Find the mean for the following data.
`{:("Marks obtained",10-20,20-30,30-40,40-50,50-60,60-70),("Number of students",8,6,12,5,2,7):}`

Answer

To find the mean of the given data, we will follow these steps:

### Step 1: Identify the Class Intervals and Frequencies
The class intervals and their corresponding frequencies (number of students) are provided as follows:

- Marks obtained (class intervals): 10-20, 20-30, 30-40, 40-50, 50-60, 60-70
- Number of students (frequencies): 8, 6, 12, 5, 2, 7

### Step 2: Calculate the Midpoint (X) for Each Class Interval
The midpoint for each class interval can be calculated using the formula:
\[ X = \frac{\text{Lower limit} + \text{Upper limit}}{2} \]

Calculating the midpoints:
- For 10-20: \( X = \frac{10 + 20}{2} = 15 \)
- For 20-30: \( X = \frac{20 + 30}{2} = 25 \)
- For 30-40: \( X = \frac{30 + 40}{2} = 35 \)
- For 40-50: \( X = \frac{40 + 50}{2} = 45 \)
- For 50-60: \( X = \frac{50 + 60}{2} = 55 \)
- For 60-70: \( X = \frac{60 + 70}{2} = 65 \)

### Step 3: Calculate the Product of Midpoint and Frequency (XF)
Now, we will calculate the product of each midpoint (X) and its corresponding frequency (F):

- For 10-20: \( XF = 15 \times 8 = 120 \)
- For 20-30: \( XF = 25 \times 6 = 150 \)
- For 30-40: \( XF = 35 \times 12 = 420 \)
- For 40-50: \( XF = 45 \times 5 = 225 \)
- For 50-60: \( XF = 55 \times 2 = 110 \)
- For 60-70: \( XF = 65 \times 7 = 455 \)

### Step 4: Calculate the Total of Frequencies (F) and Total of XF
Now, we will sum up the frequencies and the XF values:

- Total of frequencies (F):
\[ F = 8 + 6 + 12 + 5 + 2 + 7 = 40 \]

- Total of XF:
\[ \text{Total } XF = 120 + 150 + 420 + 225 + 110 + 455 = 1480 \]

### Step 5: Calculate the Mean
The mean can be calculated using the formula:
\[ \text{Mean} (\bar{X}) = \frac{\sum XF}{\sum F} \]

Substituting the values we calculated:
\[ \bar{X} = \frac{1480}{40} = 37 \]

### Conclusion
The mean of the given data is **37**.

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