Probability Interval from Measured Data

How to use

1. This calculator has one analysis mode: a two-sided central normal interval estimated from measured data.
2. Enter 2–10 measurements of the same quantity. Use one consistent unit for every value.
3. Enter the desired central probability R as a percentage strictly between 0% and 100%.
4. Select Calculate to update the interval, mean, sample standard deviation, z value, and sample count. Select Reset to restore the five-value 90% example.
5. Read the lower and upper bounds in the same unit as the measurements. R is the modelled share of an approximately normal population between those bounds.
6. Correct warnings for too few or too many values, nonnumeric entries, or invalid probability. Review unusual values before relying on the result.
7. Do not use this result as a confidence interval for the population mean or as a finite-sample statistical tolerance interval with a stated confidence level. Do not use it for dependent, strongly non-normal, censored, or mixed-population data.

Equations used

For n measurements x_i, let x̄ be their arithmetic mean, s the Bessel-corrected sample standard deviation, and R the requested central probability in percent.

For the lower tail, z(p) is negative. The constants are a₁ = 2.30753, a₂ = 0.27061, a₃ = 0.99229, and a₄ = 0.04481. The displayed positive two-sided value is z_R = −z(p), rounded to four decimals.

Definitions, units, and limits: x_i, x̄, s, x_min, and x_max share the user’s unit; n is a count; R is percent; p and z are dimensionless. Calculations retain full floating-point precision; result displays use up to seven significant digits. Sign convention: s and z_R are non-negative magnitudes, and the lower/upper offsets are −z_Rs and +z_Rs. The model assumes independent observations from one approximately normal population.