Quartile deviation for a frequency distribution is

Options

A. `Q=Q_(3)-Q_(1)`
B. `Q=1/2 (Q_(3)-Q_(i))`
C. `Q=1/3 (Q_(3)-Q_(1))`
D. `Q=1/4 (Q_(2)-Q_(1))`

Answer

To find the quartile deviation for a frequency distribution, we follow these steps:

### Step 1: Understand Quartiles
Quartiles divide a dataset into four equal parts. The first quartile (Q1) is the value below which 25% of the data fall, and the third quartile (Q3) is the value below which 75% of the data fall.

### Step 2: Calculate Q1 and Q3
For a frequency distribution, we need to calculate Q1 and Q3 using the cumulative frequency.

1. **Find the cumulative frequency** for the data.
2. **Locate Q1**: This is found at the position \( \frac{N}{4} \), where \( N \) is the total number of observations.
3. **Locate Q3**: This is found at the position \( \frac{3N}{4} \).

### Step 3: Use the Quartile Deviation Formula
The formula for quartile deviation (QD) is given by:

\[ QD = \frac{1}{2} (Q3 - Q1) \]

### Step 4: Substitute Values
Once you have the values of Q1 and Q3 from the previous step, substitute them into the formula to calculate the quartile deviation.

### Step 5: Interpret the Result
The quartile deviation gives a measure of the spread of the middle 50% of the data. A smaller quartile deviation indicates that the data points are closer to the median.

### Example Calculation
Assuming you have calculated Q1 = 10 and Q3 = 20:

1. Substitute into the formula:
\[ QD = \frac{1}{2} (20 - 10) = \frac{1}{2} \times 10 = 5 \]

### Final Answer
Thus, the quartile deviation for the frequency distribution is 5.

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