The algebraic sum of the deviations of 20 observations measured from 30 is 2. What would be the mean of the observations?

Options

A. 30
B. 32
C. 30.2
D. 30.1

Explanation

To find the mean of the observations given that the algebraic sum of the deviations of 20 observations from 30 is 2, we can follow these steps:

### Step 1: Understand the concept of deviations
The deviation of each observation from a certain value (in this case, 30) is calculated as the difference between the observation and that value. The algebraic sum of deviations is the total of all these differences.

### Step 2: Set up the equation
Let \( n \) be the number of observations, which is given as \( n = 20 \). The algebraic sum of the deviations from 30 is given as 2. This can be expressed mathematically as:

\[ \sum (X_i - 30) = 2 \]

where \( X_i \) represents each observation.

### Step 3: Rewrite the equation
We can rewrite the equation as:

\[ \sum X_i - n \cdot 30 = 2 \]

Substituting \( n = 20 \):

\[ \sum X_i - 20 \cdot 30 = 2 \]

### Step 4: Calculate the sum of observations
Now, calculate \( 20 \cdot 30 \):

\[ 20 \cdot 30 = 600 \]

So, substituting this back into the equation:

\[ \sum X_i - 600 = 2 \]

### Step 5: Solve for the sum of observations
Now, we can solve for \( \sum X_i \):

\[ \sum X_i = 600 + 2 = 602 \]

### Step 6: Calculate the mean
The mean \( \bar{X} \) of the observations can be calculated using the formula:

\[ \bar{X} = \frac{\sum X_i}{n} \]

Substituting the values we have:

\[ \bar{X} = \frac{602}{20} \]

### Step 7: Perform the division
Now, perform the division:

\[ \bar{X} = 30.1 \]

Thus, the mean of the observations is **30.1**.