Calculate the mean for the following data.

Answer

To calculate the mean for the given data, we will follow these steps:

### Step 1: Understand the formula for mean
The mean (average) of a data set is calculated using the formula:
\[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \]
where \(x_i\) is the value of the data point and \(f_i\) is the frequency of that data point.

### Step 2: Identify the data points and their frequencies
Assuming we have the following data points and their corresponding frequencies:
- \(x_1 = 2\), \(f_1 = 2\)
- \(x_2 = 4\), \(f_2 = 2\)
- \(x_3 = 6\), \(f_3 = 4\)
- \(x_4 = 8\), \(f_4 = 5\)
- \(x_5 = 10\), \(f_5 = 3\)
- \(x_6 = 12\), \(f_6 = 2\)
- \(x_7 = 14\), \(f_7 = 1\)
- \(x_8 = 16\), \(f_8 = 1\)

### Step 3: Calculate \(x_i \cdot f_i\) for each data point
Now, we will multiply each data point by its frequency:
- \(2 \cdot 2 = 4\)
- \(4 \cdot 2 = 8\)
- \(6 \cdot 4 = 24\)
- \(8 \cdot 5 = 40\)
- \(10 \cdot 3 = 30\)
- \(12 \cdot 2 = 24\)
- \(14 \cdot 1 = 14\)
- \(16 \cdot 1 = 16\)

### Step 4: Sum the products \(x_i \cdot f_i\)
Now, we will sum all the products calculated:
\[ 4 + 8 + 24 + 40 + 30 + 24 + 14 + 16 = 160 \]

### Step 5: Calculate the total frequency \(\sum f_i\)
Next, we will sum all the frequencies:
\[ 2 + 2 + 4 + 5 + 3 + 2 + 1 + 1 = 20 \]

### Step 6: Substitute values into the mean formula
Now, we can substitute the sums into the mean formula:
\[ \text{Mean} = \frac{160}{20} = 8 \]

### Final Answer
The required mean is \(8\).

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