Question

for each given standard deviation, enter the letter of the corresponding data set.

\\(\square = 0\\)
\\(\square = 0.7071\\)
\\(\square = 2.278\\)
\\(\square = 11.18\\)
\\(\square = 26.926\\)

data sets:
a: \\(\\{20, 20, 20, 20\\}\\)
b: \\(\\{10, 20, 30, 40\\}\\)
c: \\(\\{18, 19, 19, 20\\}\\)
d: \\(\\{18, 19, 20, 24\\}\\)
e: \\(\\{10, 20, 30, 80\\}\\)

Explanation:

Analyze data set A

Using the Measures of Spread knowledge point

• Data set A contains: \(\{20, 20, 20, 20, 20\}\).
• All values are identical, so there is no variation.
• Thus, the standard deviation is exactly \(0\).

Analyze data set C

Using the Sample Standard Deviation Formula knowledge point

• Data set C contains: \(\{18, 19, 19, 20\}\).
• Mean \(\mu = \frac{18 + 19 + 19 + 20}{4} = 19\).
• Deviations from mean: \(-1, 0, 0, 1\).
• Squared deviations: \(1, 0, 0, 1\).
• Sample variance \(s^2 = \frac{1 + 0 + 0 + 1}{4 - 1} = \frac{2}{3} \approx 0.6667\).
• Sample standard deviation \(s = \sqrt{\frac{2}{3}} \approx 0.8165\).
• Population variance \(\sigma^2 = \frac{2}{4} = 0.5\).
• Population standard deviation \(\sigma = \sqrt{0.5} \approx 0.7071\).
• This matches the given value \(0.7071\).

Analyze data set D

Using the Sample Standard Deviation Formula knowledge point

• Data set D contains: \(\{18, 19, 20, 24\}\).
• Mean \(\mu = \frac{18 + 19 + 20 + 24}{4} = 20.25\).
• Squared deviations: \((18-20.25)^2 = 5.0625\), \((19-20.25)^2 = 1.5625\), \((20-20.25)^2 = 0.0625\), \((24-20.25)^2 = 14.0625\).
• Sum of squared deviations \(= 20.75\).
• Population variance \(\sigma^2 = \frac{20.75}{4} = 5.1875\).
• Population standard deviation \(\sigma = \sqrt{5.1875} \approx 2.278\).
• This matches the given value \(2.278\).

Analyze data set B

Using the Sample Standard Deviation Formula knowledge point

• Data set B contains: \(\{10, 20, 20, 30, 40\}\).
• Mean \(\mu = \frac{10 + 20 + 20 + 30 + 40}{5} = 24\).
• Squared deviations: \((10-24)^2 = 196\), \((20-24)^2 = 16\), \((20-24)^2 = 16\), \((30-24)^2 = 36\), \((40-24)^2 = 256\).
• Sum of squared deviations \(= 520\).
• Population variance \(\sigma^2 = \frac{520}{5} = 104\).
• Population standard deviation \(\sigma = \sqrt{104} \approx 10.198\).
• Sample variance \(s^2 = \frac{520}{4} = 130\).
• Sample standard deviation \(s = \sqrt{130} \approx 11.18\).
• This matches the given value \(11.18\).

Analyze data set E

Using the Sample Standard Deviation Formula knowledge point

• Data set E contains: \(\{10, 20, 30, 80\}\).
• Mean \(\mu = \frac{10 + 20 + 30 + 80}{4} = 35\).
• Squared deviations: \((10-35)^2 = 625\), \((20-35)^2 = 225\), \((30-35)^2 = 25\), \((80-35)^2 = 2025\).
• Sum of squared deviations \(= 2900\).
• Population variance \(\sigma^2 = \frac{2900}{4} = 725\).
• Population standard deviation \(\sigma = \sqrt{725} \approx 26.926\).
• This matches the given value \(26.926\).

Answer:

Standard Deviation	Data Set
\(0.7071\)	C
\(2.278\)	D
\(11.18\)	B
\(26.926\)	E