1Precision
Closeness of repeated measurements to one another.
2Accuracy
Closeness of a measured value to the true or accepted value.
3Standard deviation (SD)
Measure of the spread or uncertainty of observations.
4Probable error
Error limit within which an error is expected to fall with a specified probability, commonly 50% for normally distributed errors.
5Uncertainty
Quantified doubt associated with a measurement or estimated value.
6Random error
Error that varies unpredictably between observations.
7Systematic error
Error that follows a consistent pattern or bias.
8Blunder
Gross mistake caused by an observer, procedure, or recording error.
9Redundancy
Additional observations beyond the minimum required to determine unknown parameters.
10Degrees of freedom
Number of independent redundant observations available for statistical evaluation
11Degrees of freedom
commonly n − u for a full-rank model with n observations and u independent unknowns.
12Least squares
Adjustment method that minimizes the weighted sum of squared residuals.
13Mathematical model
Representation of the relationships between observations and unknown parameters.
14Observation model
Model describing the expected values and statistical properties of observations.
15Functional model
Describes the mathematical relationship between observations and parameters.
16Stochastic model
Describes the variances, covariances, and uncertainties of observations.
17Residual
Difference between an observed value and its adjusted or computed value, depending on the adopted convention.
18Adjustment
Process of estimating the most consistent parameter values from redundant observations.
19By-products of adjustment
Additional results such as standard deviations, covariance matrices, and precision estimates.
20Covariance
Measure of how two random variables vary together.
21Variance
Measure of the variability of a single random variable
22Variance
is the covariance of a variable with itself.
23Variance-covariance matrix
Matrix containing variances on its diagonal and covariances off its diagonal.
24Correlation coefficient
Standardized measure of the strength and direction of a linear relationship between two variables.
25Pearson product-moment correlation coefficient
Common coefficient used to measure linear correlation between two variables.
26Independent variables
Variables whose joint probability distribution factors into the product of their marginal distributions.
27Uncorrelated variables
Variables with zero covariance.
28Joint distribution
Probability distribution describing two or more random variables together.
29Marginal distribution
Probability distribution of one variable considered separately from the others.
30Smaller standard deviation
means higher precision.
31High precision
does not automatically mean high accuracy.
32Functional component
mathematical relationships.
33Stochastic component
statistical properties and uncertainty.
34Positive covariance
the variables tend to increase or decrease together.
35Negative covariance
one variable tends to increase as the other decreases.
36Zero covariance
no linear association is indicated by covariance; a nonlinear relationship may still exist.
37Error propagation
Process of determining how input uncertainties affect computed quantities.
38Covariance propagation
Determination of the covariance matrix of derived quantities from the input covariance matrix.
39Jacobian matrix
Matrix of first-order partial derivatives of output functions with respect to input variables.
40Covariance matrix (C or Σ)
Matrix containing the actual variances and covariances.
41Cofactor matrix (Q)
Matrix describing the relative variance-covariance structure, separated from the variance factor.
42Weight matrix (W)
Matrix representing the relative weights of observations.
43Variance factor
Scale factor relating a cofactor matrix to a covariance matrix.
44Linear function
Function in which the variables enter linearly.
45Nonlinear function
Function in which the variables do not enter linearly.
46σ² = (SD)²
Variance formula
47σxy = E[(X − μx)(Y − μy)]
For a sample of n paired observations:
sxy = Σ[(xi − x̄)(yi − ȳ)] / (n − 1)
Covariance formula
48ρxy = σxy / (σx × σy)
Equivalently, using variances:
ρxy = σxy / √(σx² × σy²)
Pearson correlation coefficient formula
49−1 ≤ ρ ≤ +1
•+1: perfect positive linear correlation
•0: no linear correlation
•−1: perfect negative linear correlation
Range of correlation formula
50For Z = X + Y:
•Mean: μz = μx + μy
•Variance: σz² = σx² + σy² + 2σxy
For a difference, Z = X − Y:
•Mean: μz = μx − μy
•Variance: σz² = σx² + σy² − 2σxy
If X and Y are independent, their covariance is zero. Therefore:
•Var(X + Y) = σx² + σy²
The same variance formula applies to X − Y when covariance is zero.
Mean and variance of a sum formula
51For a linear transformation:
•Y = AX
The covariance matrix of the output is:
•Σyy = A Σxx Aᵀ
Where:
•Σxx = input covariance matrix
•A = transformation matrix
•Aᵀ = transpose of A
•Σyy = output covariance matrix
For nonlinear functions, use the Jacobian matrix J to approximate covariance propagation:
•Σyy ≈ J Σxx Jᵀ
Covariance propagation formula
52•C = σ₀² Q
Where:
•C = covariance matrix
•Q = cofactor matrix
•σ₀² = variance factor, or variance of unit weight
For a diagonal covariance matrix, the weight matrix is the inverse of the covariance matrix:
•W = C⁻¹
If all observations have equal variance σ²:
•C = σ² I
•W = (1/σ²) I
Here, I is the identity matrix.
Covariance and cofactor matrices formula