# Why is the standard deviation the error on the singular measurement?

I'm a beginner with the study in data analysis in Physics. I'm trying to understand the meaning, in the field of experimental Physics, of the standard deviation $\sigma$ of a series of data.

There is one fundamental thing about $\sigma$ that I read but I could not understand.

In a series of $N$ measurements of the same physical quantity (in the same conditions) the standard deviation $\sigma$ of the data represents the error on the singular measurement.

That is I should write the result of one measurement as $$x_i\pm \sigma$$

I'm aware of these facts about $\sigma$ (regarding its meaning):

• $\sigma=\sqrt{\sum \frac{(x_i-\mu)^2}{N}}$ , where $\mu$ is the theoric "true value" of the physical quantity measured
• The flexes of the Gaussian distribution are in $x_{1,2}=\pm \sigma$
• $[\bar{x}-\sigma,\bar{x}+\sigma]$ contains the $68\%$ of measurements, where $\bar{x}$ is the mean value, which is the best possible approximation of $\mu$

• There is the $68\%$ of probability to find $\mu$ in $[x_i-\sigma,x_i+\sigma]$ and, which is equivalent, to find $x_i$ in $[\mu-\sigma,\mu+\sigma]$

• There is the $99.7\%$ of probability to find $\mu$ in $[x_i-3\sigma,x_i+3\sigma]$ and, which is equivalent, to find $x_i$ in $[\mu-3\sigma,\mu+3\sigma]$

I'm ok with these facts that come from the properties of the Gaussian distribution but still I do not see why $\sigma$ is the error on the singular datum $x_i$.

In other words I do not understand why the interval of variation of $x_i$ should be $[x_i-\sigma,x_i+\sigma]$.

Does this interval have particular properties in terms of probability, linked with the error on the singular value?

The measurements are repeated $N$ times for the same physical conditions; there are thus $N$ points for this single datum, which will be reported as the mean of the sample , along with the measured standard deviation. The interval is conventional, based on common usage.