# Statistics for Environmental Engineers

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Of the 30 measurements, 12 are censored at a limit of 18 jlg/L, so the fraction censored is h = 12/30 = 0.40. The mean and variance computed from the logarithms of the noncensored values are:

X = 3.2722 and    s2x = 0.03904

The limit of censoring is also transformed:

Xc = ln (yc) = ln (18) = 2.8904

Using these values, we compute:

Y= sX/(X — Xc)2 = 0.03904/(3.2722 — 2.8904)2 = 0.2678.

which is used with h = 0.4 to interpolateX = 0.664 in Table 15.4. The estimated mean and variance of the log-transformed values are:

rjx = 3.2722 — 0.664(3.2722 — 2.8904) = 3.0187 Cf2 = 0.03904 + 0.664(3.2722 — 2.8904)2 = 0.1358

The transformation equations to convert these into estimates of the mean and variance of the untransformed y’s are:

П y = exp (n * + 0.5(72)

<7y2= fj У [ exp (a*) — 1]

Substituting the parameter estimates fjx and (7X gives:

rj y = exp[ 3.0187 + 0.5 (0.1358)] = exp (3.0866) = 21.9 ig/L 772 = (21.90)2[exp(0.1358) — 1 ] = 69.76(ig/L)6y = 8.35 ilg/L

### The Delta-Lognormal Distribution

The delta-lognormal method estimates the mean of a sample of size n as a weighted average of nc replaced censored values and n — nc uncensored lognormally distributed values. The Aitchison method (1955, 1969) assumes that all censored values are replaced by zeros (D = 0) and the noncensored values have a lognormal distribution. Another approach is to replace censored values by the detection limit (D = MDL) or by some value between zero and the MDL (U.S. EPA, 1989; Owen and DeRouen, 1980).

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