# Statistics for Environmental Engineers

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—    Xуi    A 2 £(Уг — у)

у = -p- and s = , кк

Note that s is calculated using a denominator of к instead of the usual n — 1.

For the case where the lower tail of the distribution is truncated, the estimated mean will be too high and the estimated variance will be too low. These estimates are adjusted using a factor X. The maximum likelihood estimates for the mean n and the variance a are:

П = У — X(у — yc)

2    2 л    2

a = s + /I (у — Уc )

The hat (л) indicates that the value is an estimate (n is an estimate of the true mean n). Cohen (1961) provides tables of X as a function of h and у = s’2/(у — ус)2. A portion of his table is reproduced in Table 15.4.

Example 15.4

Using Cohen’s method, estimate the mean and variance of the censored sample from Example 15.1. The sample of n = 27 observations includes 23 measured and 4 censored values. The censoring limit is 6 ^glL. The 23 uncensored values are 6.9, 7.8, 8.9, 7.7, 9.6, 8.7, 6.7, 8.0, 8.5, 6.5, 9.2, 7.4, 6.3, 7.3, 8.3, 7.2, 7.5, 6.1, 9.4, 7.6, 8.1, 7.9, and 10.1. The average and variance of the к = 23 observations with measurable values are:

у =    = 7.9 Mg/L

s2 = X(уг У- = 1.1078 (Mg/L)2

The limit of censoring is at уc = 6 and the proportion of censored data is:

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