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: For the data 1,2,3,4,4,5,5,5,8 with \( n=9 \): \( \sum x=37 \) and \( \sum x^2 = 1+4+9+16+16+25+25+25+64=185 \).
Step 2: Variance \( = \dfrac{\sum x^2}{n} - \left(\dfrac{\sum x}{n}\right)^2 = \dfrac{185}{9} - \left(\dfrac{37}{9}\right)^2 \approx 20.56 - 16.90 \approx 3.65 \).
Step 3: Standard deviation \( = \sqrt{3.65} \approx 1.9 \), closest to \( 2 \) among the given options.
\[ \boxed{\text{Standard deviation} \approx 2} \]

Answer 2

Since several values repeat, it's convenient to treat this as a grouped frequency distribution and weight each distinct value by how often it occurs, rather than listing all nine values separately.

The distinct values and their frequencies are: 1 (once), 2 (once), 3 (once), 4 (twice), 5 (three times), 8 (once), with total frequency \( n=9 \). The weighted sum is \( 1+2+3+2(4)+3(5)+8 = 1+2+3+8+15+8=37 \), giving mean \( \frac{37}{9}\approx4.11 \). The weighted sum of squares is \( 1^2+2^2+3^2+2(4^2)+3(5^2)+8^2=1+4+9+32+75+64=185 \).

\[ \text{Var}=\frac{185}{9}-\left(\frac{37}{9}\right)^2\approx20.56-16.90\approx3.65,\quad \text{SD}\approx1.9 \]

1. \( 2\sqrt{2} \): About \( 2.83 \), well above the grouped-frequency result.
2. \( \sqrt{2} \): About \( 1.41 \), below the grouped-frequency result.
3. 1.5: Still below the computed value of about \( 1.9 \).
4. 2: The nearest option to the value obtained from this weighted computation.

Grouping the repeated values by frequency and weighting them accordingly reproduces the same standard deviation of roughly 1.9, closest to 2.

Therefore, the correct answer is 2.