Question

explain in detail about the various types of errors and explain the propagation of errors​

Answer 1

Propagation of Error (or Propagation of Uncertainty) is defined as the effects on a function by a variable's uncertainty. It is a calculus derived statistical calculation designed to combine uncertainties from multiple variables, in order to provide an accurate measurement of uncertainty.

Introduction

Every measurement has an air of uncertainty about it, and not all uncertainties are equal. Therefore, the ability to properly combine uncertainties from different measurements is crucial. Uncertainty in measurement comes about in a variety of ways: instrument variability, different observers, sample differences, time of day, etc. Typically, error is given by the standard deviation ( σx ) of a measurement.

Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty. For example, lets say we are using a UV-Vis Spectrophotometer to determine the molar absorptivity of a molecule via Beer's Law: A = ε l c. Since at least two of the variables have an uncertainty based on the equipment used, a propagation of error formula must be applied to measure a more exact uncertainty of the molar absorptivity. This example will be continued below, after the derivation.

Derivation of Exact Formula

Suppose a certain experiment requires multiple instruments to carry out. These instruments each have different variability in their measurements. The results of each instrument are given as: a, b, c, d... (For simplification purposes, only the variables a, b, and c will be used throughout this derivation). The end result desired is x , so that x is dependent on a, b, and c. It can be written that x is a function of these variables:

x=f(a,b,c)(1)

Because each measurement has an uncertainty about its mean, it can be written that the uncertainty of dxi of the ith measurement of x depends on the uncertainty of the ith measurements of a, b, and c:

dxi=f(dai,dbi,dci)(2)

The total deviation of x is then derived from the partial derivative of x with respect to each of the variables:

dx=(δxδa)b,cda,(δxδb)a,cdb,(δxδc)a,bdc(3)

see the attachment one of the.

example sum