Confidence interval in the prediction of the reaction progress
The Gaussian distribution or the normal distribution (or normal curve) is assumed to occur in the situation that each measurement contains a large amount of small, independent error sources. These errors have to be of the same magnitude, and as often both, positive and negative. When measuring the progress of a reaction one tries to eliminate the systematic errors, so that only accidental errors have to be taken into account. In that case the measured values will spread around the average value, in a form of the Gaussian-type curve. It can be proved that in the case when the average value of a measured value is the 'best value' or 'central tendency', a Gaussian distribution holds. The 'mean value' is here defined as that value, for which the chance of the good reproducibility on subsequent measurements is maximal.
Taking the definition of the standard deviation (see below) it can be seen that
s is the standard deviation in the Gauss distribution of the form:
where x and y represent the coordinates (x: measured value and y:relative occurrence or frequency). The points of inflection are situated at x =
± s.
For this distribution about two of the three measurements have a distance less than
s from the maximum value. And about one of the twenty measurements has a distance of more than 2s.
'Standard deviation'.
The standard deviation is a commonly-used measure of variation. The standard deviation of a 'population' of values is computed as:
s = [S(xi-µ)2/N]1/2
where
µ is the 'population' mean i.e. the average value of a measured value
N
is the 'population' size i.e. the number of measured values.
The estimate of the population standard deviation is computed as:
s = [S(xi-xbar)2/n-1]1/2
where
xbar
is the mean deviation
n is the number of experimental points on the thermoanalytical curves.
'Mean value' and 'confidence interval'.
Probably the most often used descriptive statistic is the mean. The mean is a particularly informative measure of the 'central tendency' of the variable (reaction progress) if it is reported along with its confidence intervals. The confidence intervals for the prediction (mean) give us a range of values around the mean where we expect the 'true' mean (reaction progress or reaction rate) is located (with a given level of certainty).
Example (low-temperature decomposed substance): in Fig. 1 the mean in the prediction is 3.1 years for reaching a reaction progress of 60% (isothermal conditions, T=20°C), and the lower and upper limits of the p=.05 confidence interval are about 2.7 and 3.5 years respectively These values indicate that there is a 95% probability that the mean required time for reaching 60% reaction progress is greater than 2.7 and lower than 3.5 years.
Figure 1:
Reaction extent
(DSC, normalized signals) and confidence interval (with d =
4s
= 0.5%)
of a low-temperature decomposed substance as a function of time under isothermal conditions (T = 20°C).
Figure 2:
Starting temperature and adiabatic induction time relationship of a low-temperature exothermally decomposed substance. Determination of the adiabatic induction time is made under adiabatic conditions. The confidence interval was determined
for d =
4s = 0.5%.
Note that the width of the confidence interval depends on the number of experimental data and on the variation of data values. The larger the number of experimental data, the more reliable its mean. The larger the variation, the less reliable the mean. The calculation of confidence intervals is based on the assumption that the variable is normally distributed in the 'population'. The estimate is valid if this assumption is fulfilled, i.e. if the number of experimental data is large, e.g. n=100 or more experimental points in one thermoanalytical curve.
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