Adjustment of Observations in Geodesy: Mean Errors, Standard Deviation, and Variance.

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.

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Lesson 06 – Mean Errors, Standard Deviation, and Variance

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Objectives

1. Understand the concept of mean error in observations.
2. Compute variance and standard deviation.
3. Interpret the dispersion of measurements.
4. Relate statistical indicators to measurement precision.
5. Apply these concepts to repeated geodetic observations.

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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 v_i, 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
• v_i = 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

Observation	Residual
125.334	-0.0008
125.338	0.0032
125.331	-0.0038
125.336	0.0012
125.335	0.0002

• Step 3 – Variance

• Step 4 – Standard deviation

7. Proposed Exercise

Repeated angle observations (seconds): 32.418; 32.421; 32.416; 32.420.

Determine:

a) Mean value

b) Variance

c) Standard deviation

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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Adjustment of Observations in Geodesy: Statistical Fundamentals of Observations.

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.

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Lesson 03 – Statistical Fundamentals of Observations

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Objectives

1. Understand the role of statistics in geodetic observations.
2. Compute the arithmetic mean of repeated measurements.
3. Understand the concept of dispersion.
4. Calculate variance and standard deviation.
5. Interpret precision based on statistical measures.

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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:

• L_i = 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

Observation	Residual (m)
125.334	-0.0008
125.338	0.0032
125.331	-0.0038
125.336	0.0012
125.335	0.0002

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:

a) Mean

b) Standard deviation

c) Standard error of the mean

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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