What is the standard deviation of the following data ?
`{:("Measurement",0-10,10-20,20-30,30-40),("Frequency",1,3,4,2):}`

Options

A. 81
B. 7.6
C. 9
D. 2.26

Explanation

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

### Step 1: Create a Frequency Table
We will create a table with the given class intervals (measurements) and their corresponding frequencies.

| Measurement (X) | Frequency (F) |
|------------------|---------------|
| 0 - 10 | 1 |
| 10 - 20 | 3 |
| 20 - 30 | 4 |
| 30 - 40 | 2 |

### Step 2: Calculate Midpoints
Next, we will calculate the midpoints (X) for each class interval. The midpoint is calculated as:
\[ \text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2} \]

| Measurement (X) | Frequency (F) | Midpoint (x) |
|------------------|---------------|---------------|
| 0 - 10 | 1 | 5 |
| 10 - 20 | 3 | 15 |
| 20 - 30 | 4 | 25 |
| 30 - 40 | 2 | 35 |

### Step 3: Calculate \(F \cdot x\) and \(F \cdot x^2\)
Now, we will calculate \(F \cdot x\) and \(F \cdot x^2\) for each class.

| Measurement (X) | Frequency (F) | Midpoint (x) | \(F \cdot x\) | \(F \cdot x^2\) |
|------------------|---------------|---------------|----------------|------------------|
| 0 - 10 | 1 | 5 | 5 | 25 |
| 10 - 20 | 3 | 15 | 45 | 675 |
| 20 - 30 | 4 | 25 | 100 | 2500 |
| 30 - 40 | 2 | 35 | 70 | 2450 |

### Step 4: Calculate Summations
Now, we will calculate the summation of frequencies (\(N\)), \(F \cdot x\), and \(F \cdot x^2\).

- \(N = 1 + 3 + 4 + 2 = 10\)
- \(\Sigma F \cdot x = 5 + 45 + 100 + 70 = 220\)
- \(\Sigma F \cdot x^2 = 25 + 675 + 2500 + 2450 = 4650\)

### Step 5: Calculate the Mean (\(x̄\))
The mean (\(x̄\)) is calculated as:
\[ x̄ = \frac{\Sigma F \cdot x}{N} = \frac{220}{10} = 22 \]

### Step 6: Calculate Deviations and Their Squares
Now, we will calculate the deviations from the mean and their squares.

| Midpoint (x) | Deviation (x - x̄) | Squared Deviation \((x - x̄)^2\) |
|--------------|---------------------|-----------------------------------|
| 5 | 5 - 22 = -17 | 289 |
| 15 | 15 - 22 = -7 | 49 |
| 25 | 25 - 22 = 3 | 9 |
| 35 | 35 - 22 = 13 | 169 |

### Step 7: Calculate \(F \cdot (x - x̄)^2\)
Now, we will multiply the squared deviations by their corresponding frequencies.

| Frequency (F) | Squared Deviation \((x - x̄)^2\) | \(F \cdot (x - x̄)^2\) |
|---------------|-----------------------------------|-------------------------|
| 1 | 289 | 289 |
| 3 | 49 | 147 |
| 4 | 9 | 36 |
| 2 | 169 | 338 |

### Step 8: Calculate the Final Summation
Now, we will calculate the summation of \(F \cdot (x - x̄)^2\).

- \(\Sigma F \cdot (x - x̄)^2 = 289 + 147 + 36 + 338 = 810\)

### Step 9: Calculate the Variance and Standard Deviation
The variance (\(\sigma^2\)) is calculated as:
\[ \sigma^2 = \frac{\Sigma F \cdot (x - x̄)^2}{N} = \frac{810}{10} = 81 \]

Finally, the standard deviation (\(\sigma\)) is:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{81} = 9 \]

### Final Answer
The standard deviation of the given data is **9**.

---