of performances
• Calculate their total performances
• Compare if this total exceeds \(\frac{1}{2}\) of all performances (approx. \(25,000 ÷ 2 ≈ 12,500\))

In other words: Do the top 3 composers represent more than 50% of all performances?

For this statement, let’s sort by “Number of performances” in descending order. This instantly reveals the top performers without any searching or comparing.

After sorting, we can immediately see the top 3 composers:

• Mozart: approximately 3,000 performances
• Beethoven: approximately 2,900 performances
• Tchaikovsky: approximately 2,300 performances

Quick sum: 3,000 + 2,900 + 2,300 = approximately 8,200 total performances

For the half threshold:
Total performances: approximately 25,000
Half: \(25,000 ÷ 2 = \mathrm{approximately}\ 12,500\)

Our comparison:
8,200 is clearly less than 12,500

Statement 2 is No.

Teaching callout: Sorting by performance count made identifying the top 3 composers instantaneous. We didn’t need to compare values or search through the data at all - they appeared right at the top of our sorted table.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: “The mean number of performances exceeds the median by more than 300.”
What we’re looking for:

• Calculate the mean (average) number of performances
• Identify the median number of performances
• Compare if the difference is greater than 300

In other words: Is the average skewed at least 300 performances higher than the middle value?

Let’s keep our table sorted by “Number of performances” (descending) from the previous question. This sort already helps us find the median much more efficiently.

First, let’s calculate the mean:
\(\mathrm{Mean} = \frac{\mathrm{Total\ performances}}{\mathrm{Number\ of\ composers}}\)
\(\mathrm{Mean} = 25,000 ÷ 20 = \mathrm{approximately}\ 1,250\ \mathrm{performances}\)

Next, for the median:
With 20 composers sorted by performances, the median is the average of the 10th and 11th values.
Looking at these values in our sorted table:

• 10th value: approximately 931
• 11th value: approximately 865

\(\mathrm{Median} = \frac{931 + 865}{2} = \mathrm{approximately}\ 900\)

Now we compare:
\(\mathrm{Mean} - \mathrm{Median} = 1,250 - 900 = \mathrm{approximately}\ 350\)

Since 350 > 300, the mean exceeds the median by more than 300.

Statement 3 is Yes.

Teaching callout: Notice how our sorting from Statement 2 continued to help us here. Finding the median in an unsorted list of 20 values would require arranging all values in order - but we already had them sorted, so we could instantly look at the 10th and 11th values.

FINAL ANSWER COMPILATION

Reviewing our analysis:

• Statement 1: No - Composers from the 1700s account for about 5,000 performances, which is less than one-third of the total.
• Statement 2: No - The top 3 composers account for about 8,200 performances, which is less than half of the total.
• Statement 3: Yes - The mean (1,250) exceeds the median (900) by approximately 350, which is more than 300.

The answer is therefore: No, No, Yes

LEARNING SUMMARY

Skills We Used

• Strategic Sorting: We sorted by different columns to make each statement easier to evaluate
• Estimation: We used rounded numbers for faster mental math when precision wasn’t needed
• Visual Pattern Recognition: After sorting, we could visually identify patterns without manual counting

Strategic Insights

1. The Sorting Superpower: For every statement, we asked “What column should I sort by to make this question trivial?” This transformed difficult questions into simple visual tasks.
2. Efficient Information Processing:
   • For Statement 1: Sort by birth year → visually identify 1700s composers → estimate total
   • For Statement 2: Sort by performances → top 3 instantly visible → estimate total
   • For Statement 3: Keep performance sort → median values immediately accessible
3. Know When Precision Matters:
   • For Statements 1 & 2: The differences were large enough that rough estimates were sufficient
   • For Statement 3: We needed slightly more precision since the values were closer

Common Mistakes We Avoided

1. Manual Searching: We didn’t scan the entire table to find composers born in specific periods
2. Unnecessary Calculations: We didn’t calculate exact totals when estimates clearly showed the relationship
3. Inefficient Median Finding: We didn’t manually arrange 20 values to find the median

Remember: In table analysis questions, sorting is your most powerful tool for transforming complex data analysis into simple visual scanning. Always ask yourself what column you should sort by to make the question easiest to answer.