Question

The standard deviation of the observations 1; 2; 3; 4; 4; 5; 5; 5; 8 is

• A. \( 2\sqrt{2} \)
• B. \( \sqrt{2} \)
• C. 1.5
• D. 2

Hint:

To calculate the standard deviation, first find the mean, then calculate the squared differences, find the variance, and take the square root of the variance.

Answer 1

The Correct Option is D

Step 1: Calculate the mean.
First, calculate the mean of the data set: \[ \text{Mean} = \frac{1 + 2 + 3 + 4 + 4 + 5 + 5 + 5 + 8}{9} = \frac{37}{9} \approx 4.11 \]
Step 2: Calculate the squared differences from the mean.
Next, calculate the squared differences from the mean for each observation: \[ (1 - 4.11)^2 \approx 9.61, \quad (2 - 4.11)^2 \approx 4.45, \quad (3 - 4.11)^2 \approx 1.23 \] \[ (4 - 4.11)^2 \approx 0.01, \quad (4 - 4.11)^2 \approx 0.01, \quad (5 - 4.11)^2 \approx 0.79 \] \[ (5 - 4.11)^2 \approx 0.79, \quad (5 - 4.11)^2 \approx 0.79, \quad (8 - 4.11)^2 \approx 15.45 \]
Step 3: Calculate the variance.
Variance is the average of these squared differences: \[ \text{Variance} = \frac{9.61 + 4.45 + 1.23 + 0.01 + 0.01 + 0.79 + 0.79 + 0.79 + 15.45}{9} \approx 4.5 \]
Step 4: Calculate the standard deviation.
The standard deviation is the square root of the variance: \[ \text{Standard deviation} = \sqrt{4.5} \approx 2 \]
Step 5: Conclusion.
Therefore, the standard deviation is \( \boxed{2} \). The correct answer is (4) 2.