Consider the following statement:
1. A continuous random variable can take all values in an interval.
2. A random variable which takes a finite number of values is necessarily discrete.
3. Construction of a frequency distribution is based on data which are discrete.
Which of the above statements are correct ?

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Explanation

To determine which of the statements regarding random variables are correct, let's analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1:** A continuous random variable can take all values in an interval.

**Explanation:**
A continuous random variable is defined as one that can take any value within a given range or interval. This means that between any two values, there are infinitely many possible values. For example, if we consider the time taken to complete a task, it can be any value in a range (like 1.5 hours, 1.55 hours, etc.), which makes it continuous.

**Conclusion:** Statement 1 is **correct**.

### Step 2: Analyze Statement 2
**Statement 2:** A random variable which takes a finite number of values is necessarily discrete.

**Explanation:**
A discrete random variable is one that can take on a countable number of distinct values. If a random variable can only take a finite number of values (like the outcome of rolling a die), it is indeed classified as discrete. Therefore, if a random variable takes a finite number of values, it must be discrete.

**Conclusion:** Statement 2 is **correct**.

### Step 3: Analyze Statement 3
**Statement 3:** Construction of a frequency distribution is based on data which are discrete.

**Explanation:**
A frequency distribution is a summary of how often each value occurs in a dataset. While it is true that frequency distributions are commonly constructed for discrete data, they can also be created for continuous data by grouping the data into intervals (bins). Thus, this statement is misleading because it implies that frequency distributions can only be created for discrete data, which is not the case.

**Conclusion:** Statement 3 is **incorrect**.

### Final Conclusion
- **Statement 1:** Correct
- **Statement 2:** Correct
- **Statement 3:** Incorrect

Thus, the correct statements are 1 and 2.