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
- $μ = \frac{Σ x}{n}$
- $σ^{2} = \frac{Σ {(x - μ)}^{2}}{n} = \frac{Σ x^{2}}{n} - μ^{2}$
If you are using a sample to estimate the mean of a population then an unbiased estimate is given by
- $\bar{x} = \frac{Σ x}{n}$
- 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
- $s^{2} = \frac{Σ {(x - \bar{x})}^{2}}{n - 1}$
- This can be written in different ways
- $s^{2} = \frac{Σ {(x - \bar{x})}^{2}}{n - 1} = \frac{1}{n - 1} (Σ x^{2} - \frac{{(Σ x)}^{2}}{n}) = \frac{n}{n - 1} (\frac{Σ x^{2}}{n} - {\bar{x}}^{2})$
- 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 $\frac{n}{n - 1}$

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
  - $s = \sqrt{\frac{Σ {(x - \bar{x})}^{2}}{n - 1}} = \sqrt{\frac{1}{n - 1} (Σ x^{2} - \frac{{(Σ x)}^{2}}{n})}$
- 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 $Σ x$ and $Σ x^{2}$ then you can simply use the formulae in the formula booklet
- $\bar{x} = \frac{Σ x}{n}$
- $s^{2} = \frac{1}{n - 1} (Σ x^{2} - \frac{{(Σ x)}^{2}}{n})$
If you are given the raw data then you will first need to calculate $Σ x$ and $Σ x^{2}$

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.

$n = 50$ $Σ t = 6174$ $Σ t^{2} = 831581$

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

Examiner Tip

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

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

Revision Note

What is an unbiased estimator of a population parameter?

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

Is there an unbiased estimate for the standard deviation?

How do I calculate unbiased estimates?

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1. Statistical Sampling

1.1 Sampling & Data Collection

1.1.1 Sampling & Data Collection

1.2 Sampling & Estimation

1.2.1 Unbiased Estimates

1.2.2 Distribution of Sample Means

1.2.3 Confidence Intervals

2. Statistical Distributions

2.1 Poisson Distribution

2.1.1 The Poisson Distribution

2.2 Linear Combinations of Random Variables

2.2.1 Linear Combinations of Random Variables

2.3 Continuous Random Variables

2.3.1 Probability Density Function

2.3.2 E(X) & Var(X) (Continuous)

2.3.3 Continuous Uniform Distribution

2.4 Working with Distributions

2.4.1 Choosing Distributions

2.4.2 Approximations of Distributions

3. Hypothesis Testing

3.1 Hypothesis Testing

3.1.1 Hypothesis Testing

3.1.2 Type I & Type II Errors

3.2 Hypothesis Testing (Discrete Distribution)

3.2.1 Binomial Hypothesis Testing

3.2.2 Poisson Hypothesis Testing

3.3 Hypothesis Testing (Normal Distribution)

3.3.1 Normal Hypothesis Testing

Author: Dan