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 solve the problem, we need to find the mean of 50 observations given that the sum of the deviations of these observations from 30 is 50.

### Step-by-Step Solution:

1. **Understanding Deviations**:
The deviation of each observation from 30 can be expressed as \( x_i = \text{observation}_i - 30 \). Therefore, the sum of the deviations from 30 can be written as:
\[ \sum_{i=1}^{50} (x_i) = \sum_{i=1}^{50} (\text{observation}_i - 30) = \sum_{i=1}^{50} \text{observation}_i - 30 \times 50 \]

2. **Given Information**:
We are given that the sum of the deviations is 50:
\[ \sum_{i=1}^{50} x_i = 50 \]

3. **Setting Up the Equation**:
From the previous step, we can set up the equation:
\[ \sum_{i=1}^{50} \text{observation}_i - 30 \times 50 = 50 \]
Simplifying this gives:
\[ \sum_{i=1}^{50} \text{observation}_i - 1500 = 50 \]

4. **Solving for the Sum of Observations**:
To find the sum of the observations, we rearrange the equation:
\[ \sum_{i=1}^{50} \text{observation}_i = 50 + 1500 = 1550 \]

5. **Calculating the Mean**:
The mean of the observations is calculated as:
\[ \text{Mean} = \frac{\sum_{i=1}^{50} \text{observation}_i}{50} = \frac{1550}{50} = 31 \]

Thus, the mean of the observations is **31**.