The standard deviation of the observations 5, 5,5,5,5 is

Options

A. A) 0
B. B) 5
C. C) 20
D. D) 25

Answer

To find the standard deviation of the observations 5, 5, 5, 5, 5, we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the observations are 5, 5, 5, 5, 5.

**Sum of observations:**
\[ 5 + 5 + 5 + 5 + 5 = 25 \]

**Number of observations:**
There are 5 observations.

Now, substituting these values into the mean formula:
\[ \text{Mean} = \frac{25}{5} = 5 \]

### Step 2: Calculate the Deviations from the Mean
Next, we find the deviation of each observation from the mean:
\[ \text{Deviation} = \text{Observation} - \text{Mean} \]
For each observation (which is 5):
- Deviation for the first observation: \(5 - 5 = 0\)
- Deviation for the second observation: \(5 - 5 = 0\)
- Deviation for the third observation: \(5 - 5 = 0\)
- Deviation for the fourth observation: \(5 - 5 = 0\)
- Deviation for the fifth observation: \(5 - 5 = 0\)

So, the deviations are: 0, 0, 0, 0, 0.

### Step 3: Square the Deviations
Next, we square each of the deviations:
- Squared deviation for the first observation: \(0^2 = 0\)
- Squared deviation for the second observation: \(0^2 = 0\)
- Squared deviation for the third observation: \(0^2 = 0\)
- Squared deviation for the fourth observation: \(0^2 = 0\)
- Squared deviation for the fifth observation: \(0^2 = 0\)

So, the squared deviations are: 0, 0, 0, 0, 0.

### Step 4: Calculate the Variance
Variance is the average of these squared deviations. We use the formula:
\[ \text{Variance} = \frac{\text{Sum of squared deviations}}{\text{Number of observations}} \]
**Sum of squared deviations:**
\[ 0 + 0 + 0 + 0 + 0 = 0 \]

Now, substituting into the variance formula:
\[ \text{Variance} = \frac{0}{5} = 0 \]

### Step 5: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0} = 0 \]

### Final Answer
The standard deviation of the observations 5, 5, 5, 5, 5 is **0**.

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