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Unbiased Estimates (DP IB Maths: AI HL)
Revision Note
Unbiased Estimates
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
- An unbiased estimate for the mean μ of a population can be calculated using
- An unbiased estimate for the variance σ² of a population can be calculated using
- This is given in the formula booklet
- This can also be written as
- Notice that dividing by gives a biased estimate but dividing by gives an unbiased estimate
- Different calculators can use different notations for
- , , are notations you might see
- You may also see the square roots of these
Is sn-1 an unbiased estimate for the standard deviation?
- Unfortunately sn-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
- X1, X2, ..., Xn
- E(Xi) = μ
- Using the formula for a linear combination of n independent variables:
- As this shows the formula will produce an unbiased estimate for the population mean
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
- X1, X2, ..., Xn
- E(Xi) = μ
- Var(Xi) = σ2
- Using the formula for a linear combination of n independent variables:
- It can be shown that
- This comes from rearranging
- It can be shown that
- This comes from rearranging
- Using the formula for a linear combination of n independent variables:
- As this shows that the sample variance is not unbiased
- You need to multiply by
Examiner Tip
- Check the wording of the exam question carefully to determine which of the following you are given:
- The population variance:
- The sample variance:
- An unbiased estimate for the population variance:
Worked example
The times, minutes, spent on daily revision of a random sample of 50 IB students from the UK are summarised as follows.
Calculate unbiased estimates of the population mean and variance of the times spent on daily revision by IB students in the UK.
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