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.
-
As = sqrt( (ΣΔ^2) / n )
-
Bs = sqrt( (ΣΔ) / n )
-
Cs = sqrt( (ΣΔ^2) / (n - 1) )
-
Ds = (ΣΔ) / (n - 1)
-
Es = (ΣΔ^2) / n
Answer
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:
- 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.
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) )