• Often, horizontal (2D) position, can be expressed as sigma ( σ, standard error ) values which can be calculated for a normal distribution of one variable (univariate). For example, σ_N std. deviation of all Latitude (or Delta North) values with respect to the average Latitude (or North value). For various magnitudes (1σ_N, 2σ_N, etc.) the following interpretation follows from standard statistics:

1σ_N   68.3% of measured positions within 1σ_Nvalue of average position
2σ_N   95.4% of measured positions within 2σ_Nvalue of average position
3σ_N   99.7% of measured positions within 3σ_Nvalue of average position.
4σ_N   Almost 100% of measured positions within 4σ_Nvalue of average position.

These are defined for 2-D and 3-D positions respectively. Standard deviation of the 2-D/3-D distance values from the average position are given by the following relations

• The probability of being within a certain region is described by confidence ellipse (2-D) or confidence ellipsoid (3-D) and represents an approximation of the true precision measures in all directions
• Example : 95% horizontal error ellipse means that if the ellipse is centered on the true position, then there is a 95% chance of the estimated position lying within the boundary of the ellipse.
• The size, shape and orientation of the a posteriori error ellipses depend on the particular satellite geometry and the prababilistic modeling strategy that has been adopted.
• In practice, these will vary slowly with time (with satellite geometry) with step changes being seen mainly at times of satellites entering, or leaving, the configuration (i.e.; rising above, or dropping below the horizon or Elevation Mask settings- the angle of elevation below which all GPS satellites are discarded for position computation).

Figure 11.5 Confidence ellipse (GPS Tutor, 1998)

Table 11.1 summarizes various commonly used accuracy/error measures used in GPS surveying.

Table 11.1 Commonly used accuracy/error measures in GPS surveying

(1)	(2)	(3)	(4) Prob.	(5) Approximation	(6) Sketch	(7) Related Expressions
1D	rms	root mean square	68.3%*	σ		MSE - mean square error (the square of the rms)
	PE	probable error	50%*	0.674σ **
2D		error ellipse	39.4%*	defined by σx,σy & correlation
	CEP	circular error probable	50%*	radius: 0.589( σx+σy )+		CEP - also called CEP (circular probable error)
	2drms	twice distance root mean square	varies 95.4 -9 8.2%+	radius:2σ +		2nd less common definition: 2 dimensional rms (circle's radius 1σ)
3D		error ellipsoid	19.9%*	defined by σx,σy, σz  & correlations
	SEP	spherical error probable	50%*	radius: 0.513(σx+σy+ σz)*		SEP - also called SPE (spherical probable error)