Introduction to Biostatistical Applications in Health Research with Microsoft Office Excel by Hirsch Robert P

Author:Hirsch, Robert P.
Language: eng
Format: epub
ISBN: 9781119089650
Publisher: John Wiley & Sons, Inc.
Published: 2016-02-29T00:00:00+00:00

In contrast, a mean is a special type of weighted average in which each observation is given a weight equal to one.33 The use of a weighted average with weights equal to one implies that we are unable to say that one observation is closer to the population's mean than is any other observation. Thus, each observation is given the same weight as every other observation when estimating the mean.

Now, let us apply this principle of weighted averages to estimation of the population's variance of the distribution of data when we have estimates of that variance from two groups of dependent variable values.

We know from Chapter 4 that the number of degrees of freedom in a sample is used as a parameter of Student's t-distribution to reflect the degree of precision with which we can estimate the population's variance of the distribution of data. The greater the number of degrees of freedom, the less the penalty that must be paid for estimating the population's variance from the sample's observations.34 Since degrees of freedom can be used in that context to reflect how precisely we can estimate the population's variance, it makes sense for us to use degrees of freedom as the weights for the variance estimates from each of the two groups in the sample. The weighted average of these two variance estimates using degrees of freedom as the weights is called the pooled estimate of the variance. Equation (7.26) shows how to calculate the pooled estimate of the variance of the distribution of data in the population from the sample's observations.

(7.26)

where

= variance of the distribution of dependent variable values in the population

= pooled estimate of the variance of dependent variable values

= degrees of freedom in the ith group of dependent variable values

= variance estimate in the ith group of dependent variable values

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