
Most exercises in CHE 133  134 involve replicate determination of some measurable quantity. The number of repetitive measurements is usually small (3  5 determinations). We invoke simple statistical concepts in the reporting of the results of such measurements. SUPL001, Analysis of Experimental Reliability, discusses these concepts in detail.This page presents supplementary information about three of the most common quantitites. We use a concrete example based on a typical exercise:Suppose you are instructed to determine the concentration (in grams/100 mL) of a solution of sodium hypochlorite (NaOCl) by titration with sodium thiosulfate of given concentration. The titrations are performed on "precisely equal" 25.00 mL volumes of the unknown solution (aliquots delivered by a 25 mL transfer pipet). Barring errors introduced by the investigator, the volumes are "precisely equal" within the intrinsic error of the pipet.The instructions state that you should "report three determinations of the concentration that have a percent deviation less than 1%." This translates operationally into the statement: "If the percent deviation of the first three determinations is greater than 1%, you should peform a fourth determination." and so on, until the required percent deviation is achieved.Your first titration requires 21.37 mL of sodium thiosulfate. The second titration requires 21.97 mL. While the third titration has not yet been performed, it is useful to examine the results obtained thus far.To have a sense of how the exercise is proceeding, is it necessary to complete the calculation of the percent of sodium hypochlorite in g/mL at this stage? NO! The overall computation involves only fixed quantities such as molar masses and the given concentration of the sodium thiosulfate. Since the volumes of titrant are "precisely equal", the percent deviation of the final result will be exactly the same as the percent deviation of the volumes of sodium thiosulfate used.The average of our two results so
far is:


[CASE 1] We perform a third titration which requires an amount within the range of the first two measurements  21.46 mL of sodium thiosulfate. Now, the average of our three results
is: What about the average
deviation? Now, the percent
deviation is In accordance with the instructions, we should perform a fourth titration. 

[CASE 2] Suppose the fourth titration requires 21.85 mL of sodium thiosulfate  again inside the range of the first two. But, the instructions suggest reporting 3 values. Is there a basis for selecting three out of the four? If we have an experimental reason for suspecting one of the titrations, e.g., going past the end point in the titration producing the highest result (21.97) we might choose to disregard that determination. Such decisions must be documented in the laboratory notebook. I.e., it should have been noted, before performing the fourth titration, that the end point was passed in the third titration. If there is no reason to view any of the four titrations as unreliable, our average
is: Our percent deviation is still 1.2%, the same % deviation that we obtained after 3 determinations. If we have no basis for chosing three of the four, the only strategy is to do a fifth determination and hope that it will fall within the range. 

[CASE 3] If, on the other hand, the titration
producing the result 21.97 mL was
suspect (i.e., we recognized we passed the end point in that titration),
then we can exclude that value and use the three lower values for the
calculation of the average and average deviation. Namely,
which is well below the specified limit of 1% 



[CASE 4] Suppose, instead, that we choose to exclude the lowest measurement 21.37 mL. The average
of our remaining 3 determinations is now: The likelihood that the large value ( 21.97 mL ) was incorrect becomes much less plausible. The results do not suggest a specific problem with that value. The lower values are now just as likely to be incorrect as the larger values. 

Suppose, in the above example, the "true" value of the volume was 21.67. What is the accuracy of our determination in each of the above instances in percent?
Note that, while the % deviation in the fourth case is within the prescribed precision, the deviation from the true value is comparatively large. We have sacrificed accuracy by increasing the reported precision. 

Lessons to be learned:

CHE 133 HOME PAGE CHE 134 HOME PAGE
Last Update: RFS
20150703
