Complete the following activity to find the mean of the data.
`{:("Class","Class mark","Frequency",),(," "(x_(i))," "(f_(i)),x_(i) xx f_(i)),(0-20," "10," "6," "60),(20-40," "......," "......," "120),(40-60," "50," "5," "250),(60-80," "70," "7," "490),(80-100," "90," "......," "......),("Total",,sum f_(i) = 25,sum x_(i) f_(i) = 1190):}`

Answer

To find the mean of the given data, we need to complete the missing values in the table and then use the formula for the mean. Let's break down the steps systematically.

### Step 1: Identify the Class Marks and Frequencies
We have the following classes and their corresponding class marks and frequencies:

| Class | Class Mark (x_i) | Frequency (f_i) |
|-----------|------------------|-----------------|
| 0 - 20 | 10 | 6 |
| 20 - 40 | 30 | b |
| 40 - 60 | 50 | 5 |
| 60 - 80 | 70 | 7 |
| 80 - 100 | 90 | c |
| **Total** | | **25** |

### Step 2: Calculate the Missing Frequency (b)
We know the total frequency is 25. The frequencies we have are:
- 6 (for 0-20)
- b (for 20-40)
- 5 (for 40-60)
- 7 (for 60-80)
- c (for 80-100)

So, we can write the equation:
\[ 6 + b + 5 + 7 + c = 25 \]
This simplifies to:
\[ b + c + 18 = 25 \]
Thus,
\[ b + c = 7 \] (Equation 1)

### Step 3: Calculate the Missing Frequency (c)
Next, we need to calculate the total of \( x_i \times f_i \) for the known values:
- For 0-20: \( 10 \times 6 = 60 \)
- For 40-60: \( 50 \times 5 = 250 \)
- For 60-80: \( 70 \times 7 = 490 \)

Now, we can express the total of \( x_i \times f_i \):
\[ 60 + (30 \times b) + 250 + 490 + (90 \times c) = 1190 \]
This simplifies to:
\[ 800 + 30b + 90c = 1190 \]
Thus,
\[ 30b + 90c = 390 \]
Dividing the entire equation by 30 gives us:
\[ b + 3c = 13 \] (Equation 2)

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( b + c = 7 \) (Equation 1)
2. \( b + 3c = 13 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( b \) in terms of \( c \):
\[ b = 7 - c \]

Substituting this into Equation 2:
\[ (7 - c) + 3c = 13 \]
This simplifies to:
\[ 7 + 2c = 13 \]
Thus,
\[ 2c = 6 \]
So,
\[ c = 3 \]

Now substituting \( c \) back into Equation 1:
\[ b + 3 = 7 \]
Thus,
\[ b = 4 \]

### Step 5: Calculate \( x_i \times f_i \) for the Missing Frequencies
Now we can calculate the missing values:
- For 20-40: \( 30 \times 4 = 120 \)
- For 80-100: \( 90 \times 3 = 270 \)

### Step 6: Complete the Table
Now, we can fill in the completed table:

| Class | Class Mark (x_i) | Frequency (f_i) | \( x_i \times f_i \) |
|-----------|------------------|-----------------|-----------------------|
| 0 - 20 | 10 | 6 | 60 |
| 20 - 40 | 30 | 4 | 120 |
| 40 - 60 | 50 | 5 | 250 |
| 60 - 80 | 70 | 7 | 490 |
| 80 - 100 | 90 | 3 | 270 |
| **Total** | | **25** | **1190** |

### Step 7: Calculate the Mean
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\sum (x_i \times f_i)}{\sum f_i} \]
Substituting the values:
\[ \text{Mean} = \frac{1190}{25} = 47.6 \]

Thus, the mean of the data is **47.6**.