Measurements are inherently uncertain. Given any two people measuring a certain quantity using the same measuring device but with different precision will report to us different readings. Much would be the case if there were more people measuring the same quantity. But why are measurements uncertain, and how does one approximate his/her measurement to lessen this uncertainty?
Generally speaking, measurements are uncertain for three main reasons. One, being the nature of the quantity being measured, another dealing the judgement of the people measuring, and lastly, the limitations brought by the measuring device used.
If I gave you a task to measure the exact position and momentum of an electron at any given time, you would have a hard time because according to Heisenberg's Uncertainty Principle, the product of the uncertainty in the electron's position \(\Delta x\) and momentum \(\Delta p\) would always be greater than or equal, half of the rationalized Planck's Constant \(\hbar\), or:
$$\Delta x\Delta p\geq\dfrac{\hbar}{2}$$
This means that you can measure the electron's position in space with a great deal of precision, however, you run the risk of increasing the uncertainty of its measured momentum. So the nature of the electron being in constant movement, together with other factors, would affect any measurements done on it, hence making such prone to have a degree of error.
Okay, so what if I didn't tell you to measure the electron's position or momentum. Instead, I will give you a map, like the one shown (don't mind the incompleteness of the map), and tell you to go from point \(A\) to \(B\), measuring the time it took you to get to \(B\). Given this information, will, or can you expect that when I let somebody else do this, I will receive the same, or similar data? One thing we can argue at first is on which route to take. Each path will obviously affect the distance we will travel, and hence, time of travel. Say you have chosen a path. What we can think of now would be the speed at which we would be travelling. For the sake of argument, say you and I would traverse the same path, however, I would be travel faster than you. Again, we can say that since I was faster, then it took me shorter time to travel the same path. We would appreciate this much more if we let other people do this. So, in the end, we ask ourselves, how long would it take to travel from \(A\) to \(B\)? Others would argue that since it took them a shorter time to get to \(B\), they are correct. Then again, another group of people would say that they are correct since they traveled for a longer time. In truth, both of them are correct. There will always be people who would travel that distance in a short span of time, and people who would take a long time. So it would really depend on the people's judgement on which measurement is correct.
Now, I will tell you to measure the length of a simple line. The catch is, you have to use both of these rulers.
Since these rulers have no mark for 0 in., then when we have to keep in mind that the actual measurement is 1 minus the reading we got.
So, after measuring the line, as shown:
what length will you report, as a whole? Take note that the two rulers have different levels of precision, one being calibrated to \(\pm\)0.25 in., and the other \(\pm\)1 in. Also, we have to take note that the line misses the tick of the 5.25 in. reading in one ruler.
Technically speaking, there are three cases for this: 1) you report it as 4.25 in., 2) you report it as 4.50 in., or 3) you go with your instinct that it is in between 4.25 in. and 4.50 in. (or 4.37 in.). Again, either value you report would be correct. However this tells us that even the measuring devices have limitations to the extent they can measure certain quantities.
And this also brings us to the difference of measured quantities and numbers entirely. If I give you 1 rose (because it is the season of hearts right? just kidding), you will have only received 1 rose. However if I tell you that I have 1 kg and 1.00 kg of pasta for our dinner later and tomorrow, you have to take note that 1 kg \(\neq\) 1.00 kg. Because measurements are inherently uncertain, 1 kg means that I don't have 0 kg or 2 kg of pasta - the same way that 1.00 kg means that I don't have 1.01 kg or 0.99 kg of pasta. So numbers are very different compared to measurements because numbers are definite while measurements are not.
So, you really cannot take out the uncertainty in measurement as provided by the examples above - what you can do is to lessen it. How does one do so? Well, we have what we call The Orders of Approximation. These are the Order of Magnitude Approximation, Significant Figures, Limited Measurements, and Based on the Distribution Function.
