Question 28.2: A silty soil containing 14 percent moisture was mixed in a l......
A silty soil containing 14 percent moisture was mixed in a large muller mixer with 10.00 weight percent of a tracer consisting of dextrose and picric acid. After 3 min of mixing, 12 random samples were taken from the mix and analyzed colorimetrically for tracer material. The measured concentrations in the sample were, in weight percent tracer, 10.24, 9.30, 7.94, 10.24, 11.08, 10.03, 11.91, 9.72, 9.20, 10.76, 10.97, 10.55. Calculate the mixing index I_p and the standard deviation s.
For this test \mu=0.10 \text { and } N=12 ; \bar{x}=\sum x_i / N=1.2194 / 12=0.101617 . \text { Also } \sum \bar{x}_i^2= 0.1251028. (For accurate calculations a large number of decimal places must be retained.) Substitution in Eq. (28.20) gives
I_p=\frac{\sigma_0}{s}=\sqrt{\frac{(N-1) \mu(1-\mu)}{\sum_{i=1}^N x_i^2-\bar{x} \sum_{i=1}^N x_i}} (28.20)
\begin{aligned}& I_p^2=\frac{(12-1) \times 0.10(1-0.10)}{0.1251028-(0.101617 \times 1.2194)} \\& I_p=28.8\end{aligned}The standard deviation from Eq. (28.18) is 0.0104, which is 10.2 percent of \bar{x}.
s=\sqrt{\frac{\sum_{i=1}^N\left(x_i-\bar{x}\right)^2}{N-1}}=\sqrt{\frac{\sum_{i=1}^N x_i^2-\bar{x} \sum_{i=1}^N x_i}{N-1}} (28.18)