Probability between bounds and interpretation
All three outputs are percentages: inside the interval, below its lower bound, and above its upper bound. They describe the specified continuous normal model, not an empirical count of observations.
To standardize a single observation against a supplied mean and standard deviation, use the Z-Score Calculator and follow its own input conventions.
The formula
Standardize each bound by subtracting the mean and dividing by the standard deviation. Numerical normal-tail approximations supply the interval probability; direct tails reduce cancellation for distant bounds.
Verified worked example
For a standard normal distribution with mean 0 and standard deviation 1, the probability between −1 and 1 is approximately 68.268949%.
How to use this calculator
- Enter the assumed mean and a strictly positive standard deviation.
- Set lower and upper bounds in the same measurement unit, with lower no larger than upper.
- Read the interval probability and two outside-tail percentages, considering whether a normal model fits the application.
Normal Distribution input conventions
This calculation describes an assumed normal distribution, not the fraction of entered observations in a sample. The mean locates its center and the positive standard deviation sets its scale. Standardizing each bound gives a z value, after which the cumulative normal function determines probability. Bounds can be negative and need not be symmetric around the mean.
Interpreting probability between bounds
The normal model has unbounded tails, even when a real measurement cannot be negative. Choose a distribution that matches your application rather than applying this model automatically. The numerical approximation used here is intended for ordinary probability estimates; displayed percentages are rounded and very small tail probabilities can vanish at numeric precision. Equal bounds have zero probability because a continuous variable assigns zero mass to one exact point.
Assumptions & limitations
What this calculation assumes
- The variable follows the specified continuous normal distribution with positive standard deviation.
What to keep in mind
- Normal-model interval probabilities only; approximation error is about 0.000008 percentage points in ordinary tails.
Common questions
Can standard deviation be zero?
No. A continuous normal distribution requires positive standard deviation. A deterministic value needs a different probability model.
Does this check whether my data are normal?
No. It computes probabilities under a distribution you specify. Assessing normality requires observations and distributional diagnostics.
Why is the probability at one exact value zero?
A continuous normal distribution assigns probability to intervals rather than point masses. Equal lower and upper bounds therefore return zero interval probability.
Sources & further reading
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