It has been assumed in these rules that the titration has yielded proportional results; but these are not always obtained. There can be no doubt that in any actual re-action the proportion between any two re-agents is a fixed one, and that if we double one of these then exactly twice as much of the other will enter into the re-action; but in the working it may very well be that no re-action at all will take place until after a certain excess of one or of both of the re-agents is present. In titrating lead with a chromate of potash solution, for example, it is possible that at the end of the titration a small quantity of the lead may remain unacted on; and it is certain that a small excess of the chromate is present in the solution. So, too, in precipitating a solution of silver with a standard solution of common salt, a point is reached at which a small quantity of each remains in solution; a further addition either of silver or of salt will cause a precipitate, and a similar phenomenon has been observed in precipitating a hydrochloric acid solution of a sulphate with baric chloride. The excess of one or other of the re-agents may be large or small; or, in some cases, they may neutralise each other. Considerations like these emphasise the necessity for uniformity in the mode of working. Whether a process yields proportional results, or not, will be seen from a series of standardisings. Having obtained these, the results should be arranged as in the table, placing the quantities of metal used in the order of weight in the first column, the volumes measured in the second, and the standards calculated in the third. If the results are proportional, these standards will vary more or less, according to the delicacy of the process, but there will be no apparent order in the variation. The average of the standards should then be taken.
| Weight. | Volume found. | Standard |
| 0.2160 gram | 72.9 c.c. | 0.2963 |
| 0.2185 " | 73.9 " | 0.2957 |
| 0.2365 " | 79.9 " | 0.2959 |
| 0.2440 " | 82.3 " | 0.2964 |
| 0.2555 " | 85.9 " | 0.2974 |
Any inclination that may be felt for obtaining an appearance of greater accuracy by ignoring the last result must be resisted. For, although it would make no practical difference whether the mean standard is taken as 0.2961 or 0.2963, it is well not to ignore the possibility that an error of 0.4 c.c. may arise. A result should only be ignored when the cause of its variation is known.
In this series the results are proportional, but the range of weights (0.216-0.2555 gram) is small. All processes yield fairly proportional results if the quantities vary within narrow limits.
As to results which are not proportional, it is best to take some imaginary examples, and then to apply the lesson to an actual one. A series of titrations of a copper solution by means of a solution of potassic cyanide gave the following results:—
| Copper taken. | Cyanide used. | Standard. |
| 0.1 gram | 11.9 c.c. | 0.8403 |
| 0.2 " | 23.7 " | 0.8438 |
| 0.3 " | 35.6 " | 0.8426 |
| 0.4 " | 47.6 " | 0.8403 |
These are proportional, but by using a larger quantity of acid and ammonia in the work preliminary to titration, we might have had to use 1 c.c. of cyanide solution more in each case before the finishing point was reached. The results would then have been:
| Copper taken. | Cyanide used. | Standard. |
| 0.1 gram | 12.9 c.c. | 0.7752 |
| 0.2 " | 24.7 " | 0.8097 |
| 0.3 " | 36.6 " | 0.8191 |
| 0.4 " | 48.6 " | 0.8230 |
It will be noted that the value of the standard increases with the weight of metal used; and calculations from the mean standard will be incorrect.
By subtracting the lowest standardising from the highest, a third result is got free from any error common to the other two; thus:—