How this calculation works
Convert both bounds into standard-deviation units, then integrate the standard-normal density between them. The two outside probabilities complete the whole distribution. A continuous model assigns zero probability to one exact point.
z = (x−μ)/σ; interval probability = normal area between z boundsA worked example
With mean 100 and standard deviation 15, the interval 85 to 115 spans one standard deviation on either side and contains about 68.268949% of the model probability.
Good to know
A mathematical probability model, not a fit test or an empirical guarantee. Inputs must match the model assumptions. Floating-point arithmetic is used; extremely small probabilities may underflow or round to zero.
A probability is an area rather than a curve height
The bell curve is a density. Its height at one point is not the probability of observing exactly that value. For a continuous normal variable, a single exact point has probability zero; an interval can contain positive probability.
The mean sets the center and the standard deviation sets the scale. With mean 100 and standard deviation 15, the interval 85 to 115 is the same standardized interval as −1 to 1 in the standard normal distribution.
The two tails outside the entered bounds complete the interval probability. Together, the left tail, central interval and right tail sum to one before ordinary numerical rounding.
A familiar shape does not establish a normal model
A data histogram that looks roughly symmetric does not automatically justify normal probabilities in every tail. Outliers, mixtures of groups, bounded measurements or skewed processes can matter especially when you ask about unusual values.
The diagram illustrates a central one-standard-deviation interval, independent of your entered bounds. The result numbers use the actual mean, standard deviation and bounds you enter. A reversed interval is rejected; equal bounds return zero interval area.
The tool computes tail differences directly when both bounds lie on the same side. This reduces one common source of cancellation in small interval probabilities, while retaining the limits of floating-point arithmetic for extreme tails.
A few questions, answered
Can the standard deviation be zero?
No. This calculator requires a positive spread. A point mass with zero variance is a different, degenerate model.
Is an endpoint included?
For a continuous normal model, including or excluding either endpoint does not change the probability.
Is this a confidence interval calculator?
No. It finds probabilities for a specified normal variable. Inference about an estimated mean requires additional sampling assumptions and is a separate task.