Order of Magnitude Approximation generally refer to our "powers of 10." Exa-, peta-, tera-, giga-, etc., otherwise known as \(10^{18}\), \(10^{15}\), \(10^{12}\), and \(10^9\) respectively, is an example of such approximation. This is, in particular, a pretty simple way of approximation because some measurements we do everyday is intuitive (in such that we don't need to use \(10^x\)). Nobody measures the length of cloth in \(10^{12}\) or \(10^{-9}\). We know that when we buy cloth, we measure it by meters, and that it is in the \(10^0\) order of magnitude.
Despite its being so simple, this form of approximation is used a lot in some experiments and analysis. For example, when talking about units of length in space, we do not use \(10^0\), we use \(10^{15}\) as a base (technically speaking \(9.5\times10^{15}\) m for one light year), so any studies regarding space will be, somewhat immediately, in terms of petameters (\(10^{15}\)). Or about experiments regarding atoms and atomic collisions - they are so small that \(10^0\) prove to be very inappropriate. Instead (and I only based this according to the Bohr radius), they measure quantities in terms of picometers (\(10^{-12}\)). So there are big differences even in this simple method of approximation.
Significant Figures simply (and intuitively) mean "numbers that matter." The 'significance' of this order of approximation has already been revealed above, however not totally discussed. So what exactly is this concept, and why is it so important? In relation to the order of approximation discussed previously, this concept is, per say, the other half of the 'whole' thing. Because in scientific papers, most of the calculation and numbers are expressed in scientific notation. So we know that in scientific notation, we have a number \(x\) multiplied to 10 raised to the \(n\)-th power. Technically speaking, '10 raised to the \(n\)-th power' is the order of magnitude approximation, and \(x\) is the number, expressed with significant figures. Hence, in the speed of light in a vacuum \(\approx3.00\times10^8\ ^\text{m}/_\text{s}\), \(10^8\) is your order of magnitude approximation, and \(3.00\) is the number expressed with significant figures. So, again, as given in the example (although it is a constant), \(3.00\times10^8\ ^\text{m}/_\text{s}\) means that the speed of light is not \(3.01\times10^8\ ^\text{m}/_\text{s}\) nor \(2.99\times10^8\ ^\text{m}/_\text{s}\), but instead, is \(3.00\times10^8\ ^\text{m}/_\text{s}\).
Limited Measurements entail that we have repeated measurements of the same quantities, and here we recognize how wide the error truly is. If you were to measure an object using different measuring devices, then, on average, you should get pretty much the 'same' value. Even upon repeated measurements, surely the length of that object would be constant using the same measuring device, so the uncertainty we can report would, by convention, be one half the smallest division of that measuring device. For example, if the smallest division of the ruler you have is 1 cm, then your degree of uncertainty would be one-half of 1 or \(\pm\)0.5 cm. However, if upon measurement and the measured quantity changed upon repetition, we have to use another convention. We have to get the average of the measurements \(\langle Q\rangle\) (where \(Q\) is the quantity being measured), and the uncertainty \(\Delta Q\). We all know how to get the average. The uncertainty is calculated by the absolute value of the maximum difference between the data sample and the average, explicitly:
$$\Delta Q=|\langle Q\rangle-Q_i|_\text{max}$$
Where \(Q_i\in\{Q_1,Q_2,Q_3,\dotsc,Q_n\}\).
In my opinion, basing our approximation under a Distribution Curve provides an excellent guess to the actual value we want. Again, it is 'formatted' similar to the way limited measurements do, however with different values. The expected value is still the average \(\langle Q\rangle\), however, this time, the measurement for uncertainty is the standard deviation \(\sigma\). This provides a pretty much 'spot on' estimate to the actual value because, besides the beauty of how the distribution curve would look like, the data becomes normalized in such a way that the graph peaks at where the more probable or frequent value was measured, and the uncertainty is how spread the data points are to that measure of central tendency. So in essence, once you got this, you are really close to getting the actual value you want.
