In a series of 2n observations, half of them equals to a and remaining half equals to -a. If the standard deviation of the observations is 2, then find |a|.

Answer

To solve the problem, we need to find the value of |a| given that the standard deviation of a series of observations is 2. The observations consist of half being equal to 'a' and the other half being equal to '-a'.

### Step-by-Step Solution:

1. **Understand the Observations**:
- We have a total of 2n observations.
- Half of them (n observations) are equal to 'a'.
- The other half (n observations) are equal to '-a'.

2. **Calculate the Mean (X̄)**:
- The mean (X̄) of the observations can be calculated as follows:
\[ X̄ = \frac{\text{Sum of all observations}}{\text{Total number of observations}} = \frac{n \cdot a + n \cdot (-a)}{2n} = \frac{0}{2n} = 0 \]
- Thus, the mean of the observations is 0.

3. **Standard Deviation Formula**:
- The formula for the standard deviation (σ) is given by:
\[ σ = \sqrt{\frac{\sum (X - X̄)^2}{N}} \]
- Here, N is the total number of observations (2n), and X̄ is the mean (which we found to be 0).

4. **Substituting Values into the Formula**:
- Since X̄ = 0, we can rewrite the formula as:
\[ σ = \sqrt{\frac{\sum X^2}{N}} = \sqrt{\frac{n \cdot a^2 + n \cdot (-a)^2}{2n}} \]
- Simplifying this gives:
\[ σ = \sqrt{\frac{n \cdot a^2 + n \cdot a^2}{2n}} = \sqrt{\frac{2n \cdot a^2}{2n}} = \sqrt{a^2} \]

5. **Setting the Standard Deviation**:
- We know from the problem statement that the standard deviation is 2:
\[ \sqrt{a^2} = 2 \]
- This implies:
\[ |a| = 2 \]

### Final Answer:
\[ |a| = 2 \]