Difficulty: Medium
Correct Answer: s = sqrt( (ΣΔ^2) / (n - 1) )
Explanation:
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:
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.
Step-by-Step Solution:
Define deviations: Δ_i = x_i − x̄.Compute sum of squares: ΣΔ^2 = Σ(x_i − x̄)^2.Divide by (n − 1) to obtain unbiased variance: s^2 = ΣΔ^2 / (n − 1).Take square root for s: s = sqrt(ΣΔ^2 / (n − 1)).Verification / Alternative check:
Check a simple case (n = 2): using /n would understate spread; / (n − 1) correctly scales dispersion.Why Other Options Are Wrong:
Using /n yields the population standard deviation, not the sample estimator used in control charts.ΣΔ alone ignores squaring and will cancel to ~0; not a spread measure.Linear forms (ΣΔ)/(n − 1) or (ΣΔ^2)/n without square root are not standard deviation.Common Pitfalls:
Forgetting the square root step; mixing variance with standard deviation.Using biased /n denominator in small-sample concrete control.Final Answer:
s = sqrt( (ΣΔ^2) / (n - 1) )
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