I was studying about the concepts of Relative , Absolute and percentage errors in Numerical Analysis. I was reading from the book, "Numerical Mathematical Analysis" by J Scarborough.
There was a section titled " Relation between Relative Error and the Number of Significant Figures ".
I am having a hard time contemplating it. It went on as follows:
The belief is widespread, even in scientific circles, that the accuracy of a measurement or of a computed result is indicated by the number of decimals required to express it. This belief is erroneous, for the accuracy if a result is indicated by the number of significant figures required to express it. The true index of the accuracy of a measurement or of a calculation is the relative error.
For example, if the diameter of a $2-$inch steel shaft is measured to the nearest thousandth of an inch the result is less accurate than the measurement of a mile of railroad track to the nearest foot. For although the absolute errors in the two measurements are $0.0005$ inch and $6$ inches, respectively, the relative errors are $0.0005/2=1/4000$ and $1/10560.$ Hence in the measurement of the shaft we make an error of one part in $4000,$ whereas in the case of the railroad we make an error of one part in $10560.$ The latter measurement is clearly the more accurate, even though its absolute error is $12000$ times as great.
First of all, what I understood was, that the number of decimal places used to measure a quantity is not proportional to the accuracy. For example, if one measures the length of glass slab to be $4.0000001$ metres, it does not necessarily imply that this is an accurate measure of the lemgth of the glass slab. The comparison, is usually done with the number of significant digits in the measured value. The more the number of significant digits present in a measurement the more the accuracy.
We also consider the relative error to be a measure of the accuracy. The less the relative error the more the accuracy is.
I hope my understanding upto this, is correct.
But then, the author proceeds to give an example. It's written,
" If the diameter of a $2-$inch steel shaft is measured to the nearest thousandth of an inch the result is less accurate than the measurement of a mile of railroad track to the nearest foot."
We dissect the line part by part :
By the phrase : "diameter of a $2-$inch steel shaft is measured to the nearest thousandth of an inch" , I think what is implied is, that we are using a device that can measure the diameter of the 2 inches long steel shaft in three decimal places, i.e it measures the diameter by the number of 0.001 inches long sticks needed to span the diameter of the steel shaft when they are just placed one on top of another (along the diameter). Now, it might so, happen that after placing a certain number of sticks, a small amount of space is left, whose length is shorter than $0.001$ inches. In that case, we measure the number of sticks of $0.0001$ inch required to span the space when the sticks are placed one on top of another along the space (remaining). Then, say, if we get a length of, $72\times 0.001$ inches and $2\times 0.0001$ inches, we just round off, the measured quantity (i.e $0.072+0.0002=0.0722$ inches) to $0.72$ inches.
By the phrase, "measurement of a mile of railroad track to the nearest foot" it means, that the length of a $1$ mile long railway track is measured in foots, i.e more sepecifically, by calculating the number of $1$ foot long sticks needed to span the $1$ mile long railway track when they are just placed one on top of another (along the railway track). Now, it might so, happen that after placing a certain number of sticks, a small amount of space is left, whose length is shorter than $1$ foot. In that case, we measure the number of sticks of $1$ inch required to span the space when sticks are place one on top of another along the space (remaining). Then, say, if we get a length of, $234$ feets and $7$ inches, we just round off, the measured quantity (i.e $243.7$ feet) to $244$ feets.
Now, if I am not wrong, the author asserts that the measured length of diameter of the shaft i.e $0.72$ inches is less accurate than the measured length of the railway track which is in this case, is, $244$ feets.
Then the author tries to give a reason for the last point, about the accuracy, i.e " Why is the measured length of diameter of the shaft i.e $0.72$ inches is less accurate than the measured length of the railway track which is in this case, is, $244$ feets ?"
His reason was:
For although the absolute errors in the two measurements are $0.0005$ inch and $6$ inches, respectively, the relative errors are $0.0005/2=1/4000$ and $1/10560.$ Hence in the measurement of the shaft we make an error of one part in $4000,$ whereas in the case of the railroad we make an error of one part in $10560.$ The latter measurement is clearly the more accurate, even though its absolute error is $12000$ times as great.
Now, I have two questions from these lines of reasonings.
- Firstly, how are they asserting, that, " the absolute errors in the two measurements are $0.0005$ inch and $6$ inches, respectively." From what I know till now, is that the absolute error of a measurement is the modulus of the numerical differenc between the calculated value and the true value. Also, I am aware of a principle in error analysis, that ' A number is correct to $n$ significant figures, if its absolute error is not greater than half a unit in the $nth$ place. ' From this particular principle, I can say, that since, the length of the diameter of the shaft is measured by the number of approximate parts of 0.001 inches required to span the diameter (of the shaft) so, the measured diameter is correct upto the $3rd$ digit from the decimal point, in the measured length. So, the maximum absolute error is, $(1/2)\times(0.001)=0.0005$, but it is not the absolute error. Similarly, the maximum absolute error, in the measured length of $1$ mile of railway track in the nearest foot, is, $1/2\times (1)=0.5$
So, I don't really understand what's implied in this particular statement?
- Secondly, I really don't, understand how are they getting the relative errors ? The given information, as it stands is insufficient to calculate the relative. As far as I know, the relative error si equal to the ratio of the absolute error to the true value of the quantity. But in both the cases, of the shaft and the railway track, the measured value is varying and so, absolute value is unknown and the true length of the diameter of a shaft is not given. Due to all these, it seems weird how are they asserting the given fractions (in the quoted portions) as the relative errors in the two measurements carried out in here.