Source lideplayer.com/fundamental of Analytical Chemitry, F.J. Holler, S.R.Crouch Chapter 6: Random Error in Chemical Analyi
Random error are preent in every meaurement no matter how careful the experimenter. Random, or indeterminate, error can never be totally eliminated and are often the major ource of uncertainty in a determination. Random error are caued by the many uncontrollable variable that accompany every meaurement. The accumulated effect of the individual uncertaintie caue replicate reult to fluctuate randomly around the mean of the et. In thi chapter, we conider the ource of random error, the determination of their magnitude, and their effect on computed reult of chemical analye. We alo introduce the ignificant figure convention and illutrate it ue in reporting analytical reult.
6A The nature of random error - random error in the reult of analyt and 4 i much larger than that een in the reult of analyt 1 and 3. - The reult of analyt 3 how outtanding preciion but poor accuracy. The reult of analyt 1 how excellent preciion and good accuracy. Figure 6-1 A three-dimenional plot howing abolute error in Kjeldahl nitrogen determination for four different analyt.
Random Error Source - Small undetectable uncertaintie produce a detectable random error in the following way. - Imagine a ituation in which jut four mall random error combine to give an overall error. We will aume that each error ha an equal probability of occurring and that each can caue the final reult to be high or low by a fixed amount ±U. - Table 6.1 give all the poible way in which four error can combine to give the indicated deviation from the mean value. * Note that only 1 combination lead to a deviation of +4 U, 4 combination give a deviation of + U, and 6 give a deviation of 0 U. * Thi ratio of 1:4:6:4:1 i a meaure of the probability for a deviation of each magnitude
If we make a ufficiently large number of meaurement, we can expect a frequency ditribution like that hown in Figure below. Figure 6- Frequency ditribution for meaurement containing (a) Four random uncertaintie, (b) ten random uncertaintie, The mot frequent occurrence i zero deviation from the mean. At the other extreme, a maximum deviation of 10 U occur only about once in 500 reult.
When the ame procedure i applied to a very large number of random uncertaintie, a bell-haped curve reult. Such a plot i called a Gauian curve or a normal error curve. Figure 6--c Frequency ditribution for meaurement containing a very large number of random uncertaintie,
Ditribution of Experimental Reult The ditribution of replicate data from mot quantitative analytical experiment approache that of the Gauian curve. Exp: Calibration of a 10-mL pipet. In thi experiment a mall flak and topper were weighed. Ten milliliter of water were tranferred to the flak with the pipet, and the flak wa toppered. The flak, the topper, and the water were then weighed again. The temperature of the water wa alo meaured to determine it denity. The ma of the water wa then calculated by taking the difference between the two mae. The ma of water divided by it denity i the volume delivered by the pipet. The experiment wa repeated 50 time.
Conider the data in the table for the calibration of a 10-mL pipet. The reult vary from a low of 9.969 ml to a high of 9.994 ml. Thi 0.05-mL pread of data reult directly from an accumulation of all random uncertaintie in the experiment. The pread in a et of replicate meaurement can be defined a the difference between the highet and lowet reult. The frequency ditribution data i hown in the table.
The frequency ditribution data are plotted a a bar graph, or hitogram. A the number of meaurement increae, the hitogram approache the hape of the continuou Gauian curve. Figure 6-3 A hitogram (A) howing ditribution of the 50 reult in Table 6-3 and a Gauian curve (B) for data having the ame mean and tandard deviation a the data in the hitogram.
6B Statitical treatment of random error - Statitical analyi only reveal information that i already preent in a data et. -Statitical method, do allow u to categorize and characterize data and to make objective and intelligent deciion about data quality and interpretation. -Sample and Population A population i the collection of all meaurement of interet to the experimenter, while a ample i a ubet of meaurement elected from the population. Statitical ample i different from the analytical ample.
Propertie of Gauian Curve Gauian curve can be decribed by an equation that contain two parameter, the population mean µ and the population tandard deviation. The term parameter refer to quantitie uch a µ and that define a population or ditribution. Data value uch a x are variable. The term tatitic refer to an etimate of a parameter that i made from a ample of data. Figure 6-4 Normal error curve.
