Theory of Errors, Chemistry tutorial


Analytical chemistry is basically a specialized feature of chemistry which mainly deals with both the qualitative analysis (that is, investigating what are the component of a given sample) and quantitative analysis (that is, investigating how much of components are in the sample). Whenever reporting the quantitative measurement of an experiment, there are usually some controllable and uncontrollable variations which accompany such measurement. Such forms of variations are known as Error.

Definitions of an Error:

Error is a controllable and uncontrollable variation noticed whenever comparing the measured values to the true value. The error influences the precision and accuracy of a measured quantity.

Types of error:

There are fundamentally two kinds of errors termed as: systematic errors and random error

1) Systematic Errors:

These are or else termed as determinate errors as they can be determined and corrected.

Example: By using pH meter which has been incorrectly standardized. If the pH buffer employed is 14.07 though mistaken for 14.00, thus a medium measured as 12.48 is in reality 12.41 and a value put as 13.74 is in reality 13.67. Any measurement by such pH meter should be buffered by a factor of 0.07.

=> Commonly encountered Systematic Errors:

Systemic Errors generally encountered in the course of laboratory exercises comprise:

a) Instrumental Error: This takes place whenever faulty equipment and weight and also glassware employed are not calibrated or wrongly calibrated.

b) Operative Errors: Such are errors traced to the operators (that is, personnel error), either as an outcome of the experience of the personnel comprised or the operator not been careful adequate. It might be mathematical error in the computation or prejudice in estimating the measurement.

c) Methodic Error: These are the errors innate in the analytical procedure or method used. This is a very serious problem for an analyst. Such comprise errors such as co-precipitation having impurities, incomplete reaction, impurities in the reagents employed and so on. The methodical error is as well correctable.

=> Prediction and Correction of the systematic Error:

There are different ways to detect and correct systematic error:

a) Analyze samples of known composition: Your process must reproduce the recognized answer; if not there is a problem by the method or equipment employed.

b) Analyze blank Samples: Samples having none of the parameter been sought. If you notice a non zero result, it signifies your process is responsible for more than what you intended.

c) Use various analytical processes to measure the similar quantity. If the results don't tally, it signifies there is an error related by one of the methods.

d) Let various operators of varying capabilities in different laboratories (by using the similar process or different methods) carry out the similar analyses. Disagreement or variations of high magnitude point out error traceable to operators or equipment employed.

2) Random Errors:

These errors are as well termed as indeterminate errors. They are errors due to the restrictions of physical measurement and can't be ignored. A better experiment or duplicated experiment might decrease the magnitude of such kinds of errors, however can't remove it completely. These kinds of errors are as well termed as accidental errors. The errors are pointed out by small differences in successive measurements made by the similar analyst under almost similar experimental conditions.  Random errors can't be predicted or estimated. They can either be positive or negative.


a) The kind of variation related by similar analyst reading the same absorbance scales numerous times.

b) Variation related with the three or four different analyst reading the similar measuring scale or reading the lower measurement of the volumetric flask.

Apparently they would report varying values reflecting the subjective interpolations between markings.

This has been noticed that these kinds of errors for all time follow random distribution; therefore mathematical laws of probability can assist in arriving at conclusions regarding the most probable outcomes in a sequence of measurements.

On a general note, errors influence the precision and correctness of a measured quantity thus raising questions on the integrity of the reported values.

Expressing Accuracy of a measurement:

There are many ways via which accuracy of a measurement can be expressed, these include:

a) Absolute Error or Absolute Uncertainty:

This is variation or difference expressed between the true value and the measured value. This is reported in the similar units as the measurement

Example: If a 4.97mg of an analyte is analyzed as 4.91mg, the absolute error is 0.06

This becomes mean error if the measured value is the average of some measurements.

b) Relative Error or Relative Uncertainty:

This is an expression comparing the absolute uncertainty to the size of its related measurement or absolute error deduced as percentage of the true value.

Example: From the above, the relative error in the analysis is:

(0.06/4.97) x (100/1) = 1.21

The relative accuracy can then be expressed as follows:

(4.91/4.97) x (100/1) % = 98.79

Note that:

i) The relative accuracy and relative error for all time provide 100% if summed altogether.

ii) Relative errors can be deduced (as illustrated above) as parts per hundred (that is, in %) or parts per thousand (ppt)

Illustration: If the outcome of an analysis, is 29.74µg is compared to the true value of 30.15 µg. Compute the relative error in part per hundred and part per thousand.


Absolute Error = 29.74 - 30.15 = - 0.41

Relative Error in pph = (0.41/30.15) x 100 % = 0.0135 pph

Relative error in ppt = (0.41/30.15) x 1000 % = 0.135 ppt

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