GELE Adjustment Computations (Least Squares) — Theory of Errors, Weights and Most Probable ValueConcept Map

Concept

Error Classification

Sub Concepts

• Blunders (Mistakes)
• Systematic Errors
• Random Errors

Relationship To Central

Foundation for understanding what can and cannot be adjusted

Concept

Random Errors & Statistical Measures

Sub Concepts

• Standard Deviation (σ)
• Variance (σ²)
• Probable Error (0.6745σ)
• Normal Distribution

Relationship To Central

Core input to adjustment theory; quantified by standard deviation, variance, and probable error

Concept

Most Probable Value (MPV)

Sub Concepts

• Arithmetic Mean
• Residuals (v)
• Standard Deviation of Single Observation
• Standard Deviation of Mean

Relationship To Central

The best single estimate from multiple equal-reliability observations

Concept

Weights

Sub Concepts

• Weight Definition (w = 1/σ²)
• Inverse Variance Relationship
• Leveling Weights (w ∝ 1/K)
• Repetition Weights (w ∝ n)
• Relative Weight Assignment

Relationship To Central

Express relative reliability of observations; essential for handling unequal observations

Concept

Weighted Mean

Sub Concepts

• Weighted Average Formula
• Weight Application
• Reliability Adjustment

Relationship To Central

Best estimate when observations have different reliabilities

Concept

Residual Analysis

Sub Concepts

• Residual Calculation (v = x - x̄)
• Sum of Squared Residuals
• Error Propagation

Relationship To Central

Measures how each observation deviates from the MPV; basis for error quantification

Concept

Sample vs Population Statistics

Sub Concepts

• Sample Standard Deviation (n-1)
• Population Standard Deviation (n)
• Bias Correction

Relationship To Central

Determines whether to use n or n-1 in standard deviation calculations

Concept

Improvement by Repetition

Sub Concepts

• σ of Mean Formula (σ/√n)
• Improvement Rate
• Optimal Number of Observations

Relationship To Central

Shows how σ of mean improves with √n; justifies multiple measurements

To

Random Errors & Statistical Measures

From

Error Classification

Strength

strong

Relationship

Only random errors (after eliminating blunders and correcting systematic) are described by statistical measures

To

Most Probable Value

From

Random Errors & Statistical Measures

Strength

strong

Relationship

Standard deviation and variance quantify the spread that the MPV represents as the central tendency

To

Residual Analysis

From

Most Probable Value

Strength

strong

Relationship

Residuals (v = x - x̄) are calculated from the MPV; their sum of squares determines error measures

To

Random Errors & Statistical Measures

From

Residual Analysis

Strength

strong

Relationship

Sum of squared residuals (Σv²) is the basis for computing standard deviation and variance

To

Weighted Mean

From

Weights

Strength

strong

Relationship

Weights are directly applied in the weighted mean formula (x̄w = Σwx/Σw)

To

Weights

From

Random Errors & Statistical Measures

Strength

strong

Relationship

Weight definition w = 1/σ² inversely relates weight to the variance of each observation

To

Weighted Mean

From

Most Probable Value

Strength

moderate

Relationship

Arithmetic mean is special case of weighted mean when all weights are equal

To

Most Probable Value

From

Improvement by Repetition

Strength

strong

Relationship

The σ/√n formula shows that MPV precision improves predictably with observation count

To

Random Errors & Statistical Measures

From

Sample vs Population Statistics

Strength

strong

Relationship

Determines whether n or n-1 divisor is used in calculating standard deviation from sample data

To

Most Probable Value

From

Sample vs Population Statistics

Strength

moderate

Relationship

n-1 divisor is used in field surveying (sample) to get unbiased estimate of population standard deviation

To

Most Probable Value

From

Weights

Strength

moderate

Relationship

When weights differ, weighted calculations replace simple arithmetic to find the best MPV

To

Most Probable Value

From

Error Classification

Strength

strong

Relationship

Only after blunders are removed and systematic errors corrected can MPV be meaningfully computed

To

Weighted Mean

From

Improvement by Repetition

Strength

moderate

Relationship

Repetition weights (w ∝ n) embody the principle that more repeats reduce uncertainty

To

Improvement by Repetition

From

Residual Analysis

Strength

moderate

Relationship

Residual analysis determines the actual standard deviation, which feeds into σ/√n calculations