The mean of 5 observations is 10. If each observation of the data is increased by 5, the new mean is :

Options

A. 50
B. 15
C. 5
D. No change,10

Answer

To solve the problem step by step, we will first analyze the given information and then calculate the new mean after increasing each observation by 5.

### Step 1: Understand the given information
We know that the mean of 5 observations is 10.

### Step 2: Calculate the sum of the observations
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Let the sum of the observations be \( S \). Since the mean is given as 10 and the number of observations is 5, we can set up the equation:
\[ 10 = \frac{S}{5} \]
Multiplying both sides by 5 gives:
\[ S = 10 \times 5 = 50 \]

### Step 3: Increase each observation by 5
If each observation is increased by 5, the new observations can be represented as:
\[ x_1 + 5, \, x_2 + 5, \, x_3 + 5, \, x_4 + 5, \, x_5 + 5 \]

### Step 4: Calculate the new sum of observations
The new sum of the observations will be:
\[ (x_1 + 5) + (x_2 + 5) + (x_3 + 5) + (x_4 + 5) + (x_5 + 5) \]
This can be simplified to:
\[ (x_1 + x_2 + x_3 + x_4 + x_5) + (5 + 5 + 5 + 5 + 5) \]
Thus, the new sum \( S' \) is:
\[ S' = S + 25 \]
Substituting the value of \( S \):
\[ S' = 50 + 25 = 75 \]

### Step 5: Calculate the new mean
Now, we can find the new mean using the new sum:
\[ \text{New Mean} = \frac{S'}{\text{Number of observations}} = \frac{75}{5} \]
Calculating this gives:
\[ \text{New Mean} = 15 \]

### Final Answer
The new mean after increasing each observation by 5 is **15**.

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