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Are $3.43\pm 0.04$ $\frac{\mathrm{m}}{\mathrm{s}}$ and $3.48$ $\frac{\mathrm{m}}{\mathrm{s}}$ within expected range of values?

The answer is yes, but I do not clearly see why this is so. I appreciate if you can give me a hint on this.

This is what I could think of:

3.48 could be either $3.47\fbox{5-9}\cdots$ or it could be $3.48\fbox{0-4}\cdots$, 3.43 could be either $3.42\fbox{5-9}\cdots$ or it could be $3.43\fbox{0-4}\cdots$, and 0.04 could be either $0.03\fbox{5-9}\cdots$ or it could be $0.04\fbox{0-4}\cdots$. So if for example, we say that 3.48 was $3.476101$ and 3.43 was $3.43404$ and 0.04 was 0.04304, then $3.43+0.04$ would be 3.47708 and $3.43-0.04$ would be 3.391. So 3.476101 lies between 3.391 and 3.47708. So it could be possible that $3.43\pm 0.04$ and 3.48 be within expected range of values.

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2 Answers 2

That depends entirely on what you consider to be "expected range of values."

When you see a value like $3.43\pm 0.04$ (I will omit units for brevity), in many cases, it actually represents a normal probability distribution with a mean of $3.43$ and a standard deviation of $0.04$. If the $3.43\pm 0.04$ is the result of an experiment, for example, then the probability distribution applies to the true value of the thing the experiment was trying to measure. In particular, the experimenters are saying there is a 68% chance that the true value of the quantity is between $3.39$ and $3.47$, the so-called $1\sigma$ range. There is a 95% chance that the true value is between $3.35$ and $3.51$, the $2\sigma$ range. And so on.

Since the result with uncertainty really defines a probability distribution, when you want to compare another number to this result, the question you need to ask is not "is this number within the acceptable range?" (because there is no hard boundary to the range), but rather "what is the probability that the experiment would be at least this far off?" In the example in your question, the difference between the two values, $3.43$ and $3.48$, is $1.25\sigma$ (that's $1.25\times 0.04$). You can calculate that the probability of being off by $1.25\sigma$ or more is 21%; conversely, the probability of being within $1.25\sigma$ is 79%. So you could say there is a 79% probability that the two values are compatible. That's a fairly large probability, so it seems reasonable to say that these two values are, in fact, compatible (which is the more precise version of saying $3.48$ is within the "expected range").

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Errors in the form you display in your question follow the rules of statistics and associated probability estimates. The +/- value is the standard deviation/sigma and values are considered quite probable within 1 sigma. If you look at the plot in the link, values that differ by 2 sigma are not that improbable either. While 1 sigma deviations from truth have a 68% probability, 2 sigma have 27% which is not improbable either.

When working with probabilities to choose various hypothesis one requires over 4sigma deviations to consider a value incompatible.

In high energy physics a resonance is accepted as probable if the signal is over 4 sigma incompatible with the background.

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