SIGNIFICANT FIGURES (4SF) (4SF) (5SF) (3 SF) 62.4 (3 SF) x 10 4 (5 SF) Figure 0

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1 SIGIFICAT FIGURES Wth any calbrated nstrument, there s a lmt to the number of fgures that can be relably kept n the answer. For dgtal nstruments, the lmtaton s the number of fgures appearng n the dsplay. For analog (non-dgtal) nstruments, the general rule for keepng fgures s to read the nstrument to one fgure beyond the smallest graduaton markng. For eample, a balance wth a beam marked to the nearest 0. g should be read to the nearest 0.0 g, the last fgure beng estmated. In Fgure, a lne s shown net to two rulers wth dfferent calbratons. How long s the lne? Measured wth the lower ruler, ts length s 0.8 unt. Measured wth the upper ruler, the lne s 0.77 unt. The length of the lne has obvously not changed; however, our ablty to measure ts length has changed wth our use of a dfferent measurng nstrument. In both cases, the last fgure retaned s an estmated one. Estmatng the last fgure does ntroduce Fgure 0 some uncertanty, but ganng the etra dgt (wth the resultant ncrease n precson) more than offsets any error. Standard practce n vrtually all scentfc measurements s to make the measurements n ths fashon, so that your answer has a group of fgures whose numercal value you know eactly, plus one fgure at the end whose value has an mplct uncertanty of ±l or ± from the stated value. The number of fgures kept n the measurement usng ths gudelne s called the number of sgnfcant fgures. Snce all measurements are presumed to have uncertanty n the last fgure, wrtng a number n whch all the fgures are eactly known wll result n the under-estmaton of how precsely the measurement was performed. When countng the sgnfcant fgures n data, whch you dd not personally read, the followng rules are used:. All nonzero dgts are assumed to be sgnfcant fgures.. Zeros between two non-zero dgts (sometmes called captve or nternal zeros) are sgnfcant, and zeros to the left of all the nonzero dgts (called ntal zeros) are not counted as sgnfcant. 3. Zeros to the rght of all the nonzero dgts (the tralng zeros) are sgnfcant f the number contans a decmal pont.. Tralng zeros are generally not sgnfcant f the number has no decmal pont (although some people mght or mght not count them, dependng on the stuaton). It s often advantageous to wrte numbers n scentfc notaton to elmnate ambguty. A number n scentfc notaton s wrtten as a multpler (the dgt term) tmes a power of 0 (the eponental term). The dgt term should have only one fgure to the left of the decmal pont. For eample, the number can be wrtten as The number 67,000 (three sgnfcant fgures) s wrtten as 6.7 0, or to ndcate four sgnfcant fgures. Ths last eample llustrates another advantage to wrtng numbers n scentfc notaton. It s possble to show eplctly the number of sgnfcant fgures n a number wth tralng zeros. For eample, 67,000 would normally be thought to have three sgnfcant fgures. If t were actually measured wth an nstrument capable of measurng to the nearest 00 unts, you would be justfed n keepng four sgnfcant fgures. Wrtng t as shows ths eplctly wth no room for msnterpretaton, snce the number now contans a decmal pont. Some eamples of the countng of sgnfcant fgures:.0 (SF) (SF).0060 (SF) 00 (3 SF) 6. (3 SF).00 0 ( SF)

