Analysis of Experimental Data
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1 Aalysis of Experimetal Data ± g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both accurate ad precise. Accuracy refers to the proximity of a measuremet to the true value of a quatity. Precisio refers to the proximity of several measuremets to each other, that is, the reproducibility of a measuremet or set of measuremets. For Example: Two studets, Raffaella ad Barbara, measured the temperature of boilig water, which by defiitio should be 00 C uder atmosphere of pressure. Each studet made 0 temperature measuremets, show below as red (Raffaella) ad blue (Barbara) dots. The average of Raffaella's temperature measuremets is 00. C ad the average of Barbara's is also 00. C. Give the actual value for b.p. both studets had good accuracy O the other had, you ca see from the figure that the precisio of Raffaella's measuremets was far better tha Barbara's. So, what caused the two studets to get values that were ot equal to the true boilig poit of water. Ad eve if they did t get the true value, why did t they get the same umber every time they took the measuremet of the same sample? The aswer: Experimetal Errors Experimetal Errors There are three types of experimetal errors affectig values:. Systematic error errors affectig the accuracy i measuremets which have a defiite value that ca, i priciple, be measured ad accouted for. Systematic errors ca be corrected for but oly after the cause is determied. Examples may iclude a icorrect calibratio of a balace that always reads.000 g higher tha the actual mass or the presece of a iterferig substace i calorimic studies. I our example, systematic error is why the studets did ot get exactly 00.0 o C for their measuremets.. Radom error errors affectig precisio i every measuremet which fluctuate radomly ad do ot have a defiite value; They caot be positively idetified. To further uderstad radom errors, cosider the weight of a object obtaied by doig five differet weighigs o a four place aalytical balace. trial : g trial : g trial 3: g trial 4: g trial 5: g The first three digits are the same i all cases. The last digit has a ucertaity associated with it. This ucertaity is a fuctio of the type of sample, the coditios uder which it is beig weighed, the balace, ad the perso doig the weighig.
2 Eve whe all factors are optimized, there will still be some variatio i the weight. This variatio or ucertaity is the result of pushig the balace to its limit. We could cut the last figure off; the all the weights would be the same, but the weight would be kow oly to the earest milligram. We obtai more iformatio if we keep that last figure but remai aware of its ucertaity. That ucertaity arises because of radom error; ad is idicative of the precisio of the measuremet. I our example, radom error is why the studets did ot get the same measuremet every time. 3. Gross error errors producig values that are drastically differet from all other data. These errors are the result of a mistake i the procedure, either by the experimeter or by a istrumet. A example would be misreadig the umbers or miscoutig the scale divisios o a buret or istrumet display. A istrumet might produce a gross error if a poor electrical coectio causes the display to read a occasioal icorrect value. If you are aware of a mistake at the time of the procedure, the experimetal result should be discouted ad the experimet repeated correctly. Gross errors result i values that make o sese ad greatly affect the overall precisio of the experimet. It is impossible to perform a chemical aalysis i such a way that the results are totally free of errors. All oe ca hope is to miimize these errors ad to estimate their size with acceptable accuracy. Rarely is it easy to estimate the errors of experimetal data; However, we must make such estimates because data of ukow precisio ad accuracy are worthless. The questio ow becomes, how do we describe the accuracy ad precisio quatitatively? Estimatig Accuracy i Measuremet Accuracy ca be estimated easily whe a true value is kow. For example the desity of iro is 7.87 g/cm 3. I the laboratory, you fid the mass ad volume of a iro block to calculate the desity as 7.3 g/cm 3. There is obvious systematic error i accuracy sice you did ot derive the theoretical desity of pure iro. Note: Sice all values are limited by the techology to describe them a better term for true value is accepted value. Estimatig Accuracy i Measuremet Absolute error (AE) The differece betwee a experimetal value ad accepted value A.E. = exp kow % Absolute Error (%AE) exp kow %AE 00 kow % Accuracy Percetage your value differs from 00. % Accuracy = 00 - %AE Back to our example. There are two ways we could describe the accuracy of the data collected. ) we could calculate the accuracy for every data poit idividually. ) we could calculate the accuracy of each experimeter by treatig the data sets as sigle values, or average values. I our example, the average value for each studets results were 00. o C ad 00. o C..Calculate the absolute error, % absolute error ad % accuracy for each experimeter.
