The mean of 12 values of a data is calculated as 19.25. If one more value is included in the data, then for the 13 values of the new data the mean becomes 20. Find the value of 13th observation.

Options

A. 29
B. 30
C. 39
D. 20

Answer

To find the value of the 13th observation, we can follow these steps:

### Step 1: Calculate the sum of the first 12 observations.
Given that the mean of the first 12 observations is 19.25, we can use the formula for the mean:

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

Let the sum of the 12 observations be \( S_{12} \). Therefore, we have:

\[ 19.25 = \frac{S_{12}}{12} \]

To find \( S_{12} \), we can multiply both sides by 12:

\[ S_{12} = 19.25 \times 12 \]

Calculating this gives:

\[ S_{12} = 231 \]

### Step 2: Set up the equation for the new mean with the 13th observation.
Let the 13th observation be \( x \). When this observation is added, the new sum of the observations becomes:

\[ S_{13} = S_{12} + x = 231 + x \]

The new mean for 13 observations is given as 20. Therefore, we can set up the equation:

\[ 20 = \frac{S_{13}}{13} \]

Substituting \( S_{13} \) into the equation gives:

\[ 20 = \frac{231 + x}{13} \]

### Step 3: Solve for \( x \).
To eliminate the fraction, multiply both sides by 13:

\[ 20 \times 13 = 231 + x \]

Calculating the left side gives:

\[ 260 = 231 + x \]

Now, isolate \( x \) by subtracting 231 from both sides:

\[ x = 260 - 231 \]

Calculating this gives:

\[ x = 29 \]

### Conclusion:
The value of the 13th observation is \( 29 \).

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