Despite exerting all your efforts to lessening the uncertainty, there are times when you really cannot help but have/obtain errors in your data. There are two kinds of errors that we can experience, the Random Error, and Systematic Error. The main difference in the two is that Systematic Errors occur generally because of the system used in measurement. Maybe there was a misread in the measurements, or part of the procedure was manipulated or not followed. This usually results to the average of the data to be greater, or less, than the accepted value. Another way to describe this kind of error is an experiment that is precise, however, not accurate. This can be visualized as a set of darts clumped in one side of a dart board but not hitting the bull's eye mark. A Random Error on the other hand, gives an average that is almost or is the accepted value, however, the data is scattered and does not give a definite trend. This now is said to be an experiment that has low precision, and low accuracy.
However, despite the uncertainty and chances to get errors during experiments, experimentation as a way to measuring quantities still prove to be the best ways to discover new things. Without experimental data from the Ultraviolet Catastrophe, Planck would've never discovered that energy is quantized. Without data from the photoelectric experiment, Einstein wouldn't have explained why such phenomenon happen. Truly, by measurements and experimentation, we have come to see, realize, and discover so many wonderful and amazing things in all fields. Huzzah!
Generally speaking, measurements are uncertain for three main reasons. One, being the nature of the quantity being measured, another dealing the judgement of the people measuring, and lastly, the limitations brought by the measuring device used.
If I gave you a task to measure the exact position and momentum of an electron at any given time, you would have a hard time because according to Heisenberg's Uncertainty Principle, the product of the uncertainty in the electron's position \(\Delta x\) and momentum \(\Delta p\) would always be greater than or equal, half of the rationalized Planck's Constant \(\hbar\), or:
$$\Delta x\Delta p\geq\dfrac{\hbar}{2}$$
This means that you can measure the electron's position in space with a great deal of precision, however, you run the risk of increasing the uncertainty of its measured momentum. So the nature of the electron being in constant movement, together with other factors, would affect any measurements done on it, hence making such prone to have a degree of error.
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| Figure 1: A Simple Map |
Now, I will tell you to measure the length of a simple line. The catch is, you have to use both of these rulers.
 |
| Figure 2: Two Rulers with Different Levels of Precision |
Since these rulers have no mark for 0 in., then when we have to keep in mind that the actual measurement is 1 minus the reading we got.
So, after measuring the line, as shown:
 |
| Figure 3: Same Line Measured by the Two Rulers |
what length will you report, as a whole? Take note that the two rulers have different levels of precision, one being calibrated to \(\pm\)0.25 in., and the other \(\pm\)1 in. Also, we have to take note that the line misses the tick of the 5.25 in. reading in one ruler.
Technically speaking, there are three cases for this: 1) you report it as 4.25 in., 2) you report it as 4.50 in., or 3) you go with your instinct that it is in between 4.25 in. and 4.50 in. (or 4.37 in.). Again, either value you report would be correct. However this tells us that even the measuring devices have limitations to the extent they can measure certain quantities.
And this also brings us to the difference of measured quantities and numbers entirely. If I give you 1 rose (because it is the season of hearts right? just kidding), you will have only received 1 rose. However if I tell you that I have 1 kg and 1.00 kg of pasta for our dinner later and tomorrow, you have to take note that 1 kg \(\neq\) 1.00 kg. Because measurements are inherently uncertain, 1 kg means that I don't have 0 kg or 2 kg of pasta - the same way that 1.00 kg means that I don't have 1.01 kg or 0.99 kg of pasta. So numbers are very different compared to measurements because numbers are definite while measurements are not.
So, you really cannot take out the uncertainty in measurement as provided by the examples above - what you can do is to lessen it. How does one do so? Well, we have what we call The Orders of Approximation. These are the Order of Magnitude Approximation, Significant Figures, Limited Measurements, and Based on the Distribution Function.
Order of Magnitude Approximation generally refer to our "powers of 10." Exa-, peta-, tera-, giga-, etc., otherwise known as \(10^{18}\), \(10^{15}\), \(10^{12}\), and \(10^9\) respectively, is an example of such approximation. This is, in particular, a pretty simple way of approximation because some measurements we do everyday is intuitive (in such that we don't need to use \(10^x\)). Nobody measures the length of cloth in \(10^{12}\) or \(10^{-9}\). We know that when we buy cloth, we measure it by meters, and that it is in the \(10^0\) order of magnitude.
