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The median is a key concept in descriptive statistics. It represents the middle value in a dataset when the numbers are arranged in order. To find the median, you first organize the data from the smallest to largest value. If you have an odd number of values, the median is simply the middle number. However, when dealing with an even number of data points, the median is determined by averaging the two middle numbers.

In the context of vacation days from different countries, the ordered list is: 13, 25, 26, 28, 34, 35, 37, and 42. There are 8 values in total, an even number. Thus, the median is calculated by averaging the 4th and 5th values:

• 4th value is 28 (Britain)
• 5th value is 34 (Brazil)

Average these two numbers to get the median: \[\frac{28 + 34}{2} = 31\]Therefore, the median number of vacation days is 31. This means half of the countries have more than 31 days of vacation, and half have fewer.

The first quartile, often denoted as \( Q_1 \), is a measure that helps you understand the dataset’s spread by identifying the lower 25% of the data. Essentially, it is the median of the lower half of the dataset, excluding the overall median.

For our dataset of vacation days, we first focus on the lower half: 13, 25, 26, and 28 days. To find the first quartile, determine the median of these numbers:

• The 2nd value is 25 (Japan)
• The 3rd value is 26 (Canada)

Average these two values:\[\frac{25 + 26}{2} = 25.5\]This result tells us that 25% of the countries have 25.5 or fewer vacation days each year. It gives us a sense of how data is spread in the lower segment of the dataset.

The third quartile, \( Q_3 \), is another useful measure in statistics that captures the higher end of the data distribution. By identifying this upper 25%, it provides insight into the top portion of the dataset. The third quartile is the median of the upper half of the dataset, excluding the overall median.

In our exercise, the upper half of the data is: 34, 35, 37, and 42 days. To find the third quartile, we take the median of these four numbers:

• The 2nd value in this half is 35 (Germany)
• The 3rd value is 37 (France)

Calculating the average:\[\frac{35 + 37}{2} = 36\]Thus, the third quartile stands at 36 days. This means that 75% of the countries listed have less than or equal to 36 vacation days annually. Understanding this helps in identifying the countries with a higher number of vacation days and how they compare with the rest.