Analytical Chem istry - DePauw University

88 Analytical Chemistry 2.0Figure 4.8 Normal distributioncurve showing the area under thecurve for several different rangesof values of X. As shown here,68.26% of the members of a normallydistributed population havevalues within ±1σ of the population’sexpected mean, and 13.59%have values between μ–1σ andu–2σ. The area under the curvebetween any two limits can befound using the probability tablein Appendix 3.-3σ -2σ -1σ μ +1σ +2σ +3σwithin the range m ± 1s, and that 95.44% of population’s members havevalues within the range m ± 2s. Only 0.17% members of a population havevalues exceeding the expected mean by more than ± 3s. Additional rangesand probabilities are gathered together in a probability table that you willfind in Appendix 3. As shown in Example 4.11, if we know the mean andstandard deviation for a normally distributed population, then we can determinethe percentage of the population between any defined limits.Example 4.1134.13% 34.13%13.59 %13.59 %2.14 % 2.14 %Value of XThe amount of aspirin in the analgesic tablets from a particular manufactureris known to follow a normal distribution with m = 250 mg ands 2 = 25. In a random sampling of tablets from the production line, whatpercentage are expected to contain between 243 and 262 mg of aspirin?So l u t i o nWe do not determine directly the percentage of tablets between 243 mgand 262 mg of aspirin. Instead, we first find the percentage of tablets withless than 243 mg of aspirin and the percentage of tablets having more than262 mg of aspirin. Subtracting these results from 100%, gives the percentageof tablets containing between 243 mg and 262 mg of aspirin.To find the percentage of tablets with less than 243 mg of aspirin or morethan 262 mg of aspirin we calculate the deviation, z, of each limit from min terms of the population’s standard deviation, szX= −µσwhere X is the limit in question. The deviation for the lower limit is