Calculating the Third Quartile for 15 Observations

Understanding how to calculate quartiles is a fundamental concept in statistics. Quartiles divide a dataset into four equal parts. The first quartile (\(Q_1\)) represents the 25th percentile, the second quartile (\(Q_2\)) is the median (50th percentile), and the third quartile (\(Q_3\)) represents the 75th percentile.

To find the quartiles of a dataset, the first and most important step is to arrange the observations in ascending order.

For a dataset with \(n\) observations, the position of the quartiles can be found using specific formulas. The position of the third quartile (\(Q_3\)) is given by the formula:

\(\text{Position of } Q_3 = \frac{3(n+1)}{4}\)

Applying the Third Quartile Formula

In this question, the number of observations in the series is given as \(n = 15\). We need to find the third quartile.

Using the formula for the position of the third quartile (\(Q_3\)) with \(n=15\):

\(\text{Position of } Q_3 = \frac{3(15+1)}{4}\)

First, calculate the value inside the parentheses:

\(15+1 = 16\)

Now substitute this back into the formula:

\(\text{Position of } Q_3 = \frac{3 \times 16}{4}\)

Multiply 3 by 16:

\(3 \times 16 = 48\)

Finally, divide 48 by 4:

\(\text{Position of } Q_3 = \frac{48}{4} = 12\)

This result, 12, indicates that the third quartile (\(Q_3\)) is the 12th observation in the dataset.

Importance of Ordering the Data

It is crucial to remember that when calculating quartiles, the data must be arranged in ascending order. If the data is not ordered, calculating the position will not give the correct quartile value.

Analyzing the Options

Let's look at the given options based on our calculation that the third quartile's position is the 12th observation:

• Option 1: 9^th observation after arranging in descending order. This is incorrect. The position is 12, not 9, and the order should be ascending, not descending.
• Option 2: 11^th observation after arranging in ascending order. This is incorrect. The position is 12, not 11.
• Option 3: 10^th observation after arranging in descending order. This is incorrect. The position is 12, not 10, and the order should be ascending, not descending.
• Option 4: 12^th observation after arranging in ascending order. This matches our calculated position (12th) and the requirement for the data to be in ascending order.

Therefore, the third quartile is the 12th observation in the series after arranging the data in ascending order.

This method is commonly used for calculating quartiles when the number of observations is relatively small. If the position results in a fractional value, the quartile is usually found by interpolating between the two nearest observations.

Revision Table: Key Concepts for Quartiles

Additional Information on Quartile Calculation

Different methods exist for calculating quartiles, particularly regarding how fractional positions are handled or alternative formulas for position. The formula \(\frac{k(n+1)}{4}\) for the \(k\)-th quartile's position is one common method, often called Method 1 or the exclusive method.

Another method (often called Method 3 or the inclusive method) uses the formula \(\frac{kn}{4}\) for the position of the \(k\)-th quartile within the *sorted* data indices (starting from 1), sometimes rounded up, or using interpolation based on \(\frac{kn}{100}\) for the \(k\)-th percentile. However, the \(\frac{k(n+1)}{4}\) method is straightforward and leads directly to the position within the ordered list as \(k=1, 2, 3\).

For \(n=15\) and using the \(\frac{k(n+1)}{4}\) method:

• \(Q_1\) position: \(\frac{1(15+1)}{4} = \frac{16}{4} = 4\). So \(Q_1\) is the 4th observation in ascending order.
• \(Q_2\) position: \(\frac{2(15+1)}{4} = \frac{32}{4} = 8\). So \(Q_2\) is the 8th observation in ascending order.
• \(Q_3\) position: \(\frac{3(15+1)}{4} = \frac{48}{4} = 12\). So \(Q_3\) is the 12th observation in ascending order.

This calculation confirms the result for the third quartile.