#4 Estimators

Estimation of Parameters and Properties of Estimators

This tutorial explains estimation methods used in statistics. Civil engineers use estimation when analyzing rainfall, traffic flow, soil strength, or material properties.

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Module 1: Estimation of Parameters (Sample Mean)

The most common estimator for population mean is the sample mean.

$$ \bar{x} = \frac{\sum x_i}{n} $$

where

• xᵢ = observations
• n = sample size

Worked Example 1

Concrete strength (MPa): 28, 30, 27, 29, 31

$$ \bar{x} = \frac{28+30+27+29+31}{5} $$

$$ \bar{x} = 29 $$

Estimated mean strength = 29 MPa

Worked Example 2

Rainfall (mm): 42, 45, 40, 44, 39

$$ \bar{x} = \frac{42+45+40+44+39}{5} $$

$$ \bar{x} = 42 $$

Worked Example 3

Traffic flow (veh/hr): 520, 510, 495, 505, 515

$$ \bar{x} = \frac{520+510+495+505+515}{5} $$

$$ \bar{x} = 509 $$

Worked Example 4

Soil density (kN/m³): 18, 19, 20, 18, 21

$$ \bar{x} = \frac{18+19+20+18+21}{5} $$

$$ \bar{x} = 19.2 $$

Worked Example 5

Bridge load values (kN): 100, 105, 110, 95, 90

$$ \bar{x} = \frac{100+105+110+95+90}{5} $$

$$ \bar{x} = 100 $$

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Module 2: Estimators (Sample Variance)

Estimator for population variance:

$$ S^2 = \frac{\sum (x_i-\bar{x})^2}{n-1} $$

Worked Example 1

Data: 4, 6, 8

Mean

$$ \bar{x} = \frac{4+6+8}{3} = 6 $$

Variance

$$ S^2 = \frac{(4-6)^2+(6-6)^2+(8-6)^2}{2} $$

$$ S^2 = 4 $$

Worked Example 2

Data: 2, 4, 6, 8

Mean

$$ \bar{x} = 5 $$

Variance

$$ S^2 = \frac{9+1+1+9}{3} $$

$$ S^2 = 6.67 $$

Worked Example 3

Data: 10, 12, 14

Mean

$$ \bar{x} = 12 $$

Variance

$$ S^2 = \frac{4+0+4}{2} $$

$$ S^2 = 4 $$

Worked Example 4

Data: 5, 7, 9, 11

Mean

$$ \bar{x} = 8 $$

Variance

$$ S^2 = \frac{9+1+1+9}{3} $$

$$ S^2 = 6.67 $$

Worked Example 5

Data: 3, 3, 3, 3

Mean

$$ \bar{x} = 3 $$

Variance

$$ S^2 = 0 $$

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Module 3: Bias of an Estimator

Bias formula:

$$ Bias(\hat{\theta}) = E(\hat{\theta}) - \theta $$

Worked Example 1

True mean = 50

Estimated mean = 50

$$ Bias = 50 - 50 = 0 $$

Worked Example 2

True mean = 60

Estimated mean = 58

$$ Bias = 58 - 60 = -2 $$

Worked Example 3

True value = 100

Estimated value = 105

$$ Bias = 105 - 100 = 5 $$

Worked Example 4

True rainfall mean = 40 mm

Estimator average = 42 mm

$$ Bias = 42 - 40 = 2 $$

Worked Example 5

True traffic flow = 500

Estimated mean = 495

$$ Bias = 495 - 500 = -5 $$

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Module 4: Precision

Precision depends on estimator variance.

$$ Var(\hat{\theta}) $$

Worked Example 1

Estimator A: 50, 51, 49

Values are close → High precision

Worked Example 2

Estimator B: 40, 60, 50

Large variation → Low precision

Worked Example 3

Variance comparison

Estimator A variance = 2

Estimator B variance = 6

A is more precise.

Worked Example 4

Estimator values: 100, 101, 99, 100

Small variation → High precision

Worked Example 5

Estimator values: 80, 120, 90, 110

Large variation → Low precision

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Module 5: Mean Square Error (MSE)

MSE formula:

$$ MSE(\hat{\theta}) = Var(\hat{\theta}) + [Bias(\hat{\theta})]^2 $$

Worked Example 1

Variance = 4

Bias = 0

$$ MSE = 4 $$

Worked Example 2

Variance = 2

Bias = 1

$$ MSE = 2 + 1^2 $$

$$ MSE = 3 $$

Worked Example 3

Variance = 5

Bias = 2

$$ MSE = 5 + 4 $$

$$ MSE = 9 $$

Worked Example 4

Variance = 1

Bias = 0.5

$$ MSE = 1 + 0.25 $$

$$ MSE = 1.25 $$

Worked Example 5

Variance = 3

Bias = 1

$$ MSE = 3 + 1 $$

$$ MSE = 4 $$

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Practice Questions

1. Find mean of 45, 48, 50, 52, 47.
2. If variance = 6 and bias = 2, find MSE.
3. Define unbiased estimator.
4. Explain precision with example.
5. Calculate variance for data: 2, 4, 6.