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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.  SUPL-001, 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:
                                    avg       = ( 21.37 + 21.97 ) / 2 = 21.67

What about the average deviation?

                                    avg dev = ( | 21.37 - 21.67 |+ | 21.97 - 21.67 | ) / 2 = ( 0.30 + 0.30 ) / 2 = 0.30
Remembering that we must take the absolute value of the deviations

The percent deviation is
                                    % dev  =  100 X 0.30 / 21.67 = 1.4 % ( larger than the prescribed 1.0 %)


[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:
                                    avg       = ( 21.37 + 21.97 + 21.46 ) / 3 = 21.60

Note that the average has decreased slightly because our new value was less than the previous average.

What about the average deviation?
                                    avg dev = ( |21.37 - 21.60|+|21.97 - 21.60|+|21.46 - 21.60| ) / 3 = (0.23 + 0.37 + 0.14) / 3 = 0.25

Note that the average deviation has decreased slightly because our new result lies between the two earlier ones

Now, the percent deviation is
                                    % dev  =  100 X 0.25 / 21.60 = 1.2 % ( still larger than the prescribed 1.0 %)

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:
                                    avg       = ( 21.37 + 21.97 + 21.46 + 21.85) / 4 = 21.66

The average deviation is:

                                    avg dev = ( 0.29 + 0.34 + 0.20 + 0.19 ) / 4 = 0.26

and the % deviation is:

                                    % dev  =  100 X 0.26 / 21.66 = 1.2 % (still larger than the prescribed 1.0 %)

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,
 

                                    avg       = ( 21.37 + 21.46 + 21.85) / 3 = 21.56

The average deviation is:

                                    avg dev = ( 0.19 + 0.10 + 0.29 ) / 3 = 0.17

and the %deviation is:

                                    % dev  =  100 X 0.17 / 21.56 = 0.79%

which is well below the specified limit of 1%

 

The process of "throwing out" the result of a determination must be based on more than the desire to meet the suggested precision.  There is an obvious risk in eliminating a result.  We must be certain that the eliminated result is likely to be wrong (i.e., far from the true value).


[CASE 4]

Suppose, instead, that we choose to exclude the lowest measurement 21.37 mL.

The average of our remaining 3 determinations is now:
                                    avg       = ( 21.46 + 21.85 + 21.97) / 3 = 21.76
The average deviation is:

                                    avg dev = ( 0.30 + 0.09 + 0.21) / 3 = 0.20

and the % deviation is:

                                    % dev  =  100 X 0.20 / 21.76 = 0.92 % ( which is now lower than the prescribed 1.0 %)

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?

 
Average Reported
Pct Dev

Deviation from
"True" Value

% Deviation from
"True Value
CASE 1
21.60 +/- 0.25
1.4
-0.07
-0.32%
CASE 2
21.66 +/- 0.26
1.2
-0.01
-0.05%
CASE 3
21.56 +/- 0.07
1.2
-0.11
-0.51%
CASE 4
21.76 +/- 0.20
0.92
+0.09
+0.42%
   

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:

    • It is dangerous to exclude experimental values unless there is a sound experimental reason to do so. Reporting all of your results will generally improve accuracy, even though the precision may be worse.
    • With small numbers of determinations, the fact that some results cluster near one another does not indicate that they are likely to be closer to the true value than an outlying value.
    • If an exercise describes a target precision, it is helpful to calculate the precision as the experiment progresses rather than to wait until the end to see if the desired precision has been achieved.

 

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Last Update: RFS 2015-07-03