For the data : 28, 17, 12, 25, 26, 19, 13, 27, 21, 16, the third quartile is

Options

A. 25.5
B. 26
C. 26.25
D. 26.75

Answer

To find the third quartile (Q3) of the given data set, we will follow these steps:

### Step 1: Arrange the Data in Ascending Order
First, we need to arrange the data in ascending order.

Given data: 28, 17, 12, 25, 26, 19, 13, 27, 21, 16

Arranging in ascending order:
12, 13, 16, 17, 19, 21, 25, 26, 27, 28

### Step 2: Determine the Number of Observations (n)
Next, we need to count the number of observations in the data set.

In this case, n = 10 (since there are 10 numbers in the data set).

### Step 3: Calculate the Position of Q3
The formula to find the position of the third quartile (Q3) is given by:
\[ Q3 = \frac{3(n + 1)}{4} \]

Substituting n = 10 into the formula:
\[ Q3 = \frac{3(10 + 1)}{4} = \frac{3 \times 11}{4} = \frac{33}{4} = 8.25 \]

### Step 4: Interpret the Position
The position 8.25 indicates that Q3 is located between the 8th and 9th observations.

### Step 5: Identify the 8th and 9th Observations
From the arranged data:
- 8th observation = 26
- 9th observation = 27

### Step 6: Calculate Q3
To find Q3, we will use the formula:
\[ Q3 = \text{8th observation} + 0.25 \times (\text{9th observation} - \text{8th observation}) \]

Substituting the values:
\[ Q3 = 26 + 0.25 \times (27 - 26) \]
\[ Q3 = 26 + 0.25 \times 1 \]
\[ Q3 = 26 + 0.25 = 26.25 \]

### Final Answer
Thus, the third quartile (Q3) is **26.25**.

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