Today our topic is error in measurement actually we are human made for mistake machine does’t make mistake but this mistake is the reason of so many development in science so don’t worry with mistake because mistake makes a man perfect. we know science is subject of experiment for this reason we experiment our any observation in lab and make error during measurement of input data we can take any observation (Reading ) in lab for length, mass and time.
Error in Measurement Absolute Relative
Now we have taken an observation it can be anything mass, length and time whatever we want to calculate in lab suppose we have take one observation that is a₁ anything we going to measure is a true value and other is a measured value true value is that value which we are trying to find out measured value is that value which we are measuring with the help of instrument if we are measuring value of length then its error will be also length if we are measuring mass error will be mass and so on now error will be denoted by ∆a₁ hence our measured value is a₁ and error in measured value is ∆a₁ then what will be true value answer is measured value plus or minus error value that is (a₁±∆a₁) then what will be error that is equal to true value -measured value hence Error = (true value-measured vale ) now we have measured a single value then what is error value so for error value we have to make a large number of observation like a₁,a₂,a₃,a₄,a₅,a₆ …..a𝑛 for example simple pendulum time period calculation for many observation taken with help of stop watch. now sum all observation a₁,a₂,a₃,a₄,a₅,a₆ …..a𝑛 it will be ∑a𝔫 and divide it by n for its average value that is ∑a𝔫/n this average value must be close to true value because during observation we have made some positive error and some negative error and we added all this observation hence positive error and negative error cancelled each other and we have added n times and divided by n so its value will be true and this is denoted by ∑a𝔫/n =a this mean or average value is accepted as true value. now we will use word Absolute error in a₁ will be equal to true value -measured value that is (Absolute error a₁) = (∑a𝔫/n –a₁) or (a-a₁) hence it is written as ∆a₁ therefore absolute error in second, third observation (a-a₂), (a-a₂) … (a𝔪 – a𝔫) and so on ∆a₁,∆a₂,∆a₃.. ∆a𝑛 here am denote mean value of a hence this is absolute error now for mean absolute error of experiment will be adding all this without sign consideration that is |∆a₁|+|∆a₂|+|∆a₃|…..+|∆a𝑛 | here note some error may be positive or some error may be negative but we add all error with sign because one mistake not cancel other mistake it increases error so added for total error hence mean absolute error will be |∆a₁|+|∆a₂|+|∆a₃|…..+|∆a𝑛 |/n = ∆am now other type of error is percentage error like told you to measure 10 m cloth then you measured 11 m in place of 10 m then how much error in measurement of course 1 m then i told other person to measure 100 m cloth and he made 2 m error now question is which one error is more of course 1 m error person is more error we can calculate it with the help of mathematics percentage error for first person will be 1/10*100 = 10% or for second person percentage error = 2/100*100 = 2% percentage error is also called relative error hence relative error is better measurement than absolute error
Relative Error it is also called fractional error or percentage error now how it is measured for this we use error divided by value or ∆am/am absolute mean error divided by mean value now for fractional error multiply by 100 it will became percentage error now take an example to calculate error suppose a simple pendulum observation was done for its time period for one oscillation and noted as 2.63 s, 2.56 s, 2.42 s, 2.71 s, 2.80 s with stop watch then calculate mean value, mean absolute error, relative error, percentage error so in our process we first calculate mean value after calculating mean value we can found error note without finding mean value we can’t find out any error. hence mean value for the above case will be add all values and divide by 5 because five observation is only here hence mean value = (2.63+2.56+2.42+2.71+2.80)/5 = 13.12/5 = 2.60 hence mean value is 2.60 now for mean absolute error it is define as mean value subtract individual observation that will be |(2.60-2.63)|+| 2.60-2.56|+|2.60-2.42|+|2.60-2.71|+|2.60-2.80| hence (0.03+.04+.18+.11+.20 ) or 0.56 hence for mean 0.56/5 = 0.10 this is mean Absolute error now for Fractional error define as mean Absolute error divided by mean value that is 0.10/2.60 = 1/26 now for percentage error this fractional error multiply by 100 so 1/26*100 and it will be approx 4% so all value is calculated easily.
Now how to write a measurement with error express its limit value and it is written as (a𝔪±∆a𝔪) example (110±5) maximum value (110+5) and minimum value (110-5) it is also called its tolerance value or in other way (mean value±%) for our case (2.60±4%) now we will study propagation of Error.
Propagation of Errors its mean that go ahead of errors when we do measurement in certain quantity more than one quantities are involved for operation like when we do observation in lab for acceleration due to gravity g we find a equation as g= 4𝛑² l/T² where l is length of the pendulum string T is time period of oscillation we measured in lab its length and time period for different set changing length of string so we have done length measurement and time measurement both having error so what will be that error and due to this error how much error will be in gso we have to calculate that value of error and this is called propagation in error for that important point the error are always added and never cancelled in any operation suppose we have two observation like (a±∆a) and (b±∆b) now add these two we will get (a+b)±(∆a+∆b) so here (a+b) is true value and (∆a+∆b) is error or tolerance now similarly for multiplication of two error lets its multiplication is p hence (p±∆p)= (a±∆a)×(b±∆b)= ab±∆ab±∆ba±∆a∆b here product of ∆a∆b is very small so it is neglected then final (p±∆p) = ab±∆ab±∆ba now here p is equal to ab without error mean value so put value ab±∆p = ab±∆ab±∆ba now equation become ∆p = ∆ab±∆ba we can also write it as divide by ab on both site p=ab so we will get ∆p/p = ±(∆a/a+∆b/b) so here we know that ∆a/a is relative error of a and ∆b/b is relative error of b and ∆p/p is relative error of product hence percentage error in product is equal to sum of percentage error of constituent take an example for clarity in an experiment voltage is observed as (100±5) volt and current (10±0.2) A find percentage error in R so for the solution we know that R = V/I here we see that this is division type of equation so we know that for division or multiplication both case sum of relative error is applicable so we can write as R/∆R = ±(∆V/V+∆I/I) so it will be equal to R/∆R = ±(5/100+0.2/10) hence R/∆R = ±0.07 = 7/100 this is fractional error so for percentage error multiply by 100 which will be equal to 7% now for calculation of Absolute error ∆R = 7/100×R here for R =V/I that is 100/10= 10 so put R = 10 now ∆R = .±0.7 ohm now for next we will see for power quantity like if we have an observation a±∆a for a³ so for this how we can write a×a× a now what our process say about multiplication quantity simply sum relative error hence it will be a/∆a = ±(∆a/a+∆a/a+∆a/a) here power is 3 so three times addition if power will be n then n times addition so our formula will be an/∆an = ±n∆a/a hence for percentage error multiply both side by 100 take an example for circle area A we measured and percentage error in radius is 2% then how much percentage error in area simple we know that A = 𝛑r² so percentage error in area = 2×2 = 4% now i hope that error measurement concept is cleared thanks for reading