2 SIGIFICAT FIGURES I CALCULATIOS: ROUDIG OFF The aspect of dealng wth sgnfcant fgures whch causes the most trouble for begnnng students s n the handlng of sgnfcant fgures n a calculaton. The prncple s that the results of a calculaton cannot mply greater accuracy or precson than s n the numbers used n the calculaton. How to carry out ths prncple depends on the type of mathematcal operatons nvolved n the calculaton. Suppose only multplcaton or dvson are nvolved. The number of sgnfcant fgures n the result s equal to the smallest number of sgnfcant fgures n any of the numbers used n the calculaton. For eample, n the calculaton = (shown to the number of fgures on a calculator dsplay), the factor wth the fewest sgnfcant fgures s 0.009, wth three, so the answer can be epressed to only three sgnfcant fgures, as 8.. In other words, t must be rounded off. Smlarly, ( has only SF) (607 has 3 SF) 7 (.09 ( ( ( 3.7 = = When roundng numbers off, f the amount to be dropped s less than half of the last fgure to be kept, delete t. If the amount to be dropped s greater than half of the last fgure to be kept, ncrease the last fgure by. If the amount to be dropped s eactly half of the last fgure to be kept, then do nothng f t s even and ncrease t by f t s odd. The effect s that the value of the overall result wll be lowered n half the cases and rased n the other half, meanng that any errors ntroduced by roundoff wll cancel out n the long run. For eample,.0 would be rounded to.6, whle 9. would be rounded to 9. Round numbers off n one step, not n a seres of steps n whch you cut back one number at a tme. We now return to the subject of keepng track of sgnfcant fgures n calculatons. If the calculaton nvolves only addtons and subtractons, the number of sgnfcant fgures n each term n the calculaton s less mportant than the poston of the last sgnfcant fgure relatve to the decmal pont n the number. The poston (relatve to the decmal pont) of the last sgnfcant fgure n the answer s the last poston n whch all the terms n the calculaton have a sgnfcant fgure. Study the followng eamples: = = = = 0.6 The same rules for roundng off apply. ote that subtractng two numbers farly close n value can lead to the 6

3 loss of sgnfcant fgures. In etreme cases, almost all of the sgnfcant fgures n your data can be lost n because of ths. When rasng a number to a power or takng a root, the same rules apply as for multplcaton and dvson. When takng a logarthm or antlogarthm, the number of sgnfcant fgures n the number should equal the number of fgures n the mantssa (that s, the fgures after the decmal pont) of ts logarthm. If the calculaton nvolves several mathematcal operatons, keep track of the sgnfcant fgures as you perform each step. For eample, consder the calculaton [.9 ( ) ] (.37 (9. ) Workng from the nner parentheses outward, we carry out the followng seres of steps, preservng the proper number of sgnfcant fgures at each step: (.9.) (.30 = 3.7 (.37 ( = = 9 A few words concernng sgnfcant fgures and calculators (or spreadsheets) are n order. Calculators (spreadsheets) are not programmed to keep track of sgnfcant fgures. You are totally responsble for that. Occasonally (especally when they are zeros), the calculator (spreadsheets) wll drop off the fnal dgts, gvng an answer that has too few sgnfcant fgures. More commonly, though, the number of fgures shown on the calculator (spreadsheets) s many more than s justfed by the data used. The calculator (spreadsheets) wll do eactly what t s told-no more, no less. If you gve t wrong nformaton, t wll do ts job of processng the nformaton. Ths wll ndeed gve an ncorrect result, but ths s not the fault of the calculator (spreadsheets). It s the fault of the person usng the calculator (spreadsheets). ACCURACY AD PRECISIO We have used the terms accuracy and precson wthout defnng them. These terms are used almost nterchangeably by some people, but mean very dfferent thngs. The accuracy of a measurement or epermental result shows how close that result s to some "correct," "true," or "accepted" value for that quantty. The precson of a seres of measurements shows how closely that seres of measurements agrees nternally that s, how close one measurement s to another n the set. The precson of a set of measurements s almost totally a functon of how well that set of measurements was performed by the epermenter; n other words, t s a measure of the epermenter's technque. The accuracy of the measurement s partally a functon of technque, but t s also related to the qualty of the measurng nstrument and the procedure that the epermenter uses. To llustrate these concepts, Fgure shows four targets. The precson of the attempt to ht the bullseye s measured by how Fgure 3

