In geodetic measurements, repeated observations are used to reduce the influence of random errors and to estimate the precision of the results. Statistical indicators such as mean error, variance, and standard deviation allow surveyors and geodesists to evaluate the quality of observations and quantify their uncertainty. These measures form the statistical basis for the Least Squares Adjustment.
Lesson 06 – Mean Errors, Standard Deviation, and Variance
Objectives
- Understand the concept of mean error in observations.
- Compute variance and standard deviation.
- Interpret the dispersion of measurements.
- Relate statistical indicators to measurement precision.
- Apply these concepts to repeated geodetic observations.
1. Mean Error of Observations
In repeated measurements, each observation differs slightly from the true value due to random errors.
The mean error represents the expected magnitude of these random deviations.
For a set of residuals vi, the mean square error is related to the variance of the observations.
2. Variance
Variance measures the dispersion of the observations around the mean.
Where
- s2 = variance
- vi = residuals
- n = number of observations
A smaller variance indicates higher precision.
3. Standard Deviation
The standard deviation is the square root of the variance:
Interpretation:
- small standard deviation → high precision
- large standard deviation → low precision
In geodetic practice, standard deviation is the most commonly used indicator of measurement precision.
4. Relationship Between Residuals and Precision
Residuals represent the differences between observations and the estimated value.
These residuals are used to compute variance and standard deviation.
The smaller the residuals, the better the consistency of the observations.
5. Importance in Geodesy
Variance and standard deviation are essential for:
- evaluating measurement precision
- defining observation weights
- assessing the reliability of geodetic networks
- performing statistical tests in adjustment
These indicators are used extensively in leveling, GNSS processing, and network adjustment.
6. Solved Example
Repeated distance measurements (meters): 125.334; 125.338; 125.331; 125.336; 125.335.
- Step 1 – Mean value
- Step 2 – Residuals
- Step 3 – Variance
- Step 4 – Standard deviation
7. Proposed Exercise
Repeated angle observations (seconds): 32.418; 32.421; 32.416; 32.420.
Determine:
7.1 Answer
- Mean ≈ 32.4188
- Variance ≈ 0.0000048
- Standard deviation ≈ 0.0022
8. Conclusion
Mean error, variance, and standard deviation are fundamental statistical indicators for evaluating the precision of geodetic observations. These measures quantify the dispersion of measurements and provide the statistical basis for the Least Squares Adjustment.
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