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Daniele Naviglio » 23.Treatment of experimental data


Treatment of experimental data

Analytical methods and procedures in Analytical Chemistry and Instrumental Analytical Chemistry enable us to obtain a numerical result which is reported at the end of the analysis of a particular sample. This must be legible, and must contain all the information relative to the analysis carried out. The analytical result is as important as all the other operations involved in the method, and thus must be accurately recorded. Statistical analysis of experimental data dictates the guidelines for the treatment of experimental data, and indicates the way in which the final result of the analysis is expressed.

Analytical methods: carried out according to procedures necessary in order to obtain an analytical result. In general the methods follow precise indications which are summarised in the form of “recipes”. At the conclusion of an analytical procedure, an analytical result is reached.

Analytical result: a number which expresses the quantity of a substance found in an unknown matrix. This is obtained by the application of statistical analysis.

Errors in analytical determination

Reliability: the correspondance between the experimental result and “true” value. This is influenced by factors depending on the method, execution and efficiency of the instrumentation.

Factors affecting reliability of a result

  1. Sensitivity: minimum quantity of a substance determinable;
  2. Specificity: possibility of dosing with a given method without interference of one species in the presence of another;
  3. Accuracy: indicates the closeness of the average of a number of results to the “true” value;
  4. Precision: an index of repeatability of a measurement (the smaller the interval the more reliable the result).

Error

Definition: the difference between “true” value and the result of analysis.

Classification of error

  1. Determinate or Systematic Errors follows from:
  • errors of method;
  • errors due to reactivity;
  • errors due to instrument use;
  • operating errors.

These may be eliminated since they go in a single direction (upwards or downwards).

  1. Indeterminate: these result from the effect of variables not controlled for at the moment of measurement. They cannot be eliminated since they proceed in both directions. Indeterminate errors are inherent in all physical and chemical procedures of measurement, and are linked to the same material extension of the measuring instruments. Statistical analysis helps us to minimise errors associated with experimental measurement.

Notions of Statistical Analysis

Statistical Analysis is a branch of mathematics applied to the study of data drawn from a variety of sources: experimental measurements, demographic phenomena, meteorological forecasting, etc.

A series of experimental results can be grouped in a certain number of classes to study their development.

Class: comprises a series of result in a predetermined interval known as Extent of class.

Frequency of class: is the number of results grouped in classes.

Histogram: a graph recording the frequency compared to extent of classes.

Gaussian Curve: is obtained by carrying out infinite determinations. It is easily demonstrable that, for experimental measurements, which are not a random phenomenon like the throw of a die, the histogram obtained for an increasing number of measurements tends to approximate to a bell shape which is typical of the Gaussian Curve.

Principal characteristics of the Gaussian Curve

  • A symmetrical bell curve around a central value (median or average);
  • Possesses two flex points which are always symmetrical to the central value;
  • The positive deviations and relative negative deviations are equally probable;
  • Small deviations are more frequent than large deviations.
  • The probability of error associated with the average value is nil.

The characteristics of the Gaussian curve assure us above all that it is possible to give a result ahead of a series of experimental measurements, and this is guaranteed by the bell shape with its highest point precisely in the proximity of the average (mean) value. In addition, it can be deduced from the curve that it is possible to minimize only indeterminate errors, while the complete removal of error is for an infinite number of measurements.

Gaussian Curve

Source:Software

Source:Software


Arithmetical Average (mean) and standard deviation

Average: if the experimental results have equal statistical weight, the average value of, or arithmetical average is more probable than each result.

x= (x1+x2+…xN) /N = 1/N ∑ xi

It is demonstrated that the average N value, where each is measured equally accurately,  is √N times more probable than any single measurement. This is the perfectly mathematical reason why the average of a set of experimental results is reported, and not one single value.
True standard deviation (o): the distance between each of two points of infliction and the average value x. The smaller the value, the greater the precision of the measurement.

Calculation of variance, standard deviation and estimated standard deviation

Variance: is the sum of the squares of single variances of the average divided by N, when the number of measurement is infinite, divided by N-1 in other cases.

Vx= ∑ (xi-x)2 /N-1 di= xi-x

Standard Deviation: for an infinite number of measurements:

σ = √Vx = √( ∑ di2 /N)

Estimated standard deviation: for a finite number of measurements:

∂= √Vx = √( ∑ di2 /N-1)

Mean deviation

Mean deviation: is the arithmetic mean of a difference, in absolute values, between single values and the average (mean).

∂ = (ld1l+ld2l+…ldN-1l+ldNl)/N
= 1/N ∑(di)

di= lxi-xl
The mean deviation is an estimate of the dispersion of values which is less efficient than the standard deviation of a single measurement S.

Method of 4∂ for discarding a questionable result

The mean deviation is calculated, excluding the dubious result. If the deviation of this result is greater than four times the mean deviation, the dubious result is rejected.

The mean deviation is calculated, excluding the dubious result. If the deviation of this result is greater than four times the mean deviation, the dubious result is rejected.


Absolute and Relative Error

Absolute Error: difference between the experimental and the “true” result.

Relative Error: ratio between absolute error and effective size multiplied by 100 (%).

Error propagation

c= analytical result; w= constant value; a= experimental value; b= experimental value.

  1. C= w*a*b ……………. Err(c) = Err (a) + Err(b);
  2. C= w(a/b) …………… Err( c) = Err (a) + Err (b);
  3. C= w(a±b) …………… Err (c) = w(a Err(a)/c+bErr(b)/c);
  4. C= w*an*bm ………… Err (c) = nErr(a) +m Err(b).

The propagation of error shows that the greater the number of operations that are carried out, the greater the associated error in the final result of the analysis. This means that when a method is fine-tuned, a minimum number of operations must be performed in order to obtain the result.

Significant numbers and rounding off

Significant numbers

Are the numbers necessary to express the result of a measurament, with the same degree of accuracy with which it has been carried out. Only the final figure in a result may be in doubt. When the last figure of a result is in doubt, and its elimination would result in lowered accuracy, this figure is reported in brackets.

Rounding off

If the figure following the last significant figure is <5, then it is estimed downwards. If the figure following the last significant figure is >5, then it is approximated upwards. If the figure following the last significant figure is =5 followed by an infinite number of zeros, then this passes on to the next even number (in the case of an odd number). If the last digit is an even number, it remains the same.

Treatment of experimental data

A set of experimental data can be recorded in the following ways:

  • table;
  • graph.

The information of experimental data reported as a graph is greater compared to that reported in a table. From observation of a graph, the relation between the dependent and independent variable is immediately evident.

Function

If a precise analytical relation exists between the dependent variable (y), and the independent variable (x), then it is possible to apply statistical methods in order to reduce the set of experimental data to a pure analytical expression, such as:

y=f(x)

This is the best method for dealing with experimental data since the information is complete. To best interpolate the experimental points, the method of minimum squares is used. A simple method to obtain an interpolation of experimental data is to use Excel spreadsheets, which provides both a graph of the best curve to interpolate the experimental points, and the mathematical equation.

 

 

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