`D_(5)` is always equal to

Options

A. `P_(1)`
B. `P_(10)`
C. `P_(25)`
D. `P_(50)`

Answer

To solve the question regarding the value of \( D_5 \) (the 5th decile), we can follow these steps:

### Step 1: Understand the Definitions
- **Deciles**: Deciles divide a data set into ten equal parts. The 5th decile, \( D_5 \), is the value below which 50% of the data falls.
- **Percentiles**: Percentiles divide a data set into 100 equal parts. The 50th percentile, also known as the median, is the value below which 50% of the data falls.

### Step 2: Establish the Relationship
From statistical definitions, we know that:
- The 5th decile \( D_5 \) is equivalent to the 50th percentile. This is because both represent the midpoint of the data set.

### Step 3: Identify the Options
The options given are:
- Option A: \( P_1 \) (1st percentile)
- Option B: \( P_{10} \) (10th percentile)
- Option C: \( P_{25} \) (25th percentile)
- Option D: \( P_{50} \) (50th percentile)

### Step 4: Compare \( D_5 \) with the Options
Since we established that \( D_5 \) is equal to the 50th percentile:
- \( D_5 = P_{50} \)

### Step 5: Conclusion
Thus, the correct answer is:
- **Option D: \( P_{50} \)**

### Final Answer
\( D_5 \) is always equal to \( P_{50} \).

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