The equation for a normalized Gauian curve i a follow: ( x ) The Population Mean µ and the Sample Mean Ẍ y The ample mean x i the arithmetic average of a limited ample drawn from a population of data. The ample mean i defined a the um of the meaurement value divided by the number of meaurement. N i the i the no. of meaurement in the ample et e / x N N x 1 i 1 The population mean i expreed a: where N i the total number of meaurement in the population. N i 1 N x i
The Population Standard Deviation It i a meaure of the preciion of the population and i expreed a: N i1 ( x ) i N Where N i the no. of data point making up the population. The two curve are for two population of data that differ only in their tandard deviation.
Another type of normal error curve in which the x axi i a new variable z, the quantity z repreent the deviation of a reult from the population mean relative to the tandard deviation. It i commonly given a a variable in tatitical table ince it i a dimenionle quantity. z i defined a z ( ) x A plot of relative frequency veru z yield a ingle Gauian curve that decribe all population of data regardle of tandard deviation.
The equation for the Gauian error curve i: y e ( x) / e z / Thi curve ha everal general propertie: (a) The mean occur at the central point of maximum frequency, (b) there i a ymmetrical ditribution of poitive and negative deviation about the maximum, and (c) there i an exponential decreae in frequency a the magnitude of the deviation increae.
Area under a Gauian Curve Regardle of it width, 68.3% of the area beneath a Gauian curve for a population lie within one tandard deviation (1) of the mean m. Approximately 95.4% of all data point are within of the mean and 99.7% within 3.
The Sample Standard Deviation: A Meaure of Preciion The ample tandard deviation i expreed a: N i1 ( x i x) N 1 N i1 N d i 1 - Thi equation applie to mall et of data. -The number of degree of freedom indicate the number of independent reult that enter into the computation of the tandard deviation. - When number of degree of freedom, (N - 1) i ued intead of N, i aid to be an unbiaed etimator of the population tandard deviation. - The ample variance i an etimate of the population variance.
An Alternative Expreion for Sample Standard Deviation N i1 x i N ( N i1 x N 1 i ) - Any time you ubtract two large, approximately equal number, the difference will uually have a relatively large uncertainty. - Hence, you hould never round a tandard deviation calculation until the end. - Becaue of the uncertainty in x, a ample tandard deviation may differ ignificantly from the population tandard deviation. - A N become larger, x and become better etimator of µ, and.
Standard Error of the Mean - If a erie of replicate reult, each containing N meaurement, are taken randomly from a population of reult, the mean of each et will how le and le catter a N increae. - The tandard deviation of each mean i known a the tandard error of the mean, m, i expreed a: m N - When N i greater than about 0, i uually a good etimator of, and thee quantitie can be aumed to be identical for mot purpoe.
Pooling Data to Improve the Reliability of - If we have everal ubet of data, a better etimate of the population tandard deviation can be obtained by pooling (combining) the data, auming that the ample have imilar compoition and have been analyzed the ame way. - The pooled etimate of pooled, i a weighted average of the individual etimate. - To calculate pooled, deviation from the mean for each ubet are quared; the quare of the deviation of all ubet are then ummed and divided by the appropriate number of degree of freedom. pooled N1 N ( xi xi ) ( x j x ) i1 j1 k1 N 1 N N 3... ( x N t k x 3 )...
Variance and Other Meaure of Preciion Preciion of analytical data may be expreed a: Variance ( ) i the quare of the tandard deviation. The ample variance i an etimate of the population variance and i given by: N i1 ( x x) Relative Standard Deviation (RSD) and Coefficient of Variation (CV) IUPAC recommend that the ymbol r be ued for relative ample tandard deviation and r for relative population tandard deviation. In equation where it i cumberome to ue RSD, ue r and r. i N 1 N i1 ( d i ) N 1 RSD r x
RSD inppt 1000ppt ( ) x The relative tandard deviation multiplied by 100% i called the coefficient of variation (CV). CV RSDinpercent x 100% - Spread or Range (w) i ued to decribe the preciion of a et of replicate reult. It i the difference between the larget value in the et and the mallet.
6C Standard deviation of calculated reult We mut etimate the tandard deviation of a reult that ha been calculated from two or more experimental data point, each of which ha a known ample tandard deviation.
