Find the quartile deviation for the following grouped data.
`{:("Class Interval",f),(" "0-4,1),(" "5-9,3),(" "10-14,2),(" "15-19,4),(" "20-24,5),(" "25-29,3),(" "30-34,2),(" "35-39,4),(" "40-44,3),(" "45-49,3):}`

Answer

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

### Step 1: Organize the Data
We have the following class intervals and their corresponding frequencies:

| Class Interval | Frequency (f) |
|----------------|---------------|
| 0 - 4 | 1 |
| 5 - 9 | 3 |
| 10 - 14 | 2 |
| 15 - 19 | 4 |
| 20 - 24 | 5 |
| 25 - 29 | 3 |
| 30 - 34 | 2 |
| 35 - 39 | 4 |
| 40 - 44 | 3 |
| 45 - 49 | 3 |

### Step 2: Calculate Cumulative Frequency
We will calculate the cumulative frequency (cf) for each class interval.

| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
|----------------|---------------|----------------------------|
| 0 - 4 | 1 | 1 |
| 5 - 9 | 3 | 4 |
| 10 - 14 | 2 | 6 |
| 15 - 19 | 4 | 10 |
| 20 - 24 | 5 | 15 |
| 25 - 29 | 3 | 18 |
| 30 - 34 | 2 | 20 |
| 35 - 39 | 4 | 24 |
| 40 - 44 | 3 | 27 |
| 45 - 49 | 3 | 30 |

### Step 3: Calculate Total Frequency (n)
The total frequency \( n \) is the sum of all frequencies:
\[ n = 1 + 3 + 2 + 4 + 5 + 3 + 2 + 4 + 3 + 3 = 30 \]

### Step 4: Determine Q1 and Q3
- **Q1** (First Quartile):
\[ Q1 = L + \left(\frac{n}{4} - cf\right) \times \frac{h}{f} \]
Where:
- \( L \) = lower limit of the Q1 class
- \( n/4 = 30/4 = 7.5 \)

The cumulative frequency just before 7.5 is 6 (from the class 10-14). Thus, the Q1 class is 15-19.
- \( L = 15 \)
- \( cf = 6 \) (cumulative frequency of the previous class)
- \( f = 4 \) (frequency of the Q1 class)
- \( h = 5 - 0 = 5 \) (class width)

Now substituting:
\[ Q1 = 15 + \left(7.5 - 6\right) \times \frac{5}{4} \]
\[ Q1 = 15 + 1.5 \]
\[ Q1 = 16.5 \]

- **Q3** (Third Quartile):
\[ Q3 = L + \left(\frac{3n}{4} - cf\right) \times \frac{h}{f} \]
Where:
- \( 3n/4 = 3 \times 30/4 = 22.5 \)

The cumulative frequency just before 22.5 is 20 (from the class 30-34). Thus, the Q3 class is 35-39.
- \( L = 35 \)
- \( cf = 20 \) (cumulative frequency of the previous class)
- \( f = 4 \) (frequency of the Q3 class)
- \( h = 5 \)

Now substituting:
\[ Q3 = 35 + \left(22.5 - 20\right) \times \frac{5}{4} \]
\[ Q3 = 35 + 2.5 \]
\[ Q3 = 37.5 \]

### Step 5: Calculate Quartile Deviation
The quartile deviation (QD) is given by:
\[ QD = \frac{Q3 - Q1}{2} \]
Substituting the values:
\[ QD = \frac{37.5 - 16.5}{2} \]
\[ QD = \frac{21}{2} \]
\[ QD = 10.5 \]

### Final Answer:
The quartile deviation for the given grouped data is **10.5**.

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