GPS Accuracy Circle Calculator
A GPS fix is never a perfect point; the true position lies somewhere within a circle of uncertainty around the reported location. The size of that circle depends on the horizontal position standard deviation (sigma), which receivers derive from satellite geometry and signal quality. This tool converts a horizontal sigma into the common accuracy measures: the circular error probable (CEP), the radius containing half of all fixes; the 95 percent radius (often labelled R95 or 2drms-like); and the 50 percent and 99 percent circles. It uses the Rayleigh distribution, which describes the radial error when the two horizontal error components are equal and normally distributed. Enter the horizontal sigma in metres.
GPS accuracy circle formula
radius at probability p = sigma * sqrt(-2 * ln(1 - p))
CEP (p = 0.50) = sigma * 1.1774
R95 (p = 0.95) = sigma * 2.4477
R99 (p = 0.99) = sigma * 3.0349
When the east and north position errors are independent, zero-mean, and have equal standard deviation sigma, the distance from the true point follows a Rayleigh distribution. The radius enclosing a probability p is sigma times the square root of negative two times the natural log of one minus p. Substituting the probabilities gives the familiar multipliers: about 1.18 for CEP, 2.45 for the 95 percent circle, and 3.03 for the 99 percent circle.
Worked example
A receiver reports a horizontal sigma of 5 metres. CEP = 5 * 1.1774 = 5.89 metres, so half of all fixes fall within about 5.89 metres of the true position. The 95 percent radius = 5 * 2.4477 = 12.24 metres, meaning 19 of 20 fixes lie within 12.24 metres. The 99 percent radius = 5 * 3.0349 = 15.17 metres. The 1-sigma RMS radius is 5.00 metres by definition.
Frequently asked questions
What is circular error probable?
Circular error probable (CEP) is the radius of a circle, centred on the true position, that contains half of all GPS fixes. It is a widely used single-number accuracy measure. A CEP of 5 metres means that in the long run half of your fixes land within 5 metres of the truth and half fall outside it.
Why is the 95 percent radius larger than CEP?
Because it must enclose far more of the error distribution. CEP covers 50 percent of fixes, while the 95 percent radius covers nineteen of twenty. Under the Rayleigh model the 95 percent radius is about 2.08 times the CEP, so requiring higher confidence that a fix lies inside the circle means a noticeably larger circle.
What sigma should I enter?
Use the horizontal position standard deviation reported by your receiver or estimated from its horizontal dilution of precision (HDOP) multiplied by the user-equivalent range error. Many consumer receivers report an estimated horizontal accuracy that approximates one sigma. The sigma is a user-editable input so you can match your device's figure.
Does this assume equal east and north errors?
Yes. The Rayleigh model used here assumes the east and north error components are independent with equal standard deviation, which is a common and reasonable approximation for GPS. When the error ellipse is strongly elongated, more elaborate formulas that use both axis standard deviations give a more accurate CEP, but the equal-sigma result is a good general estimate.
Official sources
- U.S. Government GPS information: GPS accuracy.
- U.S. Federal Aviation Administration: GNSS performance.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.