Statistics And Probability Question 430
Question: Let R be the range and σ be the population standard deviation of a set of observations x₁, x₂, …, xₙ. Which of the following statements is always true?
Options:
A) σ ≤ R/2
B) σ ≥ R/2
C) σ = R/2
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
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First, let’s recall the definitions:
- Range (R) is the difference between the maximum and minimum values in a dataset.
- Population standard deviation (σ) is the square root of the average squared deviation from the mean.
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There’s a well-known inequality in statistics that relates the standard deviation to the range: σ ≤ R/√12
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We can prove that R/√12 is always less than or equal to R/2: R/√12 ≈ 0.2887R R/2 = 0.5R
Clearly, 0.2887R < 0.5R
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Therefore, we can say that: σ ≤ R/√12 < R/2
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This means that the standard deviation is always less than or equal to half the range.
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Hence, option A (σ ≤ R/2) is always true.
BETA
