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For each of 20 composers, the table lists the year of birth, the year of death, and the number of performances in 2010 of works written by that composer.
| Composer | Year of birth | Year of death | Number of performances |
|---|---|---|---|
| Bach | 1685 | 1750 | 1,388 |
| Beethoven | 1770 | 1827 | 2,859 |
| Brahms | 1833 | 1897 | 1,848 |
| Chopin | 1810 | 1849 | 748 |
| Dvořák | 1841 | 1904 | 1,071 |
| Handel | 1685 | 1759 | 757 |
| Haydn | 1732 | 1809 | 989 |
| Mendelssohn | 1809 | 1847 | 865 |
| Mozart | 1756 | 1791 | 3,035 |
| Prokofiev | 1891 | 1953 | 931 |
| Puccini | 1858 | 1924 | 723 |
| Rachmaninov | 1873 | 1943 | 673 |
| Ravel | 1875 | 1937 | 864 |
| Schubert | 1797 | 1828 | 1,023 |
| Schumann | 1810 | 1856 | 1,460 |
| Shostakovich | 1906 | 1975 | 779 |
| Strauss | 1864 | 1949 | 720 |
| Stravinsky | 1882 | 1971 | 849 |
| Tchaikovsky | 1840 | 1893 | 2,295 |
| Verdi | 1813 | 1901 | 845 |
| Total | ---- | --- | 24,722 |
For each of the following statements, select Yes if the statement accurately reflects the information in the table; otherwise select No.
Composers born in the 1700s accounted for less than a third of the total performances.
Music by composers ranked first, second, and third, in terms of number of performances, together accounted for more than half of the total number of performances.
The mean number of performances per composer exceeded the median by more than 300 performances.
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:
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:
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.
Statement 1 Translation:
Original: “The composers born in the 1700s account for more than one-third of the total performances.â€
What we’re looking for:
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:
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.
Statement 2 Translation:
Original: “The three composers with the most performances account for more than half of the total performances.â€
What we’re looking for:
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:
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.
Statement 3 Translation:
Original: “The mean number of performances exceeds the median by more than 300.â€
What we’re looking for:
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:
\(\mathrm{Median} = \frac{931 + 865}{2} = \mathrm{approximately}\ 900\)
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.
Reviewing our analysis:
The answer is therefore: No, No, Yes
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.
Composers born in the 1700s accounted for less than a third of the total performances.
Music by composers ranked first, second, and third, in terms of number of performances, together accounted for more than half of the total number of performances.
The mean number of performances per composer exceeded the median by more than 300 performances.