If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is

Options

A. 50
B. 51
C. 30
D. 31

Answer

To find the mean of the 50 observations given that the sum of the deviations from 30 is 50, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Deviations**:
The deviation of each observation from 30 can be expressed as \( x_i = a_i - 30 \), where \( a_i \) is the \( i^{th} \) observation.

2. **Sum of Deviations**:
The problem states that the sum of the deviations from 30 is 50. Mathematically, this can be expressed as:
\[ \sum_{i=1}^{50} (a_i - 30) = 50 \]

3. **Expanding the Sum**:
We can expand the sum:
\[ \sum_{i=1}^{50} a_i - \sum_{i=1}^{50} 30 = 50 \]
Since there are 50 observations, we have:
\[ \sum_{i=1}^{50} a_i - 30 \times 50 = 50 \]

4. **Calculating the Total Sum**:
Simplifying the equation gives:
\[ \sum_{i=1}^{50} a_i - 1500 = 50 \]
Adding 1500 to both sides results in:
\[ \sum_{i=1}^{50} a_i = 1550 \]

5. **Finding the Mean**:
The mean \( \bar{a} \) of the observations is calculated as:
\[ \bar{a} = \frac{\sum_{i=1}^{50} a_i}{50} = \frac{1550}{50} = 31 \]

### Conclusion:
The mean of the observations is **31**.