Characteristic strength from test results

Definition: the characteristic strength of concrete is the value of the compressive strength of 150 mm cubes at 28 days below which not more than 5% of the test results are expected to fall (IS 456, Terminology). With a normal distribution, $f_{ck} = f_m - 1.65\,s$.

Calculation

Formula: characteristic strength (95% confidence) $f_{ck} = f_m - 1.65\,s$, with the sample standard deviation $s = \sqrt{\sum(x-f_m)^2/(n-1)}$. The factor 1.65 comes from the normal distribution (the same factor is used in mix design, $f_{target} = f_{ck} + 1.65\,s$, IS 10262). Assumptions: the results follow the normal distribution, and the sample standard deviation ($n-1$) is used.

$n = 10$, $\sum x = 285$, mean $f_m = 285/10 = 28.50$ N/mm$^2$.

$$\begin{aligned} s &= \sqrt{\frac{70.50}{10-1}} = 2.799\ \text{N/mm}^2 \\ f_{ck} &= 28.50 - 1.65\times2.799 = 23.88\ \text{N/mm}^2 \end{aligned}$$

Answer: characteristic strength $f_{ck} \approx$ 23.9 N/mm$^2$.