A quantity calculated from a sample that contains no unknown parameters.

When is an estimator unbiased?

When its expected value equals the parameter: E(T)=\theta.

Unbiased estimate of the population mean?

The sample mean \bar x=\frac{\sum x}{n}.

Unbiased estimate of the population variance?

s^2=\frac{1}{n-1}\sum(xi-\bar x)^2.

Computational form of s^2?

s^2=\frac{1}{n-1}\left(\sum x^2-n\bar x^2\right).

Why is \frac1n\sum(xi-\bar x)^2 not used to estimate \sigma^2?

It is biased: its expected value is \frac{n-1}{n}\sigma^2, so it underestimates \sigma^2 on average.

Mean and variance of \bar X?

E(\bar X)=\mu and \operatorname{Var}(\bar X)=\frac{\sigma^2}{n}.

If X\sim N(\mu,\sigma^2), what is the distribution of \bar X?

\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right).

What is the standard error of the sample mean?

\frac{\sigma}{\sqrt n}, estimated by \frac{s}{\sqrt n} when \sigma is unknown.

How do you standardise a sample mean?

Z=\frac{\bar X-\mu}{\sigma/\sqrt n}\sim N(0,1).

Is \frac{X1+X2+X3}{2} an unbiased estimator of \mu?

No: its expected value is \frac{3\mu}{2}, so it overestimates \mu by \frac{\mu}{2}.

Estimators, bias and the sampling distribution of the meanEdexcel International A Level Further Maths: Flashcards

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Estimator versus estimate?

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Estimator versus estimate?: An estimator is a statistic (a random variable) used to estimate a parameter; an estimate is the value it takes for one sample.
What is a statistic?: A quantity calculated from a sample that contains no unknown parameters.
Define the bias of an estimator $T$ of $\theta$ .: $E(T)-\theta$ .
When is an estimator unbiased?: When its expected value equals the parameter: $E(T)=\theta$ .
Unbiased estimate of the population mean?: The sample mean $\bar x=\frac{\sum x}{n}$ .
Unbiased estimate of the population variance?: $s^2=\frac{1}{n-1}\sum(x_i-\bar x)^2$ .
Computational form of $s^2$ ?: $s^2=\frac{1}{n-1}\left(\sum x^2-n\bar x^2\right)$ .
Why is $\frac1n\sum(x_i-\bar x)^2$ not used to estimate $\sigma^2$ ?: It is biased: its expected value is $\frac{n-1}{n}\sigma^2$ , so it underestimates $\sigma^2$ on average.
Mean and variance of $\bar X$ ?: $E(\bar X)=\mu$ and $\operatorname{Var}(\bar X)=\frac{\sigma^2}{n}$ .
If $X\sim N(\mu,\sigma^2)$ , what is the distribution of $\bar X$ ?: $\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right)$ .
What is the standard error of the sample mean?: $\frac{\sigma}{\sqrt n}$ , estimated by $\frac{s}{\sqrt n}$ when $\sigma$ is unknown.
How does the standard error change if $n$ is multiplied by 4?: It is halved.
How do you standardise a sample mean?: $Z=\frac{\bar X-\mu}{\sigma/\sqrt n}\sim N(0,1)$ .
Is $\frac{X_1+X_2+X_3}{2}$ an unbiased estimator of $\mu$ ?: No: its expected value is $\frac{3\mu}{2}$ , so it overestimates $\mu$ by $\frac{\mu}{2}$ .

Exam questions on Estimators, bias and the sampling distribution of the mean

A random sample of 8 observations of a variable $X$ gives $\sum x=100$ and $\sum x^2=1292$ .
Calculate an estimate of the standard error of the sample mean.2 marks
$X_1$ , $X_2$ and $X_3$ are independent observations from a population with mean $\mu$ and variance $\sigma^2$ .
Show that $T=\frac{X_1+X_2+X_3}{2}$ is a biased estimator of $\mu$ , and state the bias.2 marks
The masses of apples from an orchard are Normally distributed with mean $150$ g and standard deviation $20$ g. Random samples of $16$ apples are taken, and the sample mean $\bar X$ is calculated for each sample.
State the distribution of $\bar X$ , and find $P(\bar X>156)$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).