The table below gives an incomplete frequency distribution with two missing frequencies `f_(1) and f_(2)`
`|{:("Value of x","Frequency"),(0,f_(1)),(1,f_(2)),(2,4),(3,4),(4,3):}|`
The total frequency is 18 and the arithmetic mean of x is 2.
What is the standard deviation?

Options

A. `(sqrt5)/(2)`
B. `(sqrt5)/(3)`
C. `(4)/(3)`
D. `(16)/(9)`

Explanation

To solve the problem, we need to find the missing frequencies \( f_1 \) and \( f_2 \) and then calculate the standard deviation of the distribution. Let's break this down step by step.

### Step 1: Set Up the Frequency Distribution Table
We have the following values and frequencies:

| Value of x | Frequency |
|------------|-----------|
| 0 | \( f_1 \) |
| 1 | \( f_2 \) |
| 2 | 4 |
| 3 | 4 |
| 4 | 3 |

### Step 2: Write the Total Frequency Equation
The total frequency is given as 18. Therefore, we can write the equation:
\[ f_1 + f_2 + 4 + 4 + 3 = 18 \]
This simplifies to:
\[ f_1 + f_2 + 11 = 18 \]
Thus, we have:
\[ f_1 + f_2 = 7 \quad \text{(Equation 1)} \]

### Step 3: Calculate the Mean
The arithmetic mean \( \bar{x} \) is given as 2. The formula for the mean is:
\[ \bar{x} = \frac{\sum (x \cdot f)}{\sum f} \]
We need to calculate \( \sum (x \cdot f) \):
\[ \sum (x \cdot f) = 0 \cdot f_1 + 1 \cdot f_2 + 2 \cdot 4 + 3 \cdot 4 + 4 \cdot 3 \]
This simplifies to:
\[ \sum (x \cdot f) = 0 + f_2 + 8 + 12 + 12 = f_2 + 32 \]
Now substituting into the mean formula:
\[ 2 = \frac{f_2 + 32}{18} \]
Multiplying both sides by 18 gives:
\[ 36 = f_2 + 32 \]
Thus, we can solve for \( f_2 \):
\[ f_2 = 36 - 32 = 4 \quad \text{(Equation 2)} \]

### Step 4: Solve for \( f_1 \)
Now we can substitute \( f_2 \) back into Equation 1:
\[ f_1 + 4 = 7 \]
This gives us:
\[ f_1 = 7 - 4 = 3 \]

### Step 5: Complete the Frequency Distribution Table
Now we have:
\[ f_1 = 3, \quad f_2 = 4 \]
So the complete frequency distribution table is:

| Value of x | Frequency |
|------------|-----------|
| 0 | 3 |
| 1 | 4 |
| 2 | 4 |
| 3 | 4 |
| 4 | 3 |

### Step 6: Calculate the Standard Deviation
To find the standard deviation, we first need to calculate the mean again, which we already found to be 2.

Now we calculate \( \sum (x - \bar{x})^2 f \):
\[ \sum (x - 2)^2 f = (0 - 2)^2 \cdot 3 + (1 - 2)^2 \cdot 4 + (2 - 2)^2 \cdot 4 + (3 - 2)^2 \cdot 4 + (4 - 2)^2 \cdot 3 \]
Calculating each term:
- For \( x = 0 \): \( (0 - 2)^2 \cdot 3 = 4 \cdot 3 = 12 \)
- For \( x = 1 \): \( (1 - 2)^2 \cdot 4 = 1 \cdot 4 = 4 \)
- For \( x = 2 \): \( (2 - 2)^2 \cdot 4 = 0 \cdot 4 = 0 \)
- For \( x = 3 \): \( (3 - 2)^2 \cdot 4 = 1 \cdot 4 = 4 \)
- For \( x = 4 \): \( (4 - 2)^2 \cdot 3 = 4 \cdot 3 = 12 \)

Now summing these:
\[ \sum (x - 2)^2 f = 12 + 4 + 0 + 4 + 12 = 32 \]

### Step 7: Calculate the Variance and Standard Deviation
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum (x - \bar{x})^2 f}{n} \]
Where \( n = 18 \):
\[ \sigma^2 = \frac{32}{18} = \frac{16}{9} \]
Thus, the standard deviation \( \sigma \) is:
\[ \sigma = \sqrt{\frac{16}{9}} = \frac{4}{3} \]

### Final Answer
The standard deviation is \( \frac{4}{3} \).