Despite its being so simple, this form of approximation is used a lot in some experiments and analysis. For example, when talking about units of length in space, we do not use \(10^0\), we use \(10^{15}\) as a base (technically speaking \(9.5\times10^{15}\) m for one light year), so any studies regarding space will be, somewhat immediately, in terms of petameters (\(10^{15}\)). Or about experiments regarding atoms and atomic collisions - they are so small that \(10^0\) prove to be very inappropriate. Instead (and I only based this according to the Bohr radius), they measure quantities in terms of picometers (\(10^{-12}\)). So there are big differences even in this simple method of approximation.
Significant Figures simply (and intuitively) mean "numbers that matter." The 'significance' of this order of approximation has already been revealed above, however not totally discussed. So what exactly is this concept, and why is it so important? In relation to the order of approximation discussed previously, this concept is, per say, the other half of the 'whole' thing. Because in scientific papers, most of the calculation and numbers are expressed in scientific notation. So we know that in scientific notation, we have a number \(x\) multiplied to 10 raised to the \(n\)-th power. Technically speaking, '10 raised to the \(n\)-th power' is the order of magnitude approximation, and \(x\) is the number, expressed with significant figures. Hence, in the speed of light in a vacuum \(\approx3.00\times10^8\ ^\text{m}/_\text{s}\), \(10^8\) is your order of magnitude approximation, and \(3.00\) is the number expressed with significant figures. So, again, as given in the example (although it is a constant), \(3.00\times10^8\ ^\text{m}/_\text{s}\) means that the speed of light is not \(3.01\times10^8\ ^\text{m}/_\text{s}\) nor \(2.99\times10^8\ ^\text{m}/_\text{s}\), but instead, is \(3.00\times10^8\ ^\text{m}/_\text{s}\).
Limited Measurements entail that we have repeated measurements of the same quantities, and here we recognize how wide the error truly is. If you were to measure an object using different measuring devices, then, on average, you should get pretty much the 'same' value. Even upon repeated measurements, surely the length of that object would be constant using the same measuring device, so the uncertainty we can report would, by convention, be one half the smallest division of that measuring device. For example, if the smallest division of the ruler you have is 1 cm, then your degree of uncertainty would be one-half of 1 or \(\pm\)0.5 cm. However, if upon measurement and the measured quantity changed upon repetition, we have to use another convention. We have to get the average of the measurements \(\langle Q\rangle\) (where \(Q\) is the quantity being measured), and the uncertainty \(\Delta Q\). We all know how to get the average. The uncertainty is calculated by the absolute value of the maximum difference between the data sample and the average, explicitly:
$$\Delta Q=|\langle Q\rangle-Q_i|_\text{max}$$
Where \(Q_i\in\{Q_1,Q_2,Q_3,\dotsc,Q_n\}\).
In my opinion, basing our approximation under a Distribution Curve provides an excellent guess to the actual value we want. Again, it is 'formatted' similar to the way limited measurements do, however with different values. The expected value is still the average \(\langle Q\rangle\), however, this time, the measurement for uncertainty is the standard deviation \(\sigma\). This provides a pretty much 'spot on' estimate to the actual value because, besides the beauty of how the distribution curve would look like, the data becomes normalized in such a way that the graph peaks at where the more probable or frequent value was measured, and the uncertainty is how spread the data points are to that measure of central tendency. So in essence, once you got this, you are really close to getting the actual value you want.
 |
| Figure 3: Set of Darts Visualizing High Precision but Low Accuracy (source: http://imgkid.com/darts-board.shtml) |
However, despite the uncertainty and chances to get errors during experiments, experimentation as a way to measuring quantities still prove to be the best ways to discover new things. Without experimental data from the Ultraviolet Catastrophe, Planck would've never discovered that energy is quantized. Without data from the photoelectric experiment, Einstein wouldn't have explained why such phenomenon happen. Truly, by measurements and experimentation, we have come to see, realize, and discover so many wonderful and amazing things in all fields. Huzzah!
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