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 $μ$: Average of all data points in a population.
• Population Variance $σ²$: How spread out the data points are around the mean.

Calculating Unbiased Estimators

• For Population Mean:
  • Use the sample mean $X̄$ as an unbiased estimator. It's calculated as $\frac{\sum X_i}{n}$, where $X_i$ is each data point, and $n$ is the sample size.
• For Population Variance:
  • Use $S^2 = \frac{\sum (X_i - X̄)^2}{n - 1}$. This formula accounts for $n - 1$ 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 $X̄$:

2. Calculate the Unbiased Sample Variance $S²$: .

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: $\text{Unbiased Variance} = \frac{n}{n - 1} \times \text{Sample Variance}$ where $n$ is the sample size.

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

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

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