Analytical Chem istry - DePauw University

96 Analytical Chemistry 2.0Here is another way to think about degreesof freedom. We analyze samples tomake predictions about the underlyingpopulation. When our sample consists ofn measurements we cannot make morethan n independent predictions aboutthe population. Each time we estimate aparameter, such as the population’s mean,we lose a degree of freedom. If there aren degrees of freedom for calculating thesample’s mean, then there are n – 1 degreesof freedom remaining for calculating thesample’s variance.every member of the population we have complete information about thepopulation.When calculating the sample’s variance, however, we first replace m withX , which we also calculate from the same data. If there are n members inthe sample, we can deduce the value of the n th member from the remainingn – 1 members. For example, if n = 5 and we know that the first foursamples are 1, 2, 3 and 4, and that the mean is 3, then the fifth member ofthe sample must beX = ( X × n) −X −X −X − X = ( 3× 5)−1−2−3− 4=55 1 2 3 4Using n – 1 in place of n when calculating the sample’s variance ensures thats 2 is an unbiased estimator of s 2 .4D.5 Confidence Intervals for SamplesEarlier we introduced the confidence interval as a way to report the mostprobable value for a population’s mean, m,µ= X ±zσn4.11where X is the mean for a sample of size n, and s is the population’s standarddeviation. For most analyses we do not know the population’s standarddeviation. We can still calculate a confidence interval, however, if we maketwo modifications to equation 4.11.The first modification is straightforward—we replace the population’sstandard deviation, s, with the sample’s standard deviation, s. The secondmodification is less obvious. The values of z in Table 4.12 are for a normaldistribution, which is a function of s 2 , not s 2 . Although the sample’s variance,s 2 , provides an unbiased estimate for the population’s variance, s 2 ,the value of s 2 for any sample may differ significantly from s 2 . To accountfor the uncertainty in estimating s 2 , we replace the variable z in equation4.11 with the variable t, where t is defined such that t ≥ z at all confidencelevels.µ= X ±tsn4.12Values for t at the 95% confidence level are shown in Table 4.15. Note thatt becomes smaller as the number of degrees of freedom increases, approachingz as n approaches infinity. The larger the sample, the more closely itsconfidence interval approaches the confidence interval given by equation4.11. Appendix 4 provides additional values of t for other confidence levels.