We have illustrated an important property of simple random sampling, namely, that estimates of population averages are unbiased.

**WHY DOES RANDOM SAMPLING WORK?**

We have illustrated an important property of simple
random sampling, namely, that estimates of population averages are unbiased.
Under certain conditions, appropriately chosen stratified random samples can
produce unbiased estimates with better accuracy than simple random samples (see
Cochran, 1977).

A quantity that provides a description of the
accuracy of the estimate of a popu-lation mean is called the variance of the
mean, and its square root is called the stan-dard error of the mean. The symbol
*σ*^{2} is used to denote the population variance.
(Chapter 4 will provide the formulas for *σ*^{2}.) When the population size *N* is very large, the sampling variance of the sample mean is known
to be approximately *σ*^{2}/*n* for a
sample size of *n*.

In fact, as Cochran (1977) has shown, the exact
value of this sample variance is slightly smaller than the population variance
due to the finite number *N* for the
population. To correct for this slightly smaller estimate, a correction factor
is applied (see Chapter 4). If *n* is
small relative to *N*, this correction
factor can be ignored. The fact that the variance of the sample mean is
approximately *σ*^{2}/*n* tells
us that since the variance of the sample mean becomes small as *n* becomes large, individual sam-ple
means will be highly accurate.

Kuzma illustrated the phenomenon that large sample
sizes produce highly accu-rate estimates of the population mean with his
Honolulu Heart Study data (Kuzma, 1998; Kuzma and Bohnenblust, 2001). For his
data, the population size for the male patients was *N* = 7683 (a relatively large number).

Kuzma determined that the population mean for his
data was 54.36. Taking re-peated samples of *n*
= 100, Kuzma examined the mean age of the male patients. Choosing five simple
random samples of size *n* = 100, he
obtained sample means of 54.85, 54.31, 54.32, 54.67, and 54.02. All these
estimates were within one-half year of the population mean. In Kuzma’s example,
the variance of the sample means was small and *n* was large. Consequently, all sample estimates were close to one
anoth-er and to the population mean. Thus, in general we can say that the
larger the *n*, the more closely the
sample estimate of the mean approaches the population mean.

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