How do I use this calculator?
Choose an interval from a mean and positive standard deviation, or recover both from the lower and upper bounds of a symmetric 2σ interval.
Calculator Operations:
- Enter the mean (μ) and standard deviation (σ) to calculate the range.
- Enter both lower and upper 2σ bounds to recover the mean (μ) and standard deviation (σ).
- The upper bound must exceed the lower bound; use consistent units.
Both modes show the 1σ, 2σ and 3σ intervals. Their approximate 68%, 95% and 99.7% coverage assumes normal data; exact normal coverage within 2σ is about 95.45%.
Use the calculate button when all known values have been entered and use the reset button when you want to clear the values from the calculator.
Calculate the mean ± 2 standard deviation interval for approximately normal data, view 68-95-99.7 ranges, or recover the mean and standard deviation from bounds.
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2 Standard Deviation Rule Formula
The following formula is used to calculate the 2 Standard Deviation Rule.
Variables:
- Range is the mean ± 2σ interval, covering about 95% for an approximately normal distribution
- μ is the mean of the data set
- σ is the standard deviation of the data set
To calculate the 2 Standard Deviation Rule, first find the mean (μ) and the standard deviation (σ) of the data set. Then, add and subtract two times the standard deviation from the mean. For approximately normal data, this interval contains about 95% of the distribution. It is not a confidence interval for the mean or a guarantee for arbitrary data.
What is a 2 Standard Deviation Rule?
The 2 Standard Deviation Rule, also known as the Empirical Rule, is a statistical rule which states that for a normal distribution, almost all data will fall within three standard deviations of the mean. Specifically, about 68% of data falls within one standard deviation, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations. The rule describes spread after the mean and standard deviation are known; it does not calculate standard deviation directly from raw observations. See NIST normal-distribution coverage.
How to Calculate 2 Standard Deviation Rule?
The following steps outline how to calculate the 2 Standard Deviation Rule.
- First, determine the mean of the data set (μ).
- Next, determine the standard deviation of the data set (σ).
- Next, use the formula Range = μ ± 2σ to calculate the interval containing about 95% under a normal-distribution model.
- Finally, calculate the 2 Standard Deviation Rule.
- After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem :
Use the following variables as an example problem to test your knowledge.
For μ = 100 and σ = 15, the 2σ interval is 100 ± 30 = 70 to 130.
In reverse, μ = (70 + 130) / 2 = 100.
σ = (130 − 70) / 4 = 15.
