In geodetic measurements, repeated observations are commonly performed to reduce the influence of random errors. Statistical analysis allows the evaluation of data quality and the determination of the most reliable value. Understanding basic statistical measures is essential for the correct application of the Least Squares Method.
Lesson 03 – Statistical Fundamentals of Observations
Objectives
- Understand the role of statistics in geodetic observations.
- Compute the arithmetic mean of repeated measurements.
- Understand the concept of dispersion.
- Calculate variance and standard deviation.
- Interpret precision based on statistical measures.
1. Role of Statistics in Geodesy
Since every measurement contains random errors, repeated observations of the same quantity will not produce identical values.
Statistical analysis allows us to:
- determine the most probable value,
- evaluate the dispersion of measurements,
- assess the precision of the observations.
The most probable value of repeated measurements is the arithmetic mean.
2. Arithmetic Mean
For n observations:
Where:
- Li = observed values
- L̅ = mean value
The mean represents the best estimate of the true value when only random errors are present.
3. Residuals (Deviations)
The difference between each observation and the mean is called a deviation (or residual):
Properties:
Residuals indicate how each observation differs from the most probable value.
4. Variance
Variance measures the dispersion of the observations:
A smaller variance indicates higher precision.
5. Standard Deviation
The standard deviation is:
Interpretation:
- Small ( s ) → high precision
- Large ( s ) → low precision
Standard deviation is one of the most important quality indicators in geodetic measurements.
6. Standard Error of the Mean
The precision of the mean is given by:
This value represents the uncertainty of the estimated mean.
7. Solved Example
A distance was measured five times (m): 125.334; 125.338; 125.331; 125.336; 125.335.
Step 1 – Mean
Step 2 – Residuals
Step 3 – Variance
Step 4 – Standard deviation
Step 5 – Standard error of the mean
Final result:
8. Proposed Exercise
Repeated angle measurements (seconds): 32.418; 32.421; 32.416; 32.420.
Determine:
Answer
- Mean = 32.4188
- Standard deviation ≈ 0.0022
- Standard error ≈ 0.0011
9. Conclusion
Statistical analysis allows the determination of the most probable value and the evaluation of precision in repeated observations. The mean, variance, and standard deviation form the statistical foundation for the Least Squares Adjustment.
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