4 closely the holes are grouped. The accuracy of the attempt s measured by how closely the "average" shot has come to the center of the target. Good agreement of the results wth one another means that your technque was reproducble, and that, possbly, the method (or, n ths case, your rfle) needs adjustment or correcton. TYPES AD SOURCES OF EXPERIMETAL ERROR Three general categores of error are nstrumental, operator, and method errors. Instrumental errors are due to some naccuracy n the measurng nstrument: a poorly calbrated ruler or volumetrc nstrument, an electrcal malfuncton n a meter, a worn pvot n a balance. The user can sometmes control the error (for eample, a ruler that s I cm short can stll gve relable results f the epermenter always remembers to add I cm to all the measurements), but usually the epermenter s lmted by the measurng nstrument. Operator errors orgnate n the technque of the epermenter or operator of the nstrument. Msreadng the scale on a buret, falng to set the zero pont on a spectrometer, and falure to clean a balance before usng are eamples of operator errors. These are the types of errors over whch the epermenter has the most control. Method errors derve from a flaw n the method of analyss or measurement. In other words, the method used cannot produce accurate and/or precse results. A method that assumes that a materal s 00% nsoluble when t s really partly soluble wll gve poor results no matter what measurng nstrument s used or who s usng t. As wth nstrumental errors, the operator has lmted control over ths type of error, other than choosng an alternate method that gves better, more relable results. Errors can also be classfed as constant or random. A constant error s always the same sze and drecton. A buret whose frst mark after the zero s.00 ml nstead of.00 ml wll always gve a volume readng that s.00 ml too large (assumng that you zero the buret each tme), or a voltmeter may gve a readng that s too small by a constant value due to some electrcal crcumstance n the system beng studed. A random error may (concevably) be of any sze and of ether drecton. The fluctuaton of a meter readng due to varatons n the electrcal current s a random error, as are the fluctuatons n a balance caused by drafts deflectng the balance pan. Random errors can be treated by a statstcal analyss of the results. STATISTICAL MEASURES OF PRECISIO Snce there are so many ways n whch errors can creep nto any epermental measurement, t s hghly unlkely for any set of repeated measurements to have eactly the same result n each tral. There can also be varablty n the sample from one tral to the net. When a quantty has been measured repeatedly and several results obtaned, we need to epress the varablty of the results n a way that can be easly understood by anyone nterested n the overall relablty and qualty of the measurements. Statstcs provdes several ways to do ths. We wll be dealng wth a set of epermental results n whch trals were performed, all of whch were supposed to gve the same result. The ndvdual results for trals,,... wll be denoted by,,.... The frst calculaton most commonly done s to fnd the arthmetc mean, or average, of the results. Ths s calculated by takng the sum of the results and dvdng by the number of trals : = = (The Greek captal sgma (r) denotes the sum of whatever follows t.) The mean usually represents the most lkely value of the quantty beng measured. For eample, the followng set of data was obtaned for a standardzaton of 0.0 M KMnO soluton: M, M, M, M, M. The arthmetc mean of these fve trals s = = ( = = = )

5 = /_( ) = M A smple measure of the precson of the set of results s the range w for the set of trals. The range s smply the dfference between the hghest and lowest values among the results. In our eample, the range s M. Ths value does not really communcate any sgnfcant nformaton about the precson. It s only when the range (or the other measures of precson to be dscussed) are compared wth the value of the quantty beng measured that real meanng can be attrbuted to these quanttes. (For eample, knowng that a mass can have an error of ± g s not that sgnfcant by tself. If the object weghs 000 kg, the error s a very small; f the object weghs g, the error s very sgnfcant.) The relatve range of a seres of measurements s equal to the range dvded by the mean. In ths case, the relatve range s M/ M = 0.0. Relatve measures of precson are often epressed n terms of percentages, parts per thousand, and so on, to obtan numbers that are easly handled. For eample, multplyng the relatve range by 00 gves t n percent; multplyng the relatve range by 000 gves t n parts per thousand (ppt); multplyng the relatve range by 0 6 gves t n parts per mllon (ppm). In our eample, the relatve range could be epressed as.%, ppt, or,000 ppm. A more meanngful measurement of precson s the devaton of each of the ndvdual values. The devaton for tral (d ) s found by subtractng the mean of the set from the measured value for that partcular tral: d = ote that f <, the devaton for that tral s a negatve number; values greater than the mean have postve devatons from the mean. If we average the absolute values of the devatons (f we average the values wth ther sgns, we always get zero), we obtan the average devaton (d) for that set of results. Dvdng the average devaton by the mean gves the relatve average devaton (rad) for that set of results. As wth the relatve range, the relatve average devaton can be epressed n percent, parts per thousand, and so on. For the set of results gven earler, the devatons, average devaton, and relatve average devaton are d = M d = M d 3 = M d = M d = M d = M rad = = 0.33% or 3.3 ppt or 3300 ppm ote that the range or the average devaton has the same unts as the quantty beng measured, but the relatve range or relatve average devaton has no unts. As a fnal measure of precson, the standard devaton s often calculated. Ths number has ts greatest meanng for large samples (wth 0), but t can be used for smaller samples (down to = 3) wth some restrctons on ts nterpretaton. The standard devaton s s calculated by takng the sum of the squares (not the square of the sum!) of the devatons for the ndvdual trals, dvdng the result by -, and takng the square Two sgnfcant fgures have been kept n the calculaton of the average devaton, even though only one would be justfed based on the numbers. Ths practce of addng an "etra' sgnfcant fgure to the result of the calculaton of the average devaton (and the standard devaton) s generally accepted, even though, techncally, t s not justfed based on the rules for sgnfcant fgures. Any tme, n any calculaton, that you drop to just one sgnfcant fgure, a great deal of precson s lost.