3 Ofte, you will eed the average for a data set to gai a better estimate of experimetal error. There are two commo ways of expressig a average: the mea ad the media. The mea (χ) is the arithmetic average of the results, or: xi x x... x Mea x i. If the aalysis of acetamiophe tablets resulted i the presece of 48mg, 479mg, 44mg, ad 435mg active igrediet i four differet trials, what was the mea value? The media is the value that lies i the middle amog the results. Half of the measuremets are above the media ad half below. If there are ad eve umber of values, the media is the average of the two middle results. 3. What the is the media for the values foud i aalyzig the acetamiophe tablets? There are ofte advatages for usig the media i place of the mea whe a average is desired. If a small umber of measuremets are made, oe value ca greatly affect the mea. 4. Compare the mea to media i our sample set. Which oe would be a more realistic average? Whe Should a Value be Omitted as a Outlier Whe Fidig a Average? There are may differet ways to determie whether or ot a value is a outlier due to gross error. Some methods iclude: Q test 0% assumptio Stadard deviatio (4σ approximatio) Five umber summary (Box Plot) We will utilize the five umber summary. The Five Number Summary For ay group of umbers you believe may cotai a outlier: I. Arrage umbers i ascedig or descedig order ad fid the media. II. Calculate Q by fidig the middle value for the umbers left of the media. III.Calculate Q 3 by fidig the middle value for the umbers right of the media. IV.Calculate the Ier Quartile Rage (IQR) by subtractig Q from Q 3. V. Multiply IQR by.5 VI.Subtract modified IQR from Q. Aythig less tha this value is a outlier. VII.Add the modified IQR to Q 3. Aythig greater tha the result is a outlier. 5. Determie if outliers exist i the followig set of data. The fid both the mea ad media of the appropriate data., 4, 56, 6, 6, 69, 73, 07. Estimatig Precisio i Measuremet I chemistry you are ofte lookig for a ukow value ad have othig to compare your experimetal values to. I quatitative work, precisio is ofte used as a idicatio of accuracy; we assume that the average of a series of precise measuremets (which should average out the radom errors because of their equal probability of beig high or low), is accurate, or close to the true value. Agai, you may be tryig to determie the true/accepted value. 3
4 Estimatig Precisio i Measuremet Relative error (RE) The differece betwee a experimetal value ad the average (mea or media) for a set of experimetal values R.E. = exp avg % Relative Error (%RE) exp avg %RE 00 avg % Precisio The closeess of a value to a set of values i terms of percetage. % Precisio = 00 - %RE Agai, to our example. Whe lookig at the precisio of the experimet we could describe the precisio i two ways. ) the precisio of each data poit collected i respect to the data for each studet usig the previous treatmets. ) estimate the precisio of each studets experimet by ivestigatig the ucertaity of their measuremets (discussed later) 6. Calculate the relative error, % relative error ad % precisio for the third measuremet take by each studet. Whe dealig with precisio, the greater the umber of trials, the better the estimatio of the overall rage. We have estimated values for accuracy ad precisio but what cofidece is there i these values obtaied experimetally? You ca see the results above are spread over a rage of values. The width of this spread is a measure of the ucertaity caused by radom errors. The elemet of ucertaity i experimetal data ca be quatified ad should be reported alog with the actual experimetal value itself writte as the experimetal value ± some degree of ucertaity. I this case Raffaella would report a boilig poit of 00. ± 0.3 C, ad Barbara would report 00. ±.4 C. The value of the ucertaity gives oe a idea of the precisio iheret i a measuremet of a experimetal quatity; here, Raffaella is more certai of her values tha Barbara. There are may ways to quatify ucertaity, ragig from very simple techiques to highly sophisticated methods. The method used will deped upo how may measuremets of a sigle quatity are made ad o how crucial the reportig of the value of ucertaity is with regard to the iterpretatio of the experimetal data. We will cosider:. The Graduatio Method. Rage 3. Sample Stadard Deviatio 4. Cofidece Limits The Graduatio Method Whe a measuremet is made directly by the studet i lab, the ucertaity must be approximated. Usig the graduatio the ucertaity i a sigle measuremet is estimated by a value oe-half of the smallest level of graduatio i the measurig istrumet. For example, if a sigle measuremet of the legth of a object is to be made usig a meter stick marked with millimeter graduatios, the legth should be reported ±.5 mm (or ±.005 m). [some professors prefer a 0% rule] We will use ½ the lowest graduatio i our lab. You must use your ow judgmet i choosig the precisio usig the Graduatio Method. If i doubt, always be coservative; i.e. report the largest of possible ucertaities (50% vs. 0%). 