Standard Deviation of a Sum or Difference - The variance of a um or difference i equal to the um of the variance of the number making up that um or difference. - The mot probable value for a tandard deviation of a um or difference can be found by taking the quare root of the um of the quare of the individual abolute tandard deviation. y a( a ) b( b ) c( c ) y a b c - Hence, the tandard deviation of the reult y i y a b c - For a um or a difference, the tandard deviation of the anwer i the quare root of the um of the quare of the tandard deviation of the number ued in the calculation.
Standard Deviation of a Product or Quotient The relative tandard deviation of a product or quotient i determined by the relative tandard deviation of the number forming the computed reult. In cae of, y a b c The relative tandard deviation y /y of the reult by umming the quare of the relative tandard deviation of a, b, and c and then calculating the quare root of the um: y y a ( a To find the abolute tandard deviation in a product or a quotient, firt find the relative tandard deviation in the reult and then multiply it by the reult. ) b ( ) b c ( c )
Standard Deviation in Exponential Calculation - Conider the relationhip: y = a x where the exponent x can be conidered free of uncertainty. -The relative tandard deviation in y reulting from the uncertainty in a i y y x( a - The relative tandard deviation of the quare of a number i twice the relative tandard deviation of the number, the relative tandard deviation of the cube root of a number i one third that of the number, and o forth. - The relative tandard deviation of y = a 3 i not the ame a the relative tandard deviation of the product of three independent meaurement y = abc, where a = b = c. a )
Standard Deviation of Logarithm and Antilogarithm For y = log a y 0. 434 a a And for y = antilog a y y.303 a The abolute tandard deviation of the logarithm of a number i determined by the relative tandard deviation of the number; converely, the relative tandard deviation of the antilogarithm of a number i determined by the abolute tandard deviation of the number.
6D Reporting computed data One of the bet way of indicating reliability i to give a confidence interval at the 90% or 95% confidence level. Another method i to report the abolute tandard deviation or the coefficient of variation of the data. A much le atifactory but more common indicator of the quality of data i the ignificant figure convention. Significant Figure The ignificant figure in a number are all of the certain digit plu the firt uncertain digit. Rule for determining the number of ignificant figure: 1. Diregard all initial zero.. Diregard all final zero unle they follow a decimal point. 3. All remaining digit including zero between nonzero digit are ignificant.
Significant Figure in Numerical Computation Determining the appropriate number of ignificant figure in the reult of an arithmetic combination of two or more number require great care. Sum and Difference - For addition and ubtraction, the reult hould have the ame number of decimal place a the number with the mallet number of decimal place. Note that the reult contain three ignificant digit even though two of the number involved have only two ignificant figure.
Product and Quotient For multiplication and diviion, the anwer hould be rounded o that it contain the ame number of ignificant digit a the original number with the mallet number of ignificant digit. Thi procedure ometime lead to incorrect rounding. When adding and ubtracting number in cientific notation, expre the number to the ame power of ten. The weak link for multiplication and diviion i the number of ignificant figure in the number with the mallet number of ignificant figure. Ue thi rule of thumb with caution.
Logarithm and Antilogarithm The following rule apply to the reult of calculation involving logarithm: 1. In a logarithm of a number, keep a many digit to the right of the decimal point a there are ignificant figure in the original number.. In an antilogarithm of a number, keep a many digit a there are digit to the right of the decimal point in the original number. The number of ignificant figure in the mantia, or the digit to the right of the decimal point of a logarithm, i the ame a the number of ignificant figure in the original number. Thu, log (9.57 10 4 ) = 4.981. Since 9.57 ha 3 ignificant figure, there are 3 digit to the right of the decimal point in the reult.
Rounding Data In rounding a number ending in 5, alway round o that the reult end with an even number. Thu, 0.635 round to 0.64 and 0.65 round to 0.6. It i eldom jutifiable to keep more than one ignificant figure in the tandard deviation becaue the tandard deviation contain error a well.
Expreing Reult of Chemical Calculation - If the tandard deviation of the value making up the final calculation are known, apply the propagation of error method. - However, if calculation have to be performed where the preciion i indicated only by the ignificant figure convention, the reult i rounded o that it contain only ignificant digit. - It i epecially important to potpone rounding until the calculation i completed. - At leat one extra digit beyond the ignificant digit hould be carried through all of the computation in order to avoid a rounding error. Thi extra digit i ometime called a guard digit.
Suggeted Problem 6.1, 6., 6.5, 6.7, 6.8, 6.10(odd), 6.1, 6.15, 6.18