Differentiate sample vs population standard deviation
Population vs. Sample Standard Deviation in a Finite Rolling Window
Context
You are implementing a rolling standard deviation over a fixed-size window of length k for a time series. Clarify how to compute standard deviation within each window, which denominator to use and why, how that choice impacts rolling estimates (especially for small k), what to return when k = 1, and how to expose and document this choice in code and tests.
Tasks
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Differentiate population vs. sample standard deviation for a finite window and explain why the denominators are n and n−1 (Bessel’s correction).
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Explain how this choice affects rolling estimates, especially for small k.
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Specify what the function should return for k = 1.
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State which definition you would implement by default, how to make it configurable, and how to document the choice in code and tests.
Constraints & Assumptions
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Preserve the scope, facts, inputs, and requested outputs from the prompt above.
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If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
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Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.
Clarifying Questions to Ask
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Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
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Show enough derivation for the interviewer to follow the reasoning.
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Explain how you would validate the result with simulation or sensitivity checks.
What a Strong Answer Covers
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A correct setup with definitions, formulas, and boundary conditions.
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A step-by-step derivation or estimation plan.
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Interpretation of the result, including uncertainty and practical limitations.
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Checks for assumptions, edge cases, and numerical stability.
Follow-up Questions
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How would the result change if the assumptions were relaxed?
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Can you verify the answer with a simulation?
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What is the most likely source of estimation error?