How this calculator works
Build symmetric intervals by subtracting and adding multiples of the SD. The rule of thumb rounds the normal central areas; the table also gives numerical normal probabilities. For skewed or multimodal data, these intervals still describe distances from the mean but the quoted coverage may be wrong. This is about individual observations, not the confidence interval for a mean.
Formula
Bounds = μ ± kσ; normal central probability = Φ(k)-Φ(-k) = 1-2Φ(-k).
Worked example
For mean 50 and SD 4, the intervals are [46,54], [42,58] and [38,62]. Their normal areas are about 68.2689%, 95.4500% and 99.7300%.
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Frequently asked questions
Is the two-SD interval exactly 95%?
No. It contains about 95.45% under a normal model. A central 95% interval uses about 1.96 SD.
Does this apply to every distribution?
No. The quoted percentages depend on normality.
Should I divide SD by sqrt(n)?
Only when you are describing sample means with the appropriate sampling assumptions. For individual observations, use their SD.