# Why is the standard deviation the error on the singular meas

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• Soren4
In summary, the standard deviation ##\sigma## of a series of data in experimental Physics represents the error on a single measurement of the same physical quantity. It is calculated using the formula ##\sigma=\sqrt{\sum \frac{(x_i-\mu)^2}{N}}## where ##\mu## is the theoretical "true value" of the quantity. This value can also be represented as a normally distributed random variable with a range of ##[x_i-\sigma,x_i+\sigma]##, indicating that there is a 68% probability that the true value falls within this range. The standard deviation is also related to the mean value, with a sample of n measurements having a mean value that is normally distributed with a range of
Soren4
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?

It means that a single measurement can be represented as a normally distributed random variable ##N(\mu,\sigma)##.

This is in contrast to the mean of a sample of n measurements, which could be represented as a normally distributed random variable ##N(\mu,\sigma/ \sqrt{n})##

## 1. What is standard deviation?

Standard deviation is a measure of how spread out a set of data is from its average or mean value. It is calculated by finding the square root of the variance, which is the average of the squared differences from the mean.

## 2. Why is standard deviation important?

Standard deviation is important because it provides a measure of the variability or dispersion of data. It helps to understand how much the individual data points deviate from the mean and can indicate how reliable or precise the data is.

## 3. How is standard deviation related to error?

The standard deviation is often used as a measure of error or uncertainty in a set of data. It represents the amount of variation or noise in the data and can help to identify potential errors or outliers in the data set.

## 4. How is standard deviation used in scientific research?

In scientific research, standard deviation is used to analyze and interpret data, to determine the reliability of results, and to compare different data sets. It is also used in hypothesis testing and to calculate confidence intervals.

## 5. Can standard deviation be negative?

No, standard deviation cannot be negative as it represents the positive square root of the variance. However, it can be zero if all the data points in a set are identical, indicating no variability in the data.

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