Find The standard deviation of the following frequency distribution is
`{:(X,2,3,4,5,6,7),(f,4,9,16,14,11,6):}`

Options

A. 1.38
B. 1.42
C. 1.45
D. `1.60`

Explanation

To find the standard deviation of the given frequency distribution, we will follow these steps:

### Step 1: Create a frequency table
We will create a table with the values of \(X\), their corresponding frequencies \(f\), the deviation \(d\) (where \(d = X - A\) and \(A\) is the assumed mean), \(f \cdot d\), and \(f \cdot d^2\).

Given:
- \(X = \{2, 3, 4, 5, 6, 7\}\)
- \(f = \{4, 9, 16, 14, 11, 6\}\)

Assume \(A = 4\) (the mean value).

| \(X\) | \(f\) | \(d = X - A\) | \(f \cdot d\) | \(f \cdot d^2\) |
|-------|-------|----------------|----------------|------------------|
| 2 | 4 | -2 | -8 | 16 |
| 3 | 9 | -1 | -9 | 9 |
| 4 | 16 | 0 | 0 | 0 |
| 5 | 14 | 1 | 14 | 14 |
| 6 | 11 | 2 | 22 | 44 |
| 7 | 6 | 3 | 18 | 54 |

### Step 2: Calculate the sums
Now we will calculate the sums of \(f\), \(f \cdot d\), and \(f \cdot d^2\).

- \(N = \sum f = 4 + 9 + 16 + 14 + 11 + 6 = 60\)
- \(\sum (f \cdot d) = -8 - 9 + 0 + 14 + 22 + 18 = 37\)
- \(\sum (f \cdot d^2) = 16 + 9 + 0 + 14 + 44 + 54 = 137\)

### Step 3: Apply the standard deviation formula
The formula for standard deviation \(\sigma\) for a frequency distribution is given by:

\[ \sigma = \sqrt{\frac{\sum (f \cdot d^2)}{N} - \left(\frac{\sum (f \cdot d)}{N}\right)^2} \]

Substituting the values we calculated:

\[ \sigma = \sqrt{\frac{137}{60} - \left(\frac{37}{60}\right)^2} \]

### Step 4: Calculate each component
1. Calculate \(\frac{137}{60}\):
\[ \frac{137}{60} \approx 2.2833 \]

2. Calculate \(\frac{37}{60}\):
\[ \frac{37}{60} \approx 0.6167 \]

3. Now, square \(\frac{37}{60}\):
\[ \left(\frac{37}{60}\right)^2 \approx 0.3794 \]

### Step 5: Final calculation
Now substitute back into the formula:

\[ \sigma = \sqrt{2.2833 - 0.3794} \approx \sqrt{1.9039} \approx 1.38 \]

### Conclusion
The standard deviation of the given frequency distribution is approximately \(1.38\).

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