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Physics LibreTexts

1.9 Observations and Uncertainty

The core of the scientific method is evidence — observational data in some form. Scientists make explanations of their data called hypotheses, and they combine the information according to the rules of logic. These logical tools have limitations. But what can we say about the limitations of the observations themselves? It turns out that there is no such thing as a perfect measurement and every measurement has an uncertainty associated with it.

Every measurement has an associated observational error, which is the uncertainty in the measurement. In the 18th century, the great German mathematician Karl Friedrich Gauss worked out the theory of observational errors. Scientists rely heavily on his ideas in all their work with measurements. They can only understand a particular result if they know the degree of error involved.

Let’s imagine a deck of cards. Suppose you try and slide a card into a standard deck exactly halfway down the pile. By guessing, you would be unlikely to place it halfway through the deck. If you tried the experiment ten times, for example, you might find that you had placed it 24, 33, 28, 27, 23, 27, 24, 32, 26, and 31 cards down the pile. There is a large error associated with guessing. But if you measured the height of the deck with a millimeter ruler, for example, you could insert the card halfway down much more accurately. This experiment might yield results of 25, 28, 24, 27, 27, 26, 28, 26, 25 and 27 cards down the pile. The second set of results has less error due to careful measurement. The histogram of positions in the deck is narrower when a more accurate measurement is made.
 


Diagram of a star's right ascension and declination as seen from outside the celestial sphere. Depicted are the star, the Earth, lines of RA and dec, the vernal equinox, the ecliptic, the celestial equator, and the celestial poles. Click here for original source URL.

To take an astronomical example, how can we identify the "exact" position of a star in the night sky? Suppose that there are many measurements and none of them are exactly the same. Where is the star exactly? We cannot say! However, we can do two things. We can take an average of all the measurements as the best estimate of the star's position (the average is also referred to as the mean value). We can take the spread in the measurements as the standard error in that estimate, which is also referred to as the uncertainty, or standard deviation. In a way, the concept of "error" is a bit misleading, since no mistake was made in the measurement. There is just a limit to our certainty in the result of any measurement.

Where does observational error come from? Usually it just reflects the normal limitations of the measuring equipment. Suppose that a ruler is marked off with centimeters as the smallest unit. We could make a single, quick measurement with an accuracy of half of the smallest unit on the ruler — a millimeter. If we measure the width of a piece of paper, we might come up with the result 217 millimeters. Notice the number of significant figures, the digits that carry meaningful information. We could have quoted the measurement with one significant figure, as 200 millimeters, but that is needlessly rough. If we used two significant figures, we would say the width is about 220 millimeters, rounding off the last figure. We could also quote the width as 217.84 millimeters — five significant figures — but that is unrealistically precise. It makes most sense to quote the result with three significant figures, which reflects the accuracy of the measurement. Accuracy is defined as the amount by which a measurement deviates from the true value.
 


Accuracy is the proximity of measurement results to the true value; precision, the repeatability, or reproducibility of the measurement. Click here for original source URL.

Scientists make a clear and important distinction between precision and accuracy. A single measurement can be quoted with a wide range of precision, but the accuracy is set by the nature of the ruler. For example, you can easily set most calculators to display 6 or 8 or 10 decimal places, and the results of any calculation you do will be shown with that precision. But that does not mean that every calculation you do is that accurate. Imagine you used the same ruler to measure the length of the block your house is on. You might add up all the numbers as you put the ruler down end to end and get 56,794 millimeters. It sounds very precise, but is your measurement really accurate to 1 millimeter? A better indication of the precision of the measurement would be 56.8 meters. Scientists always try to quote measurements with a precision that matches the accuracy.

Recall that a scientific measurement has two components: a number and a unit. The number itself has two pieces: the best estimate and the standard error. In the example just mentioned, the full measurement would be written as 217 ± 1 millimeters. The symbol "±" (also written as +/-) is called "plus-or-minus." It means that while the true value might well be 218 millimeters or 216 millimeters, it is unlikely to be 120 or 250 or even 210 centimeters. On a graph, the best estimate is drawn as a point or as some other symbol, and the standard error or uncertainty is drawn as an "error bar." These examples with a ruler are mundane. But cosmologists use the same ideas in measuring the size and age of the universe!
 


Mirrors on modern telescopes are made to improve accuracy. This is a picture of the James Webb Space telescope to be launched in 2018. It has been made so that its results can be stated with larger numbers of significan figures and will small error bars. Click here for original source URL.

When making measurements of your own, you’ll be relieved to know that there are ways to reduce observational errors. We can use more precise measuring equipment such as a micrometer marked off in tenths of millimeters. Scientists often make progress in this way, but it is not always possible to improve on existing technology. The other way to reduce observational error or improve accuracy is to make more than one measurement. As more measurements are made, the uncertainty of the average goes down. This is reflected in the way the result is stated, with a larger number of significant figures and with a smaller error bar.