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108

I work with an old toolmaker who also worked as a metrologist who goes on about this all day. It seems to boil down to exploiting symmetries since the only way you can really check something is against itself. Squareness: For example, you can check a square by aligning one edge to the center of straight edge and tracing a right angle, then flip it over, re-...

101

This is a good and somewhat tricky question for a number of reasons. I will try to simplify things down. SI Second First, let's look at the modern definition of the SI second. The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ∆νCs, the unperturbed ground-state hyperfine transition ...

78

Significant figures are a shorthand to express how precisely you know a number. For example, if a number has two significant figures, then you know its value to roughly $1\%$. I say roughly, because it depends on the number. For example, if you report $$L = 89 \, \text{cm}$$ then this implies roughly that you know it's between $88.5$ and $89.5$ cm. That is, ...

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A "year" without qualification may refer to a Julian year (of exactly $31\,557\,600~\rm s$), a mean Gregorian year (of exactly $31\,556\,952~\rm s$), an "ordinary" year (of exactly $31\,536\,000~\rm s$), or any number of other things (not all of which are quite so precisely defined). Radioactive decay tables tend to be compiled from multiple different ...

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There are lots of different strategies that are employed by the scientific community to counteract the kind of behavior Feynman talks about, including: Blind analyses: In many experiments, it is required for the data analysis procedure to be chosen before the experimenter actually sees the data. This "freezing" of methodology ensures that nothing about the ...

61

Are random errors necessarily gaussian? Errors are very often Gaussian, but not always. Here are some physical systems where random fluctuations (or "errors" if you're in a context with the thing that's varying constitutes an error) are not Gaussian: The distribution of times between clicks in a photodetector exposed to light is an exponential distribution....

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The second and the speed of light are precisely defined, and the metre is then specified as a function of $c$ and the second. So when you experimentally measure the speed of light you are effectively measuring the length of the metre i.e. the experimental error is the error in the measurement of the metre not the error in the speed of light or the second. It ...

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The more you measure things and add or multiply those measurements, the greater your errors will become. Not necessarily. If the errors in a series of measurements are independent and there is no systemic bias in the measurements then taking the average of all the measurements will give you a more accurate value than any individual measurement.

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It's the uncertainty in the last two digits: $$8.9875517923(14) = \color{blue}{8.987\,551\,79}\color{red}{23} \pm \color{blue}{0.000\,000\,00}\color{red}{14}.$$

41

My favorite story (which I learned about recently) is about Frank Dunnington and his measurements of electron properties in about 1930. He was measuring the ratio $e/m_e$. Experiments took quite a long time (four years!). When the experimental device was constructed he asked the person who helped him to construct not to tell him some key attribute of the ...

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BIPM and TAI The International Bureau of Weights and Measures (BIPM) in France computes a weighted average of the master clocks from 50 countries. That weighted average then gives International Atomic Time (TAI), which forms the basis of the other international times (e.g., UTC, which differs from TAI by the number of leap seconds that have been inserted, ...

36

A second is a second long by definition, but if you measure any time in seconds, the number of seconds you infer will be subject to an error of at least $\mathcal O(10^{-15})$ because of the uncertainty of Caesium clocks as you correctly point out. This is true even if you make a higher precision measurement using new clocks like Quantum Clocks with ...