6 root of the result: ( ) = s The standard devaton for any set of results s always larger than the average devaton. As wth the range, the relatve standard devaton can be calculated by dvdng the standard devaton by the mean, and multplyng by 00, 000, and so on, f desred to obtan the result n percent, parts per thousand, and so forth. For our eample, the standard devaton and the relatve standard devaton are: s = ( ) + ( ) + (0.0000) + ( ) + ( ) = relatve standard devaton = = 0.7% =.7 ppt One of the most mportant uses of the standard devaton for a set of trals s that the statstcal dstrbuton of the results can be related to the magntude of the standard devaton. For a large sample ( 0), the dstrbuton of results wll be such that 68% of the results wll be wthn one standard devaton of the mean, 9% wll be wthn two standard devatons, and 99% wll be wthn. standard devatons. For smaller samples, the statstcs of ths "normal dstrbuton" wll break down and these fgures wll not apply eactly. REJECTIO OF A TRIAL In a set of results, one or more often seem suspcous n that they are qute far away from the others. Are there any crcumstances that let us legtmately throw out these suspcous values so that they don't affect the average? If there s a known epermental problem wth that tral, such as a soluton beng splled, the wrong reagent added, an end-pont overshot, you would be justfed n elmnatng that result from consderaton. But you would not be justfed n actng as though that tral dd not even est. You should report all the data and calculatons for that tral, and then eclude the results of that tral from the calculaton of the average, average devaton, and so on. Totally gnorng the result and not reportng t s dshonest reportng of what you dd n that eperment. Suppose that no vald epermental reason ests for rejectng a result? Can you reject that tral just because t "looks wrong" or t "just can't be rght"? Based smply on your "gut feelngs," the answer s no! You must have a vald bass on whch to justfy rejectng a tral. It s possble, however, that random errors n the measurements (n the nstrument, method, or operator) have compounded themselves n that tral to produce a result that s so far away from the mean that t s unlkely that the result s vald. In that case, you may be able to statstcally reject a partcular tral. Two statstcal tests are commonly used to reject a tral. The frst s called the d test. To apply ths test, calculate the mean and average devaton of the set of results wthout the suspect result. If the devaton of the suspect result from ths newly calculated mean s greater than four tmes the newly calculated average devaton, then the result can be rejected on statstcal grounds. For our data, the most suspect value s M (tral ). After elmnatng that value, the new average and new average devaton are M and M, respectvely. The suspect value s M from the new mean, whch s greater than four tmes the new average devaton. Thus, that result could be rejected. Techncally. ths formula gves the estmated standard devaton. The true standard devaton σ (Greek lower case sgma) can be calculated only for samples of nfnte sze. 6

7 The second test s the Q test, where Q s the rato of the dfference between the suspect result and ts nearest "neghbor" and the overall range of the results: suspect value nearest value Q = l arg est value smallest value Ths calculated Q s then compared wth a table of crtcal Q values. If the calculated Q value s greater than the approprate value of Q from the table, the suspect result can be rejected wth the confdence assocated wth that tabulated Q value. A table of crtcal Q values follows: =3 = = =6 =7 Q 80% Q 90% Q 9% Q 99% In ths case, our calculated Q value s 0.000/0.000 = 0.6. For =, ths value of Q s greater than the Q 80% and Q 90% values, but less than the Q 9% and Q 99% values. Hence, we can reject the value of wth 90% confdence that we are dong the rght thng, but we could not reject ths value wth 9% confdence. If you decded to reject that result, you would stll have to report that the result of M was rejected at the 90% level by usng a Q test, or some smlar wordng. 7