D. Harvey - Modern Analytical Chemistry (794078), страница 20
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When the data include an odd number of measurements, the median is the middle value. For an even number of measurements, themedian is the average of the n/2 and the (n/2) + 1 measurements, where n is thenumber of measurements.EXAMPLE 4.2What is the median for the data in Table 4.1?SOLUTIONTo determine the median, we order the data from the smallest to the largestvalue3.0563.0803.0943.1073.1123.1743.198Since there is a total of seven measurements, the median is the fourth value inthe ordered data set; thus, the median is 3.107.As shown by Examples 4.1 and 4.2, the mean and median provide similar estimates of central tendency when all data are similar in magnitude. The median,however, provides a more robust estimate of central tendency since it is less sensitive to measurements with extreme values.
For example, introducing the transcription error discussed earlier for the mean only changes the median’s value from3.107 g to 3.112 g.4A.2 Measures of SpreadIf the mean or median provides an estimate of a penny’s true mass, then the spread ofthe individual measurements must provide an estimate of the variability in the massesof individual pennies. Although spread is often defined relative to a specific measureof central tendency, its magnitude is independent of the central value. Changing allmedianThat value for a set of ordered data, forwhich half of the data is larger in value–and half is smaller in value (Xmed).551400-CH04 9/8/99 3:53 PM Page 5656Modern Analytical Chemistrymeasurements in the same direction, by adding or subtracting a constant value,changes the mean or median, but will not change the magnitude of the spread.
Threecommon measures of spread are range, standard deviation, and variance.rangeThe numerical difference between thelargest and smallest values in a data set(w).Range The range, w, is the difference between the largest and smallest values inthe data set.Range = w = Xlargest – XsmallestThe range provides information about the total variability in the data set, but doesnot provide any information about the distribution of individual measurements.The range for the data in Table 4.1 is the difference between 3.198 g and 3.056 g;thusw = 3.198 g – 3.056 g = 0.142 gstandard deviationA statistical measure of the “average”deviation of data from the data’s meanvalue (s).Standard Deviation The absolute standard deviation, s, describes the spread ofindividual measurements about the mean and is given as∑i =1 ( Xins =– X )24.1n −1–where Xi is one of n individual measurements, and X is the mean.
Frequently, therelative standard deviation, sr, is reported.sr =sXThe percent relative standard deviation is obtained by multiplying sr by 100%.EXAMPLE 4.3What are the standard deviation, the relative standard deviation, and thepercent relative standard deviation for the data in Table 4.1?SOLUTIONTo calculate the standard deviation, we obtain the difference between the meanvalue (3.117; see Example 4.1) and each measurement, square the resultingdifferences, and add them to determine the sum of the squares (the numeratorof equation 4.1)(3.080 – 3.117)2(3.094 – 3.117)2(3.107 – 3.117)2(3.056 – 3.117)2(3.112 – 3.117)2(3.174 – 3.117)2(3.198 – 3.117)2=======(–0.037)2(–0.023)2(–0.010)2(–0.061)2(–0.005)2(+0.057)2(+0.081)2=======0.001370.000530.000100.003720.000030.003250.006560.01556The standard deviation is calculated by dividing the sum of the squares byn – 1, where n is the number of measurements, and taking the square root.s=0.01556= 0.0517 −11400-CH04 9/8/99 3:53 PM Page 57Chapter 4 Evaluating Analytical Data57The relative standard deviation and percent relative standard deviation aresr =0.051= 0.0163.117sr (%) = 0.016 × 100% = 1.6%It is much easier to determine the standard deviation using a scientificcalculator with built-in statistical functions.*Variance Another common measure of spread is the square of the standard deviation, or the variance.
The standard deviation, rather than the variance, is usually reported because the units for standard deviation are the same as that for the meanvalue.EXAMPLE 4.4What is the variance for the data in Table 4.1?SOLUTIONThe variance is just the square of the absolute standard deviation. Using thestandard deviation found in Example 4.3 gives the variance asVariance = s2 = (0.051)2 = 0.00264B Characterizing Experimental ErrorsRealizing that our data for the mass of a penny can be characterized by a measure ofcentral tendency and a measure of spread suggests two questions.
First, does ourmeasure of central tendency agree with the true, or expected value? Second, why areour data scattered around the central value? Errors associated with central tendencyreflect the accuracy of the analysis, but the precision of the analysis is determined bythose errors associated with the spread.4B.1 AccuracyAccuracy is a measure of how close a measure of central tendency is to the true, orexpected value, µ.† Accuracy is usually expressed as either an absolute error–E =X – µ4.2or a percent relative error, Er.Er =X −µ× 100µ4.3*Many scientific calculators include two keys for calculating the standard deviation, only one of which corresponds toequation 4.3. Your calculator’s manual will help you determine the appropriate key to use.†The standard convention for representing experimental parameters is to use a Roman letter for a value calculated fromexperimental data, and a Greek letter for the corresponding true value.
