Analytical Chem istry - DePauw University

Chapter 4 Evaluating Analytical Data87f ( X)=12πσ2e( X )−− 2µ22σwhere m is the expected mean for a population with n members∑ Xiiµ=nand s 2 is the population’s variance.2∑( X −µ)i2 iσ =4.8nExamples of normal distributions, each with an expected mean of 0 andwith variances of 25, 100, or 400, are shown in Figure 4.7. Two featuresof these normal distribution curves deserve attention. First, note that eachnormal distribution has a single maximum corresponding to m, and that thedistribution is symmetrical about this value. Second, increasing the population’svariance increases the distribution’s spread and decreases its height;the area under the curve, however, is the same for all three distribution.The area under a normal distribution curve is an important and usefulproperty as it is equal to the probability of finding a member of the populationwith a particular range of values. In Figure 4.7, for example, 99.99%of the population shown in curve (a) have values of X between -20 and20. For curve (c), 68.26% of the population’s members have values of Xbetween -20 and 20.Because a normal distribution depends solely on m and s 2 , the probabilityof finding a member of the population between any two limits isthe same for all normally distributed populations. Figure 4.8, for example,shows that 68.26% of the members of a normal distribution have a valueFigure 4.7 Normal distributioncurves for:(a) m = 0; s 2 = 25(b) m = 0; s 2 = 100(c) m = 0; s 2 =400