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

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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
    • begin mathsize 16px style mu equals fraction numerator straight capital sigma x over denominator n end fraction end style
    • begin mathsize 16px style sigma squared equals fraction numerator straight capital sigma left parenthesis x minus mu right parenthesis squared over denominator n end fraction equals fraction numerator straight capital sigma x squared over denominator n end fraction minus mu squared end style
  • If you are using a sample to estimate the mean of a population then an unbiased estimate is given by
    • begin mathsize 16px style x with bar on top space equals fraction numerator straight capital sigma x over denominator n end fraction end style 
    • 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
    • begin mathsize 16px style s squared equals fraction numerator straight capital sigma left parenthesis x minus x with bar on top right parenthesis squared over denominator n minus 1 end fraction end style 
    • This can be written in different ways
    • begin mathsize 16px style s squared equals fraction numerator capital sigma open parentheses x minus x with bar on top close parentheses squared over denominator n minus 1 end fraction equals fraction numerator 1 over denominator n minus 1 end fraction open parentheses capital sigma x squared minus open parentheses capital sigma x close parentheses squared over n close parentheses equals fraction numerator n over denominator n minus 1 end fraction open parentheses fraction numerator capital sigma x squared over denominator n end fraction minus x with bar on top squared close parentheses end style
    • 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 begin mathsize 16px style fraction numerator n over denominator n minus 1 end fraction end style

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
      • begin mathsize 16px style s equals square root of fraction numerator straight capital sigma left parenthesis x minus x with bar on top right parenthesis squared over denominator n minus 1 end fraction end root equals square root of fraction numerator 1 over denominator n minus 1 end fraction open parentheses capital sigma x squared minus open parentheses capital sigma x close parentheses squared over n close parentheses end root end style
    • 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 begin mathsize 16px style straight capital sigma x end style and  then you can simply use the formulae in the formula booklet
    • begin mathsize 16px style s squared equals fraction numerator 1 over denominator n minus 1 end fraction open parentheses straight capital sigma x squared minus open parentheses straight capital sigma x close parentheses squared over n close parentheses end style
  • If you are given the raw data then you will first need to calculate straight capital sigma x and straight capital sigma x squared

Worked example

The times, T minutes, spent on daily revision of a random sample of 50 A Level students from the UK are summarised as follows.

          n equals 50      straight capital sigma t equals 6174        straight capital sigma t squared space equals space 831581 

Calculate unbiased estimates of the population mean and variance of the times spent on daily revision by A Level students in the UK

1-2-1-unbiased-estimates-we-solution

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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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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