What is an unbiased estimator of a population parameter?

• An?estimator?is a?random variable?that is used to?estimate a population parameter
  • An?estimate?is the value produced by the estimator when a sample is used
• An estimator is called unbiased if its expected value is equal to the population parameter
  • An estimate from an unbiased estimator is called an?unbiased estimate
  • This means that the?mean?of the?unbiased estimates?will get?closer?to the?population parameter?as?more samples?are taken

• The?sample mean?is an?unbiased estimate?for the?population mean
• The?sample variance?is?not an unbiased estimate?for the?population variance
  • On average the sample variance will?underestimate?the population variance
  • As the?sample size increases?the sample variance gets?closer to the unbiased?estimate

What are the formulae for unbiased estimates of the mean and variance of a population?

• A sample of?n?data values (x₁, x₂, ...?etc) can be used to find unbiased estimates for the mean and variance of the population

Is?s_n-1?an unbiased estimate for the standard deviation?

• Unfortunately?s_n-1?is not an unbiased estimate for the standard deviation of the population
• It is better to work with the unbiased variance rather than standard deviation
• There is not a formula for an unbiased estimate for the standard deviation that works for all populations
  • Therefore you will not be asked to find one in your exam

How do I show the sample mean is an unbiased estimate for the population mean?

• You?do not need to learn this proof
  • It is simply here to help with your understanding
• Suppose the population of X has mean?μ?and variance?σ2
• Take a sample of?n?observations
  • X_1,?X_2,?..., X_n
  • E(X_i) = μ
• Using the formula for a linear combination of?n?independent variables:Why is there a divisor of?n-1 in the unbiased estimate for the variance?

• You?do not need to learn this proof
  • It is simply here to help with your understanding
• Suppose the population of X has mean?μ?and variance?σ2
• Take a sample of?n?observations
  • X_1,?X_2,?..., X_n
  • E(X_i) = μ
  • Var(X_i) = σ²
• Using the formula for a linear combination of?n?independent variables: