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.

Answer 2

Rather than computing each deviation from the mean individually, we can use the shortcut variance formula \( \text{Var} = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2 \), which avoids working with the decimal mean at every step.

The data is 1, 2, 3, 4, 4, 5, 5, 5, 8, so \( n=9 \), \( \sum x = 1+2+3+4+4+5+5+5+8=37 \), and \( \sum x^2=1+4+9+16+16+25+25+25+64=185 \). Then:

\[ \text{Var} = \frac{185}{9} - \left(\frac{37}{9}\right)^2 \approx 20.56 - 16.90 \approx 3.65 \]\[ \text{Standard deviation} = \sqrt{3.65} \approx 1.9 \]

1. \( 2\sqrt{2}\approx2.83 \): Noticeably larger than our computed value of about \( 1.9 \).
2. \( \sqrt{2}\approx1.41 \): Noticeably smaller than about \( 1.9 \).
3. 1.5: Also smaller than the computed value of about \( 1.9 \).
4. 2: The closest of the four options to the computed value of approximately \( 1.9 \), consistent with this being the intended rounded answer.

Using the shortcut sum-of-squares formula gives a standard deviation close to 2, clearly closer to this option than to any of the others.

Therefore, the correct answer is 2.