3.5 Statistics as EstimatorsWhereas this chapter is concerned with initial data analysis, in choosing appropriate
descriptive statistics we should consider the purpose of a study and pay particular
attention to the inferences that may be required to answer particular research questions. In
designing a study we tacitly acknowledge that it is not feasible to measure the entire
population of interest, instead we draw a sample from the population. Our basic research
question may, for example, be concerned with the average A-level points score in the
population of first class honours graduates. What we have is an average score for the
obtained sample. To answer our research question we need to infer from our sample
average what the population mean is likely to be. We use statistical inference to do this.
We generally use descriptive statistics such as the sample mean and variance to
estimate the corresponding population parameter. When descriptive sample statistics are
used to yield estimates of yet other statistics that describe properties of populations, this
is referred to as estimation.
Characteristics of samples are called statistics and the comparable measures in a
population that these statistics estimate are called parameters. By convention, and to
help distinguish between sample statistics and population parameters, sample statistics
are designated by Latin letters and population parameters are designated by Greek letters.
For example: a sample average is denoted by ‘x-bar’, written as and the comparable
population parameter is symbolized by the Greek letter μ (mu). It may help you to
remember, that P’s go together e.g., ‘Population Parameters’, and S’s go together e.g.,
‘Sample Statistics’. This terminology and notation is summarized in the following table.
Table 3.5: Names, notations and explanatory
formulas for summary measures of populations and
samples
Summary Measures(^) Name Notation Read as Formulas
Sample statistics: Average
x-bar
Variance S^2 S-squared
Standard deviation S S
Population parameters: Mean μ mu
Variance σ^2 sigma squared
Standard deviation σ sigma
Note: these are explanatory formulas and are not necessarily the formulas that would be used for
computational purposes.
Initial data analysis 75