# Uncertainty And Error Analysis: Unerdenity And Error Analysis

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*Register to read the introduction…*parallax, reading error etc.

An experiment is said to be accurate if the systematic error is small; meaning, the experiment was done very carefully.

3.2 RANDOM ERRORS

Random errors are the unavoidable errors that creep into every measurement, no matter how carefully it is performed. Possible causes of random errors may be: (a) random motion of air molecules, (b) unstable voltage source from mains supplied by TNB, (c) vibrations caused by a passing motor vehicle (or even a passing person!) can unpredictably influence force measurements carried out using a sensitive electronic balance, or (d) background radiation when dealing with radioactive; e.g. cosmic rays.

4. UNCERTAINTY FROM A SINGLE MEASUREMENT

Whenever we report an experimental quantity, then we must report two

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We now introduce another technique for combining uncertainties, which represents an application of differential calculus. This may sound daunting and overly complicated but, in fact, if we can differentiate functions such as sines, cosines and logs, we will encounter few difficulties in calculating uncertainties involving these functions.

7.2.1 PARTIAL DIFFERENTIATION

Suppose V depends on the two variables, a and b. The mathematical way of writing this is:

V = V (a, b) We say the V is a function of a and b. An example of such a function would be:

V = ab[pic] If a changes by an amount (a, and b changes by an amount (b, we can write the accompanying change in V, (V, as:

(V = [pic](a + [pic](b .....(5)

[pic] is a partial derivative of V with respect to a. When finding a partial derivative, all quantities in the equation, except for the one that is being differentiated, are taken to be

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To find [pic], we treat b as a constant and a2 differentiates to 2a , to give [pic]= 2ab

To find [pic], we treat a as a constant and b differentiates to 1. This gives

[pic]= a2

In order to use equation (5) in problems of error propagation, we replace the quantities (a and (b by the uncertainties (a and (b, so that the equation is rewritten:

(V = [pic](a + [pic](b.....(6)

[pic] means that we take the magnitude of the partial differential, that is, we ignore any minus sign that may occur once we have differentiated. The consequence of not doing this is that cancellation of the terms on the right-hand side of equation (6) could occur. Note that we partially differentiate the function with respect to each quantity that possesses uncertainty. Quantities with no uncertainty are regarded as constants.

7.3 COMBINING UNCERTAINTIES: SUMS, DIFFERENCES, PRODUCTS AND QUOTIENTS

Equation (6) is applicable to any formula that you are likely to encounter. However, there are situations that are so common, such as taking the product of two quantities, that it is worth using equation (6) to determine the relationship that combines the uncertainties in the quantities.

7.3.1