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CIE A-Level Maths Study Notes

5.4.5 Unbiased Estimation

Statistics, a cornerstone of mathematical analysis, heavily relies on the concept of unbiased estimation. This key idea enables us to estimate population parameters such as mean and variance accurately, ensuring that our estimations reflect the true values as accurately as possible when averaged over numerous samples.

Understanding Unbiased Estimators

An unbiased estimator predicts a parameter's true value on average, though not necessarily in every single instance.

Key terms:

  • Population Mean μμ: Average of all data points in a population.
  • Population Variance σ2σ²: How spread out the data points are around the mean.

Calculating Unbiased Estimators

  • For Population Mean:
    • Use the sample mean Xˉ as an unbiased estimator. It's calculated as Xin\frac{\sum X_i}{n}, where XiX_i is each data point, and nn is the sample size.
  • For Population Variance:
    • Use S2=(XiXˉ)2n1S^2 = \frac{\sum (X_i - X̄)^2}{n - 1}. This formula accounts for n1n - 1 in the denominator to maintain unbiasedness.

Application Examples

Example 1: Raw Data Calculation

Consider a scenario where we have a sample of five students' heights in centimeters: 160, 165, 170, 155, 180. Calculate the unbiased estimators for the mean and variance.

Solution:

1. Calculate the Sample Mean Xˉ:

Xˉ=160+165+170+155+1805=8305=166cmX̄ = \frac{160 + 165 + 170 + 155 + 180}{5} = \frac{830}{5} = 166 cm

2. Calculate the Unbiased Sample Variance S2: .

S2=(160166)2+(165166)2+(170166)2+(155166)2+(180166)251=36+1+16+121+1964=3704=92.5cm2S^2 = \frac{(160-166)^2 + (165-166)^2 + (170-166)^2 + (155-166)^2 + (180-166)^2}{5-1} = \frac{36 + 1 + 16 + 121 + 196}{4} = \frac{370}{4} = 92.5 cm²Raw Data Calculation Graph

Example 2: Summarized Data Calculation

Imagine a scenario where a class has a mean test score of 75 with a variance of 12. Estimate the population mean and variance.

Solution:

  • Population Mean Estimate:
    • The sample mean is provided as 75. This value itself is an unbiased estimate of the population mean. Therefore, our unbiased estimate of the population mean is 75.
  • Population Variance Estimate:
    • To estimate the population variance unbiasedly, we need the sample size. Let's assume different sample sizes for demonstration, such as 10, 20, and 30.

The formula to calculate the unbiased estimate of population variance is: Unbiased Variance=nn1×Sample Variance\text{Unbiased Variance} = \frac{n}{n - 1} \times \text{Sample Variance} where nn is the sample size.

1. For Sample Size = 10 Unbiased Variance=10101×12=13.33\text{Unbiased Variance} = \frac{10}{10 - 1} \times 12 =\approx 13.33

2. For Sample Size = 20 Unbiased Variance=20201×12=12.63\text{Unbiased Variance} = \frac{20}{20 - 1} \times 12 = \approx 12.63

3. For Sample Size = 30 Unbiased Variance=30301×12=12.41\text{Unbiased Variance} = \frac{30}{30 - 1} \times 12 = \approx 12.41

Summarized Data Calculation Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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