35

Because significant figures measures uncertainty relative to the size of the number Suppose you take a measurement of something and it comes out to be 0.002 meters. You then measure something else and it comes to 345 meters. You know that $0.002$ means $0.002 \pm 0.0005$ and $345$ means between $345 \pm 0.5 .$ The uncertainty in the numbers here are $0.... 33 The digits in parentheses are the uncertainty, to the precision of the same number of least significant digits. (The meaning of the uncertainty is context-dependent but generally represents a standard deviation, or a 95% confidence interval.) So: $$e/m=1.758\,820\,1\color{blue}{50}\,\color{magenta}{(44)}×10^{11} \ \mathrm{C/kg}=\left(1.758\,820\,1\color{blue}... 29 One thing I haven't seen mentioned is amplification. Amplification: Imagine you have a lever that is 10 cm on one side of the pivot and 1 m on the other. Then any change in position on the short side is amplified 10 times on the long side. This new precision can be used to make new even more precise tools. Rinse and repeat. Even just by iterating through ... 28 Simple error analysis assumes that the error of a function \Delta f(x) by a given error \Delta x of the input argument is approximately$$ \Delta f(x) \approx \frac{\text{d}f(x)}{\text{d}x}\cdot\Delta x $$The mathematical reasoning behind this is the Taylor series and the character of \frac{\text{d}f(x)}{\text{d}x} describing how the function f(x) ... 27 We talk in terms of standard deviation because this is traditionally the quantity you use to specify the variance of a Gaußian distribution specifically and any random distribution more generally. You seem to be misinterpreting the recommendation that all uncertainties be reported as standard deviations as a guideline that this should also always be what ... 26 That's a really nice one! I'm not an expert on experiments and measurements but this is how I see it: The ultimate calibration tool is always nature. We pick special phenomena which rely on certain parameters. Take temperature, for example. The phenomenon here is the phase transition of water. You set boiling water to a 100^\circ C and freezing to 0^\circ... 25 Use the second derivative (or third, or whatever). The reason we use that formula is that$$ df \approx \frac{df}{dx} dx $$is the first order Taylor approximation to df. If the first order term vanishes, you should include higher terms:$$ df \approx \frac{df}{dx} dx+\frac{1}{2}\frac{d^2f}{dx^2} dx^2+... $$In your case, with f=x^2, and x=0, we'd ... 25 This one is tricky unless you know the magic term: ephemeris. An ephemeris gives the position of celestial bodies over time. Once you know that one, finding out information about their uncertainties is easier. The uncertainties are actually rather interesting in that they are planet specific. For example, the dominating factor for Mercury's uncertainty ... 24 While appropriate in many important contexts, LeFitz's answer can fail in one important situation, and can lead you astray, for example, when plotting graphs in logarithmic scale. More specifically, LeFit'zs answer is only valid for situations where the error \Delta x of the argument x you're feeding to the logarithm is much smaller than x itself:$$ \... 24 It basically comes from calculus (or more generally just the mathematics of change). If you have a quantity that is a product$z=x\cdot y$, then the change in this value based on the change of$x$and$y$is$^*\Delta z=x\Delta y+y\Delta x$. So then it is straightforward that $$\frac{\Delta z}{z}=\frac{x\Delta y+y\Delta x}{xy}=\frac{\Delta x}{x}+\frac{\... 24 The calculations are done in Schwartz (2019), Lecture 3: Equilibrium (https://scholar.harvard.edu/files/schwartz/files/3-equilibrium.pdf) Here's a summary of the key equations, using the notation and equation numbers from those lecture notes. Consider a classical system of spherical objects, each of radius R. Let \ell be the average distance an object ... 23 Suppose you are analysing the weights of people in the UK to see what the distribution of weights looks like. Suppose also you can measure the weight to arbitrary precision, so that no two people's weights will be exactly the same. When you're finished you plot your data on a histogram, but the trouble is that because everyone has a different weight you get ... 20 The notion of "significant figures" is meant to communicate how much you know about a number. A number with one sig fig means you know it to roughly one part in 10, two sig figs mean you know it to roughly one part in 100, and so on. This is a useful idea, because if you multiply a number with n sig figs with a number with m, the resulting number ... 20 Measurement errors can accumulate, yes. But we are not talking about measurements here, we are talking about processes and tooling. That's another deal. If you fling off a piece of flint from a flintstone by banging another stone into it, then the "precision" of that other stone doesn't matter. What matters is that you supply enough force - not ... 19 To repeat Wikipedia: The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its value is exactly 299,792,458 metres per second, a figure that is exact because the length of the metre is defined from this constant and the international standard for time. In other words, it's exact ... 19 We don't know the mass of the Moon with that level of accuracy. NASA gives only 4 significant digits. The best estimate of the mass of the Earth I could find is:$$M_\oplus = (5.9722 \pm 0.0006)\times 10^{24}\;~{\rm kg}$\$ Does this have something to do with the earth mass being a "standard unit of mass in astronomy"? Yes, most likely. Google must have ...

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