167
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Accepted
Can 1 kilogram of radioactive material with half life of 5 years just decay in the next minute?
The short answer is yes. No matter how many atoms there are, there is always a (sometimes vanishingly small) chance that all of them decay in the next minute. The fun answer is actually seeing how ...
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115
votes
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How do you make more precise instruments while only using less precise instruments?
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 ...
- 8,906
75
votes
Accepted
Why does a collection of radioactive atoms show predictable behaviour while a single one is highly random?
Law of large numbers
This law simply states that if you repeat a trial many times, the result tends to be the expected value. For example if you roll a 6-sided die, you could get any of the six ...
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65
votes
Accepted
Are random errors necessarily Gaussian?
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 ...
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64
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Can 1 kilogram of radioactive material with half life of 5 years just decay in the next minute?
TLDR: statistical models are models, and thus by definition not a perfect reflection of reality.
Nihar's answer is good but I'm going to tackle it from a different direction.
First off, if we only ...
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61
votes
How do you make more precise instruments while only using less precise instruments?
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 ...
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58
votes
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Why most distribution curves are bell shaped? Is there any physical law that leads the curves to take that shape?
First, distributions are not always bell-shaped. A very important set of distributions decrease from a maximum at $x=0$, such as the exponential distribution (delay times until a random event such as ...
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48
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How do we know that radioactive decay rates are constant over billions of years?
The comment Samuel Weir makes on the fine structure constant is pretty close to an answer. For electromagnetic transitions of the nucleus, these would change if the fine structure constant changed ...
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40
votes
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Can Second Law of Thermodynamics / Entropy override Newton's Laws?
Can the Second Law of Thermodynamics / Entropy override Newton's Laws?
No. In the example given, every particle obeys Newton's laws. There is no particle that is not obeying $F=ma$.
From the ...
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38
votes
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Line of Best Fit with or Without Constant Term
You should almost always include the intercept. Not including the intercept can lead to bias in your estimate of the slope in your model as well as other problems.
it is generally a safe practice not ...
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34
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How do we know that radioactive decay rates are constant over billions of years?
There are various questions that one would have to answer, if one wished to claim that there had been large changes in decay rates over geological time. Here is what I think might be the best ...
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32
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How do you make more precise instruments while only using less precise instruments?
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 ...
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31
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How do we know that radioactive decay rates are constant over billions of years?
The basic point here is that we don't "know" anything about "the real world". All we have is a model of the world, and some measure of how well the model matches what we observe.
Of course, you can ...
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30
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Can 1 kilogram of radioactive material with half life of 5 years just decay in the next minute?
One thing to keep in mind is that this is not only a statistics question and the analogy of atoms decaying and flipping coins can be misleading.
For example, uranium 235 has a half life of more than ...
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27
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Why is the standard uncertainty defined with a level of confidence of only 68%?
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 ...
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26
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How do you make more precise instruments while only using less precise instruments?
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 ...
- 1,548
25
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Why don't we use absolute error while calculating the product of two uncertain quantities?
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 ...
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23
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Least squares fit versus averaging
That's a good question. The answer depends on how the data is collected. We will posit two simple models and furnish the best unbiased estimator for each model via the Gauss Markov theorem.
We will ...
- 330
21
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How do you make more precise instruments while only using less precise instruments?
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 ...
- 50.3k
21
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Least squares fit versus averaging
Here is the quantitative (simulation) approach @Dr. Momo was mentioning: we can simulate some points for $x_i$ and $F_i$, give them some Gaussian errors, estimate $k$ with the two approaches and see ...
- 4,907
21
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Line of Best Fit with or Without Constant Term
I cannot improve on Dale's answer, but speaking as an (ex) experimental scientist I strongly recommend you allow a non-zero intercept as it can be useful indication that you have systematic errors ...
- 351k
20
votes
Accepted
Why do coherent states have Poisson number distribution?
Short version
$\newcommand{\ket}[1]{\lvert #1 \rangle}\newcommand{\Ket}[1]{\left| #1 \right>}
%
$Because you can use beamsplitters to split a coherent state into a tensorial product of many ...
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20
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Why can't the Uncertainty Principle be broken for individual measurements if it is a statistical law?
While uncertainties can be interpreted as statements about ensembles of identically-prepared states, that's not how they are defined. You take a state $\psi$ and then the uncertainty $\sigma_\psi(A) = ...
- 122k
18
votes
Why most distribution curves are bell shaped? Is there any physical law that leads the curves to take that shape?
‘Bell curve’ often refers to a Gaussian distribution. That distribution is so common that it’s also called the normal distribution. It’s very common because it emerges any time you’re looking at the ...
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17
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Accepted
When to average in the lab for indirect measurements?
Averaging destroys information. Do it as late as is practical in your analysis.
A commenter points out that, if you are making many position/velocity measurements of the same object as it moves once, ...
rob♦
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17
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How do you make more precise instruments while only using less precise instruments?
The way I understand it, errors only accumulate.
That is not always the case. Human ingenuity found and systematically selected processes which improved a particular quality.
Two examples:
You can ...
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17
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Least squares fit versus averaging
As in all things statistical, it completely depends on your assumptions.
Why averages are useful:
If you have a variable x which you can sample, and it comes from a normal distribution with some mean $...
- 4,930
16
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Are random errors necessarily Gaussian?
The reason is probably the central limit theorem: When you add lots of independent random variables, their sum will form a normal distribution, irrespective of their individual probability ...
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