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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 :

  1. 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.

  2. 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).

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    $\begingroup$ This doesn't appear to address the question itself, which asks for the distinction between a confidence interval and a "2 sigma ... range" (which is something that is closer to a tolerance interval). $\endgroup$ – whuber May 9 '15 at 13:18
  • $\begingroup$ That's not how I understood the question : it seemed to me that it was unclear to the author why confidence intervals were not always constructed using the "2 sigma rule". Maybe @Berry could edit his question to make it clearer ? Plus, (but it might be a personal bias from being used to work with sampling) when I see $[ \hat{\mu} - 2 \hat{\sigma} ; \hat{\mu} + 2 \hat{\sigma}]$, it makes me think of a confidence interval (typically under the hypothesis of asymptotic normal distribution) more than a tolerance interval. $\endgroup$ – Antoine R May 9 '15 at 14:23
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    $\begingroup$ Confused. The answer is true if the variable of concern is a bunch of sample means, which according to central limit theorem has to be normal. The confidence interval is about +/- 2*STANDARD ERROR from the mean; I don't understand how SD will approximate SE, which also considers sample size. The question conflates the 95% of sample and 95% of sample means, and that should be addressed. $\endgroup$ – Penguin_Knight May 9 '15 at 15:15
  • $\begingroup$ Ah, I understand your comments now. I didn't know the difference between standard deviation and standard error (not a native English speaker), so I didn't spot how the "2 standard deviation" rule had to refer to a tolerance interval rather than a confidence interval. Will edit my answer. $\endgroup$ – Antoine R May 9 '15 at 15:53
  • $\begingroup$ @Penguin_Knight, in sampling, confidence intervals are constructed using ONE standard deviation (which is almost an unbiased estimator of the variance of the Horvitz-Thompson estimator). Hence my confusion. And central limit theorems don't always apply in this case, which is why I wanted to mention sometimes confidence intervals are not constructed using the "2 sigma rule". $\endgroup$ – Antoine R May 9 '15 at 15:54
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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.

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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.

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