Concrete statistics: if Δ denotes the deviation of each individual test strength from the sample mean for n results, choose the correct expression for the sample standard deviation.

Introduction / Context:
Quality control of concrete relies on statistical treatment of cube/cylinder strengths. The standard deviation gauges dispersion around the mean and feeds directly into characteristic strength calculations and compliance criteria. Using the correct formula (with the appropriate degrees-of-freedom correction) is essential, especially when the population variance is unknown and must be estimated from sample data.

Given Data / Assumptions:

• We have n strength test results with mean x̄.
• For each result, Δ = (x_i − x̄).
• We seek the sample standard deviation s applicable to concrete control per common practice (e.g., IS/ASTM approaches).

Concept / Approach:
The unbiased estimator of variance uses (n − 1) in the denominator because one degree of freedom is consumed in calculating the sample mean. Thus, sample variance s^2 = ΣΔ^2 / (n − 1) and sample standard deviation s = sqrt(ΣΔ^2 / (n − 1)). Using /n would bias the estimate low for small samples and is typically reserved for population variance when the true mean is known a priori.

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