Find the mean of the data given in the following table.
`{:("Class",0-20,20-40,40-60,60-80,80-100),("Frequency"," "6," "4," "5," "7," "3):}`

Answer

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

### Step 1: Identify the Class Intervals and Frequencies
We have the following class intervals and their corresponding frequencies:
- Class Intervals: 0-20, 20-40, 40-60, 60-80, 80-100
- Frequencies (f): 6, 4, 5, 7, 3

### Step 2: Calculate the Midpoint (xi) for Each Class Interval
The midpoint (xi) for each class interval is calculated as:
\[ xi = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]

Calculating the midpoints:
- For 0-20: \( xi = \frac{0 + 20}{2} = 10 \)
- For 20-40: \( xi = \frac{20 + 40}{2} = 30 \)
- For 40-60: \( xi = \frac{40 + 60}{2} = 50 \)
- For 60-80: \( xi = \frac{60 + 80}{2} = 70 \)
- For 80-100: \( xi = \frac{80 + 100}{2} = 90 \)

### Step 3: Create a Table of Frequencies and Midpoints
| Class Interval | Frequency (f) | Midpoint (xi) |
|----------------|---------------|----------------|
| 0 - 20 | 6 | 10 |
| 20 - 40 | 4 | 30 |
| 40 - 60 | 5 | 50 |
| 60 - 80 | 7 | 70 |
| 80 - 100 | 3 | 90 |

### Step 4: Calculate the Product of Frequency and Midpoint (fi * xi)
Now, we will calculate \( fi \times xi \) for each class interval:
- For 0-20: \( 6 \times 10 = 60 \)
- For 20-40: \( 4 \times 30 = 120 \)
- For 40-60: \( 5 \times 50 = 250 \)
- For 60-80: \( 7 \times 70 = 490 \)
- For 80-100: \( 3 \times 90 = 270 \)

### Step 5: Create a Table of Frequencies, Midpoints, and their Products
| Class Interval | Frequency (f) | Midpoint (xi) | fi * xi |
|----------------|---------------|----------------|---------|
| 0 - 20 | 6 | 10 | 60 |
| 20 - 40 | 4 | 30 | 120 |
| 40 - 60 | 5 | 50 | 250 |
| 60 - 80 | 7 | 70 | 490 |
| 80 - 100 | 3 | 90 | 270 |

### Step 6: Calculate the Sums
Now, we will calculate the total of frequencies and the total of \( fi \times xi \):
- \( \Sigma f = 6 + 4 + 5 + 7 + 3 = 25 \)
- \( \Sigma (fi \times xi) = 60 + 120 + 250 + 490 + 270 = 1190 \)

### Step 7: Calculate the Mean
The mean (x̄) is calculated using the formula:
\[ \text{Mean} = \frac{\Sigma (fi \times xi)}{\Sigma f} \]
Substituting the values:
\[ \text{Mean} = \frac{1190}{25} = 47.6 \]

### Conclusion
Therefore, the mean of the data is **47.6**.

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