The standard deviation of 26, 27, 31, 32, 35 is

Options

A. 0
B. `sqrt((16)/(5))`
C. `sqrt((274)/(25))`
D. `sqrt((58)/(5))`

Answer

To find the standard deviation of the data set \(26, 27, 31, 32, 35\), we will follow these steps:

### Step 1: Calculate the Mean
The mean (\(\bar{x}\)) is calculated using the formula:
\[ \bar{x} = \frac{\text{Sum of all values}}{n} \]
where \(n\) is the number of values.

**Calculation:**
\[ \text{Sum} = 26 + 27 + 31 + 32 + 35 = 151 \]
\[ n = 5 \]
\[ \bar{x} = \frac{151}{5} = 30.2 \]

### Step 2: Calculate the Deviations from the Mean
Next, we calculate the deviations of each value from the mean:
\[ X - \bar{x} \]

**Calculations:**
- For \(26\): \(26 - 30.2 = -4.2\)
- For \(27\): \(27 - 30.2 = -3.2\)
- For \(31\): \(31 - 30.2 = 0.8\)
- For \(32\): \(32 - 30.2 = 1.8\)
- For \(35\): \(35 - 30.2 = 4.8\)

### Step 3: Square the Deviations
Now, we square each of the deviations:
\[ (X - \bar{x})^2 \]

**Calculations:**
- For \(26\): \((-4.2)^2 = 17.64\)
- For \(27\): \((-3.2)^2 = 10.24\)
- For \(31\): \((0.8)^2 = 0.64\)
- For \(32\): \((1.8)^2 = 3.24\)
- For \(35\): \((4.8)^2 = 23.04\)

### Step 4: Calculate the Variance
The variance (\(s^2\)) is the average of these squared deviations:
\[ s^2 = \frac{\sum (X - \bar{x})^2}{n} \]

**Calculation:**
\[ \text{Sum of squared deviations} = 17.64 + 10.24 + 0.64 + 3.24 + 23.04 = 54.8 \]
\[ s^2 = \frac{54.8}{5} = 10.96 \]

### Step 5: Calculate the Standard Deviation
The standard deviation (\(s\)) is the square root of the variance:
\[ s = \sqrt{s^2} \]

**Calculation:**
\[ s = \sqrt{10.96} \approx 3.31 \]

### Final Answer
The standard deviation of the data set \(26, 27, 31, 32, 35\) is approximately \(3.31\).

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