Find the quartile deviation of the following data:
`{:(x,f," "x,f),(2,4," "13,2),(3,6," "17,4),(5,8," "19,6),(7,9," "23,6),(11,10," ",):}`

Answer

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

### Step 1: Organize the Data
We have the data in the form of (x, f) pairs:
- (2, 4)
- (3, 6)
- (5, 8)
- (7, 9)
- (11, 10)

### Step 2: Calculate Cumulative Frequency
We need to calculate the cumulative frequency (cf) for the given data.

| x | f | Cumulative Frequency (cf) |
|----|---|----------------------------|
| 2 | 4 | 4 |
| 3 | 6 | 4 + 6 = 10 |
| 5 | 8 | 10 + 8 = 18 |
| 7 | 9 | 18 + 9 = 27 |
| 11 | 10| 27 + 10 = 37 |

So, the cumulative frequency is:
- cf for x=2 is 4
- cf for x=3 is 10
- cf for x=5 is 18
- cf for x=7 is 27
- cf for x=11 is 37

### Step 3: Find n (Total Frequency)
The total frequency \( n \) is the last value of the cumulative frequency, which is \( n = 37 \).

### Step 4: Calculate Q1 (First Quartile)
To find Q1, we use the formula:
\[ Q1 = \frac{n + 1}{4} \]
Substituting the value of \( n \):
\[ Q1 = \frac{37 + 1}{4} = \frac{38}{4} = 9.5 \]

Now, we look for the cumulative frequency that exceeds 9.5. The cumulative frequency just above 9.5 is 10, which corresponds to \( x = 3 \). Therefore, \( Q1 = 3 \).

### Step 5: Calculate Q3 (Third Quartile)
To find Q3, we use the formula:
\[ Q3 = \frac{3(n + 1)}{4} \]
Substituting the value of \( n \):
\[ Q3 = \frac{3(37 + 1)}{4} = \frac{3 \times 38}{4} = \frac{114}{4} = 28.5 \]

Now, we look for the cumulative frequency that exceeds 28.5. The cumulative frequency just above 28.5 is 37, which corresponds to \( x = 11 \). Therefore, \( Q3 = 11 \).

### Step 6: Calculate Quartile Deviation
The quartile deviation (QD) is calculated using the formula:
\[ QD = \frac{Q3 - Q1}{2} \]
Substituting the values of \( Q3 \) and \( Q1 \):
\[ QD = \frac{11 - 3}{2} = \frac{8}{2} = 4 \]

### Final Answer
The quartile deviation of the given data is **4**.

---