Find the quartile deviation of the following discrete series.
`{:(x,3,5,6,8,10,12),(f,7,2,3,4,5,6):}`

Options

A. 4
B. 3
C. 3.5
D. 4.5

Explanation

To find the quartile deviation of the given discrete series, we will follow these steps:

### Step 1: Organize the Data
The given data is:
- \( x = \{3, 5, 6, 8, 10, 12\} \)
- \( f = \{7, 2, 3, 4, 5, 6\} \)

### Step 2: Calculate Cumulative Frequency
We will calculate the cumulative frequency for the given frequencies \( f \):
- Cumulative frequency for \( 3 \): \( 7 \)
- Cumulative frequency for \( 5 \): \( 7 + 2 = 9 \)
- Cumulative frequency for \( 6 \): \( 9 + 3 = 12 \)
- Cumulative frequency for \( 8 \): \( 12 + 4 = 16 \)
- Cumulative frequency for \( 10 \): \( 16 + 5 = 21 \)
- Cumulative frequency for \( 12 \): \( 21 + 6 = 27 \)

Thus, the cumulative frequency table is:
- \( 3 \) : \( 7 \)
- \( 5 \) : \( 9 \)
- \( 6 \) : \( 12 \)
- \( 8 \) : \( 16 \)
- \( 10 \) : \( 21 \)
- \( 12 \) : \( 27 \)

### Step 3: Find \( n \)
The total frequency \( n \) is the last cumulative frequency, which is \( 27 \).

### Step 4: Calculate \( Q_1 \) (First Quartile)
Using the formula for the first quartile:
\[ Q_1 = \frac{n + 1}{4} \]
Substituting \( n = 27 \):
\[ Q_1 = \frac{27 + 1}{4} = \frac{28}{4} = 7 \]
Now, we look for the cumulative frequency just greater than or equal to \( 7 \). The smallest cumulative frequency greater than \( 7 \) is \( 9 \), which corresponds to \( x = 5 \). Therefore, \( Q_1 = 5 \).

### Step 5: Calculate \( Q_3 \) (Third Quartile)
Using the formula for the third quartile:
\[ Q_3 = \frac{3(n + 1)}{4} \]
Substituting \( n = 27 \):
\[ Q_3 = \frac{3(27 + 1)}{4} = \frac{3 \times 28}{4} = \frac{84}{4} = 21 \]
Now, we look for the cumulative frequency just greater than or equal to \( 21 \). The smallest cumulative frequency greater than \( 21 \) is \( 27 \), which corresponds to \( x = 12 \). Therefore, \( Q_3 = 12 \).

### Step 6: Calculate Quartile Deviation
Now, we can calculate the quartile deviation using the formula:
\[ \text{Quartile Deviation} = \frac{Q_3 - Q_1}{2} \]
Substituting \( Q_3 = 12 \) and \( Q_1 = 5 \):
\[ \text{Quartile Deviation} = \frac{12 - 5}{2} = \frac{7}{2} = 3.5 \]

### Final Answer
The quartile deviation of the given discrete series is \( 3.5 \).

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