# Confidence intervals vs. standard deviation

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?

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).

• 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).
– whuber
Commented May 9, 2015 at 13:18
• 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. Commented May 9, 2015 at 14:23
• 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. Commented May 9, 2015 at 15:15
• 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. Commented May 9, 2015 at 15:53
• @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". Commented May 9, 2015 at 15:54

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.

My answer focuses on the distinction between estimation and prediction.

• With the interval (mean) +/- 2*(std deviation), you predict that about 95% of the data will fall in this interval (with the important proviso that the population is normal). We are answering the question: "Where are the data?"

• With the interval (mean) +/- 2*(std deviation)/sqrt(n), you estimate that the actual population mean lies in this interval, with 95% confidence. We are answering the question: "Where is the average of the data?"

The second interval is the confidence interval, and involves dividing the SD by $$\sqrt n$$, yielding what is known as the standard error of the mean. Some textbooks, software, etc. are sloppy with this terminology, adding to the confusion. But standard deviation measures variation of data, and standard error measures uncertainty in an estimate.

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.