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.

• Please consider writing descriptive question titles with appropriate punctuation, grammar, and formatting. See this meta post: How do we write good question titles?. – user191954 Oct 30 '18 at 14:11

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").