# Statistics for Environmental Engineers

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6.1

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Compute the sample mean (y) and standard deviation (s) of the resulting Winsorized sample in the usual way:

y = 7.53 jig/L    s = 1.022 jg/L

The Winsorized mean (yw) is the mean of the Winsorized sample:

yw = У = 7.53 jg/L

The Winsorized standard deviation (sw) is an approximately unbiased estimator of s and is computed from the standard deviation of the Winsorized sample as:

s

w

s(n — 1) V- 1

1.02(27- 1) 19-1

1.48 jg/L

where n is the total number of observations and V is the number of observations not replaced during Winsorization. V = 27 — 4 — 4 = 19 because the four “less-than” values and the four largest values have been replaced.

If the data are from a normal distribution, the upper and lower limits of a two-sided 100(1 — a)% confidence interval of the mean n are:

w

yw ± tV-,a/2    /—

Jn

where tV_1al2 cuts 100(a/2)% off each tail of the t distribution with V — 1 degrees of freedom. Note that the degrees of freedom are V — 1 instead of the usual n — 1, and sw replaces s.

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