# Unbiased Estimates(CIE A Level Maths: Probability & Statistics 2)

Author

Dan

Expertise

Maths

## Unbiased Estimates

#### What is an unbiased estimator of a population parameter?

• An estimator is a statistic that is used to estimate a population parameter
• When a sample is used with the estimator, the value that it produces is called an estimate
• An estimator is called unbiased if the expected value of the estimator 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

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

• If you had the data for a whole population you could find the actual population mean and variance using
• If you are using a sample to estimate the mean of a population then an unbiased estimate is given by
•
• This is the same formula for the population mean
• If you are using a sample to estimate the variance of a population then an unbiased estimate is given by
•
• This can be written in different ways
• This is a different formula to the population variance
• The last formula shows a method for finding an unbiased estimate for the variance
• Find the variance of the sample (treating it as a population)
• Multiply this by

#### Is there an unbiased estimate for the standard deviation?

• Unfortunately square rooting an unbiased variance does not result in an unbiased standard deviation
• There is not a formula for an unbiased estimate for the standard deviation that works for all populations
• Therefore it is better to just work with the variance and not the standard deviation
• If you need an estimate for the standard deviation then you can use:
• You can use the square root of your unbiased estimate for the population variance
• This won’t be unbiased but it will be a good estimate

#### How do I calculate unbiased estimates?

• If you are given the summary statistics  and  then you can simply use the formulae in the formula booklet
• If you are given the raw data then you will first need to calculate and

#### Worked example

The times,  minutes, spent on daily revision of a random sample of 50 A Level 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 A Level students in the UK

#### Exam Tip

• Always check whether you need to divide by n or n-1 by looking carefully at the wording in the question.

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