7. Use a ruler to measure the width of your text. Report your measuremet with the correct ucertaity accordig to the Graduatio Method. 8. Make a temperature readig ad report with the correct ucertaity. 9. Repeat for the volume of water i a accurately read 50 ml volumetric flask. This is how all measuremets must be recorded i your lab reports. Some istrumets or glassware may have the ucertaity prited o the tool ad should be used i place of the graduatio method. 4
5 Rage The graduatio method adequately estimates the ucertaity of a sigle value but ca ot be used whe a series of values exists for a sigle observable. I this case the ucertaity ca be crudely approximated by the rage. The rage is give as the differece betwee the maximum ad miimum values of the measured quatity. I the case of the set of five weights give: trial : ±.000 g trial : ±.000 g trial 3: ±.000 g trial 4: g trial 5: g the rage is g g = g. So i our example, the value should be recorded as the average (mea or media) ± rage/, rouded to the correct sig. figs. Or, ± g (R) If you remember our previous example, we did ot have a way to describe the precisio of the two studets data. Errors cause ucertaity, ucertaity affects precisio; therefore, ucertaity ca quatify precisio. The greater the ucertaity, or rage, the less precise the values. Here we see that, crudely, the mass of the sample should be somewhere betwee ad g. We could use rage to estimate the precisio of each studet but rage teds to be to much of a over estimatio. We should seek a better estimatio. Sample Stadard Deviatio The most commo way to describe the ucertaity, or precisio, for a set of data is by the sample stadard deviatio (s). The stadard deviatio is used to describe the likelihood that a value will fall ear the mea for a ormal set of data s i x x i For ormal data sets, 68.3 % of experimetal values have statistical probability of fallig withi oe stadard deviatio (σ) of the mea, 95.5% withi σ ad 99.7% withi 3σ. let s determie the ucertaity of our weights usig stadard deviatio: trial : g trial : g trial 3: g trial 4: g trial 5: g xi x i s s 4 = So the value should be recorded as the average (mea or media) ± the sample deviatio: ± g, or ± g (s) This tells us that there is a 68.3 % chace that the mass is betwee ad g. Our precisio, or ucertaity, for the weights measured has bee foud as: Rage ± g, or ± g (R) SD ± g, or ± g (s) Notice that the stadard deviatio method gives a smaller margi for error, however, it oly describes a 68.3% cofidece. I other words, there is a 68.3 % probability that the true mass of the weights is betwee ad g. 68.3% is a reasoable descriptio of ucertaity but a 95 % cofidece iterval is the miimum stadard for reportig ucertaity i chemistry ad physics. Cofidece Limits The sample stadard deviatio ca be adjusted to ay give cofidece iterval by utilizig the followig equatio: ts Cofidece iterval = N Where s is the sample stadard deviatio, t is a statistical costat (foud o the followig slides) ad N is the umber of samples i the data set. Let us calculate the ucertaity of our weight measuremets with a 95 % cofidece C.I. = ± (.57)(0.000)/ 5 = ± So our ucertaity is ± (95%,N=5) 5
6 Cofidece Limit T values 50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% t value Cofidece Level 0.I lab, you have titrated 8 equal volumes of a ukow acid with a stadard base. You have dispesed the followig volumes of titrat: 43.8mL, 43.30mL, 44.0mL, 43.90mL, 5.69mL, 43.87mL, 43.88mL ad 43.70mL. Do the followig aalysis: A. Determie if ay gross errors occurred. B. Fid the mea ad media. C. Choose the best value for the average; justify. D. Estimate the ucertaity of the measuremets usig the Graduatio Method (assume correct measuremets). E. Calculate the precisio of your measuremets i terms of rage ad sample stadard deviatio. F. Adjust the ucertaity to a 99% cofidece level..if the actual amout of titrat added i problem 0 should have bee 43.85mL, what was the absolute error ad % accuracy?.o a further titratio, you obtaied a volume of 44.0mL. What is the relative error ad % precisio i relatioship to the ew average volume added? 6
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