The 95% confidence interval gives you a range.
The 2 sigma of a standard deviation also gives you a range of ~95%.
Can someone shed some light on how they are different?
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There are two things here :
The "2 sigma rule" where sigma refers to standard deviation is a way to construct tolerance intervals for normally distributed data, not confidence intervals (see this link to learn about the difference). Said shortly, tolerance intervals refer to the distribution inside the population, whereas confidence intervals refer to a degree of certainty regarding an estimation.
In case you meant standard error instead of standard deviation (which is what I understood at first), then the "2 sigma rule" gives a 95% confidence interval if your data are normally distributed (for example, if the conditions of the Central Limit Theorem apply and your sample size is great enough).
May be, it will be easier to explain, to avoid confusion.
Standard deviation: With probability about 95% we will find every new sample in interval (x_mean - 2 * sigma; x_mean + 2 * sigma) what says us where to expect the location of new samples.
Confidence interval: With probability of f.e. 95% the real x_mean value will be found in the interval (x_mean - x_ci; x_mean + x_ci) which shows us quality of the measurements.
"x_ci" and "2 * sigma" are two different values, because of corresponding to two different expectations. x_ci = t * sigma / sqrt(n), where t is a multiplier according to the used theory.
This equation relies on the assumption that the errors are Gaussian. Also, the factor of 2 in front of the SE(β1) term will vary slightly depending on the number of observations n in the linear regression. To be precise, rather than the number 2, the equation should contain the 97.5 % quantile of a t-distribution with n−2 degrees of freedom.