Marks obtained (in mathematics) by 9 students are given below: 60, 67, 52, 76, 50, 51, 74, 45 and 56
a. Find the arithmetic mean.
b. If marks of each student be increased by 4, what will be the new value of arithmetic mean?

Answer

To solve the problem step by step, we will find the arithmetic mean of the marks obtained by the 9 students and then determine the new mean if each student's marks are increased by 4.

### Step 1: Calculate the Arithmetic Mean

1. **List the Marks**: The marks obtained by the 9 students are:
- 60, 67, 52, 76, 50, 51, 74, 45, 56

2. **Sum the Marks**: Add all the marks together.
\[ \text{Total Marks} = 60 + 67 + 52 + 76 + 50 + 51 + 74 + 45 + 56 \]
\[ \text{Total Marks} = 60 + 67 = 127 \]
\[ 127 + 52 = 179 \]
\[ 179 + 76 = 255 \]
\[ 255 + 50 = 305 \]
\[ 305 + 51 = 356 \]
\[ 356 + 74 = 430 \]
\[ 430 + 45 = 475 \]
\[ 475 + 56 = 531 \]
So, the total marks are 531.

3. **Count the Number of Students**: The number of students (n) is 9.

4. **Calculate the Mean**: The arithmetic mean (x̄) is calculated using the formula:
\[ x̄ = \frac{\text{Total Marks}}{n} = \frac{531}{9} \]
Performing the division:
\[ x̄ = 59 \]

### Step 2: Calculate the New Mean After Increasing Marks

1. **Increase Each Mark by 4**: If each student's marks are increased by 4, the new marks will be:
- 60 + 4 = 64
- 67 + 4 = 71
- 52 + 4 = 56
- 76 + 4 = 80
- 50 + 4 = 54
- 51 + 4 = 55
- 74 + 4 = 78
- 45 + 4 = 49
- 56 + 4 = 60

2. **New Mean Calculation**: The new mean can be calculated as follows:
- Since we added 4 to each of the 9 marks, the total increase in marks is:
\[ \text{Total Increase} = 9 \times 4 = 36 \]
- Therefore, the new total marks will be:
\[ \text{New Total Marks} = 531 + 36 = 567 \]
- The new mean (x̄') is:
\[ x̄' = \frac{\text{New Total Marks}}{n} = \frac{567}{9} \]
Performing the division:
\[ x̄' = 63 \]

### Final Answers:
a. The arithmetic mean is **59**.
b. The new value of the arithmetic mean after increasing each mark by 4 is **63**.