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Applied Statistics . Fifth year Pharmacy student Pharos University 2013. I.1 Errors in Quantitative Analysis. - Every physical measurement is subject to errors , and therefore there is an uncertainty in the results that can be minimized but never eliminated completely.
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Applied Statistics Fifth year Pharmacy student Pharos University 2013
I.1 Errors in Quantitative Analysis - Every physical measurement is subject to errors, and therefore there is an uncertainty in the results that can be minimized but never eliminated completely. - The error is represented by the equation: E = xi - µ -Where E= Error, xi = the experimental value and µ is the true value.
I.2 Types of Errors -Errors are classified into two categories: 1-Determinate (Systematic) errors : a-can be attributed to concrete causes (instrumental defects, reagent impurities, personal errors, method error etc…). b- They are unidirectional c- are fixed for a series of measurements under the same experimental conditions. These error usually can be corrected in various ways for e.g. by a- by theoretical calculation of corrections. b- calibration of an instruments.
c- analysis of standards. d- running blank and so forth…. Determinate Errors may be : a- Constant : when the absolute value of E is the same for all examples. b- Proportional : when the relative error (E/µ) is constant (i.e. , the absolute error E is proportional to the quantity µ of the determined substance ). c- Composite : are a combination of constant and proportional errors. .
2-Indeterminate (random) errors : a- Accompany every measurement and are due non-permanent causes and include noise present in the measurement. b- They are bidirectional ( positive and negative) , and therefore affect results irregularly. c- Various types of random noise may occur in measurements , such as: i. Electronic noise in the detectors. ii. Non-reproducible placement of a sample cuvettes in the cell holder of a spectrophotometer. iii. Uncertainties in measuring mass or volume. Random error are decreased to a certain extent by increasing the number of measurements, but they cannot be eliminated, since an infinite number of measurements would be required.
The normal distribution Curve • The normal distribution curve is shown in Fig. The area under the whole • Curve from x = - ∞ to x = + ∞ is UNITY. • The width of the peak denotes the reproducibility of the measurements • (the better the reproducibility, the smaller the width).
Normal distribution cont. If one divides the area between two ordinates (Plus or minus) under the normal curve by the total area under the curve, this represents the fraction of the total measurements that an experimental value will fall between these limits. The values of the area are given in statistical tables. The contents of such a table are presented in approximate form in Fig, e.g., the probability, P%, that a value fall within the range µ ± 1 σ is 68.28%, where σ is the standard deviation of an infinite population.
Accuracy and Precision • The accuracy of a measurement denotes the closenessof an experimental value (xi ) ( or of the mean value of a set of measurements) to the true value, µ. It is usually expressed by the absolute error, E. • The Precision of a set of measurements refers to the agreement among the results. • Various cases of accuracy and precision are shown in Figure: • Case 1: denotes excellent accuracy ( unbiased) and precision because the measurements are close to the true value. • Case 2.: denotes poor or low precision (impercise) because the measurements are scattered and good or high accuracy because the average mean is close to the true value, µ. Case 3.: denotes excellent precision because the measurements are close together, and poor accuracy (biased) because the mean is far from µ. This indicates the existence of a determinate error which has not been discovered. So in this case an appropriate correction cannot be made. Case 4.: denotes poor precision because the measurement are scattered and poor accuracy because the mean is far from µ .
Another e.g. ( Excersise) Four students (A–D) each perform an analysis in which exactly 10.00 ml of exactly 0.1 M sodium hydroxide is titrated with exactly 0.1 M hydrochloric acid. Each student performs five replicate titrations, with the results shown in Table
