# 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.

This makes it easier to interpret the meaning of the reported measurements. Additional statistics are reported when the sample mean is not expected to be Gaussian.