The mean of 22 numbers is 19. If two numbers 23 and 11 are removed, then what will be the mean of remaining number?

Options

A. `18.6`
B. `19.2`
C. `19.4`
D. `18.2`

Answer

To solve the problem step by step, we will follow the given information and apply the necessary calculations.

### Step 1: Calculate the sum of the 22 numbers.
The mean of 22 numbers is given as 19. To find the sum of these numbers, we use the formula for mean:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} \]

Rearranging this formula gives us:

\[ \text{Sum of observations} = \text{Mean} \times \text{Total number of observations} \]

Substituting the values:

\[ \text{Sum of observations} = 19 \times 22 = 418 \]

### Step 2: Remove the two numbers from the sum.
The two numbers to be removed are 23 and 11. First, we calculate their total:

\[ \text{Total of removed numbers} = 23 + 11 = 34 \]

Now, we subtract this total from the sum of the 22 numbers:

\[ \text{New sum} = 418 - 34 = 384 \]

### Step 3: Calculate the mean of the remaining numbers.
After removing the two numbers, we have 20 numbers left. We can now calculate the mean of these remaining numbers using the same mean formula:

\[ \text{Mean of remaining numbers} = \frac{\text{New sum}}{\text{Total number of remaining observations}} \]

Substituting the values:

\[ \text{Mean of remaining numbers} = \frac{384}{20} \]

### Step 4: Perform the division.
Now we perform the division:

\[ \frac{384}{20} = 19.2 \]

### Conclusion:
The mean of the remaining 20 numbers is **19.2**.

---