Question

Write short note on sampling in chemistry

Answer 1

statistics, a population is an entire set of objects or units of observation of one sort or another, while a sample
is a subset (usually a proper subset) of a population, selected for particular study (usually because it is impractical
to study the whole population). The numerical characteristics of a population are called parameters. Generally the
values of the parameters of interest remain unknown to the researcher; we calculate the “corresponding” numerical
characteristics of the sample (known as statistics) and use these to estimate, or make inferences about, the unknown
parameter values.
A standard notation is often used to keep straight the distinction between population and sample. The table
below sets out some commonly used symbols.
size mean variance proportion
Population: N µ σ2 π
Sample: n x s ¯
2 p
Note that it’s common to use a Greek letter to denote a parameter, and the corresponding Roman letter to denote
the associated statistic.
2 Properties of estimators: sample mean
Consider for example the sample mean,
x¯ =
1
n
Xn
i=1
xi
If we want to use this statistic to make inferences regarding the population mean, µ, we need to know something
about the probability distribution of x¯. The distribution of a sample statistic is known as a sampling distribu-
tion. Two of its characteristics are of particular interest, the mean or expected value and the variance or standard
deviation.
What can we say about E(x¯) or µx¯
, the mean of the sampling distribution of x¯? First, let’s be sure we
understand what it means. It is the expected value of x¯. The thought experiment is as follows: we sample repeatedly
from the given population, each time recording the sample mean, and take the average of those sample means. It’s
unlikely that any given sample will yield a value of x¯ that precisely equals µ, the mean of the population from
which we’re drawing. Due to (random) sampling error some samples will give a sample mean that exceeds the
population mean, and some will give an x¯ that falls short of µ. But if our sampling procedure is unbiased, then
deviations of x¯ from µ in the upward and downward directions should be equally likely. On average, they should
cancel out. In that case
E(x¯) = µ = E(X) (1)
or: the sample mean is an unbiased estimator of the population mean.
So far so good. But we’d also like to know how widely dispersed the sample mean values are likely to be,
around their expected value. This is known as the issue of the efficiency of an estimator. It is a comparative
∗Last revised 2002/01/29.
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