`{:(X,1,2,3,4),("Frequency",2,3,f,5):}` The frequency of a discrete variable X with one missing frequency f is given above. If the arithmetic mean of X is `(23)/(8)`, what is the value of the missing frequency?

Options

A. 5
B. 6
C. 8
D. 10

Explanation

To find the missing frequency \( f \) in the given data, we will follow these steps:

### Step 1: Identify the given data
We have the discrete variable \( X \) and its corresponding frequencies:
- \( X = \{1, 2, 3, 4\} \)
- Frequencies = \( \{2, 3, f, 5\} \)

### Step 2: Calculate \( \Sigma f \) (the sum of frequencies)
The sum of the frequencies is:
\[ \Sigma f = 2 + 3 + f + 5 = 10 + f \]

### Step 3: Calculate \( \Sigma xf \) (the sum of \( x \) multiplied by frequency)
Now, we calculate \( xf \) for each value of \( X \):
- For \( X = 1 \): \( 1 \times 2 = 2 \)
- For \( X = 2 \): \( 2 \times 3 = 6 \)
- For \( X = 3 \): \( 3 \times f = 3f \)
- For \( X = 4 \): \( 4 \times 5 = 20 \)

Now, summing these values:
\[ \Sigma xf = 2 + 6 + 3f + 20 = 28 + 3f \]

### Step 4: Use the formula for the arithmetic mean
The formula for the arithmetic mean \( \bar{X} \) of a discrete variable is given by:
\[ \bar{X} = \frac{\Sigma xf}{\Sigma f} \]
We know that the arithmetic mean is given as \( \frac{23}{8} \). Therefore, we can set up the equation:
\[ \frac{28 + 3f}{10 + f} = \frac{23}{8} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 8(28 + 3f) = 23(10 + f) \]

### Step 6: Expand both sides
Expanding both sides:
\[ 224 + 24f = 230 + 23f \]

### Step 7: Rearranging the equation
Now, we will rearrange the equation to isolate \( f \):
\[ 24f - 23f = 230 - 224 \]
\[ f = 6 \]

### Conclusion
The value of the missing frequency \( f \) is \( 6 \).

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