Analytical Chem istry - DePauw University

Chapter 4 Evaluating Analytical Data67So l u t i o nTo calculate the standard deviation we first calculate the difference betweeneach measurement and the mean value (3.117), square the resulting differences,and add them together to give the numerator of equation 4.1.2 2( 3. 080 − 3. 117) = ( − 0. 037) = 0.0013692 2( 3. 094−31. 17) =− ( 0. 023) = 0.00052922( 3. 107 − 3. 117) = ( −0.010) = 0.0001002 2( 3. 056 − 3. 117) = ( − 0. 061) = 00 . 037212 2( 3. 112− 3. 117) = ( − 0. 005) = 0.0000252 2( 31 . 74 − 3. 117) = ( + 0. 057) = 0.003249( 3. 198−3. 117) 2 =+ ( 0. 081) 2 = 0.0065610.015554For obvious reasons, the numerator ofequation 4.1 is called a sum of squares.Next, we divide this sum of the squares by n – 1, where n is the number ofmeasurements, and take the square root.s =0.015554= 0. 051 g7−1Finally, the relative standard deviation and percent relative standard deviationares r= 0.051 g3.117 g=0. 016 %s r = (0.016) × 100% = 1.6%It is much easier to determine the standard deviation using a scientificcalculator with built in statistical functions.Many scientific calculators include twokeys for calculating the standard deviation.One key calculates the standard deviationfor a data set of n samples drawn froma larger collection of possible samples,which corresponds to equation 4.1. Theother key calculates the standard deviationfor all possible samples. The later is knownas the population’s standard deviation,which we will cover later in this chapter.Your calculator’s manual will help you determinethe appropriate key for each.Va r i a n c eAnother common measure of spread is the square of the standard deviation,or the variance. We usually report a data set’s standard deviation, ratherthan its variance, because the mean value and the standard deviation havethe same unit. As we will see shortly, the variance is a useful measure ofspread because its values are additive.Example 4.4What is the variance for the data in Table 4.1?So l u t i o nThe variance is the square of the absolute standard deviation. Using thestandard deviation from Example 4.3 gives the variance ass 2 = (0.051) 2 = 0.0026