For example, the experimentally determined–mean is X, and its underlying true value is µ. Likewise, the standard deviation by experiment is given the symbol s, andits underlying true value is identified as σ.varianceThe square of the standard deviation (s 2).1400-CH04 9/8/99 3:53 PM Page 5858Modern Analytical Chemistrydeterminate errorAny systematic error that causes ameasurement or result to always be toohigh or too small; can be traced to anidentifiable source.Although the mean is used as the measure of central tendency in equations 4.2 and4.3, the median could also be used.Errors affecting the accuracy of an analysis are called determinate and are characterized by a systematic deviation from the true value; that is, all the individualmeasurements are either too large or too small.
A positive determinate error resultsin a central value that is larger than the true value, and a negative determinate errorleads to a central value that is smaller than the true value. Both positive and negative determinate errors may affect the result of an analysis, with their cumulative effect leading to a net positive or negative determinate error. It is possible, althoughnot likely, that positive and negative determinate errors may be equal, resulting in acentral value with no net determinate error.Determinate errors may be divided into four categories: sampling errors,method errors, measurement errors, and personal errors.heterogeneousNot uniform in composition.Sampling Errors We introduce determinate sampling errors when our samplingstrategy fails to provide a representative sample.
This is especially important whensampling heterogeneous materials. For example, determining the environmentalquality of a lake by sampling a single location near a point source of pollution, suchas an outlet for industrial effluent, gives misleading results. In determining the massof a U.S. penny, the strategy for selecting pennies must ensure that pennies fromother countries are not inadvertently included in the sample.
Determinate errors associated with selecting a sample can be minimized with a proper sampling strategy,a topic that is considered in more detail in Chapter 7.method errorAn error due to limitations in theanalytical method used to analyze asample.Method Errors Determinate method errors are introduced when assumptionsabout the relationship between the signal and the analyte are invalid. In terms of thegeneral relationships between the measured signal and the amount of analytesampling errorAn error introduced during the processof collecting a sample for analysis.Smeas = knA + Sreag(total analysis method)4.4Smeas = kCA + Sreag(concentration method)4.5method errors exist when the sensitivity, k, and the signal due to the reagent blank,Sreag, are incorrectly determined. For example, methods in which Smeas is the mass ofa precipitate containing the analyte (gravimetric method) assume that the sensitivity is defined by a pure precipitate of known stoichiometry.
When this assumptionfails, a determinate error will exist. Method errors involving sensitivity are minimized by standardizing the method, whereas method errors due to interferentspresent in reagents are minimized by using a proper reagent blank. Both are discussed in more detail in Chapter 5. Method errors due to interferents in the samplecannot be minimized by a reagent blank. Instead, such interferents must be separated from the analyte or their concentrations determined independently.measurement errorAn error due to limitations in theequipment and instruments used tomake measurements.toleranceThe maximum determinatemeasurement error for equipment orinstrument as reported by themanufacturer.Measurement Errors Analytical instruments and equipment, such as glassware andbalances, are usually supplied by the manufacturer with a statement of the item’smaximum measurement error, or tolerance.
For example, a 25-mL volumetricflask might have a maximum error of ±0.03 mL, meaning that the actual volumecontained by the flask lies within the range of 24.97–25.03 mL. Although expressedas a range, the error is determinate; thus, the flask’s true volume is a fixed valuewithin the stated range. A summary of typical measurement errors for a variety ofanalytical equipment is given in Tables 4.2–4.4.1400-CH04 9/8/99 3:53 PM Page 59Chapter 4 Evaluating Analytical DataTable 4.259Measurement Errors for Selected GlasswareaMeasurement Errors forGlasswareTransfer PipetsVolumetric FlasksBuretsVolume(mL)Class A Glassware(±mL)Class B Glassware(±mL)125102025500.0060.0060.010.020.030.030.050.0120.0120.020.040.060.060.105102550100250500100020000.020.020.030.050.080.120.200.300.500.040.040.060.100.160.240.400.601.01025500.020.030.050.040.060.10aSpecificationsfor class A and class B glassware are taken from American Society for Testing andMaterials E288, E542 and E694 standards.Table 4.3BalancePrecisa 160MA & D ER 120MMetler H54Table 4.4Measurement Errorsfor Selected BalancesCapacity(g)MeasurementError160120160±1 mg±0.1 mg±0.01 mgPipet RangeMeasurement Errorsfor Selected Digital PipetsVolume(mL or µL)aMeasurement Error(±%)10501001.00.60.620010001.50.815100.60.40.310–100 µLb200–1000 µLc1–10 mLdaUnitsfor volume same as for pipet range.bDatafor Eppendorf Digital Pipet 4710.cDatafor Oxford Benchmate.dDatafor Eppendorf Maxipetter 4720 with Maxitip P.1400-CH04 9/8/99 3:53 PM Page 6060Modern Analytical ChemistryVolumetric glassware is categorized by class.
Class A glassware is manufacturedto comply with tolerances specified by agencies such as the National Institute ofStandards and Technology. Tolerance levels for class A glassware are small enoughthat such glassware normally can be used without calibration. The tolerance levelsfor class B glassware are usually twice those for class A glassware.
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