Thinking...Why isn’t it practical to examine

a whole population?

SamplingAims: To look at how the means and standard deviations/variances of samples compare to the original population.

Lesson OutcomesName: To know what the distribution

of sample means X is and know what is meant by “unbiased estimators” of sample mean, variance and standard deviation are.

Describe: The key features of the distribution of sample means and how they relate to the original population.

Explain: How to find an unbiased estimator of the population variance.

MathsEx 5A p122, 5B p126, 5C p131Next Lesson: More SamplingAleksandr Mikhailovich Lyapunov

1857-1918: Russian mathematician who proved the central limit theorem we meet today.

SamplingIt is rarely practical to sample a

whole population.Even getting information about

all students in college is a challenge.

For this reason it is important to know how the key features (mean and standard deviation) of samples compare to the corresponding values of the “parent population”.

AveragesFor years you have been taught

that the average of a sample is representative of the data being sampled.

You have also been told that to get a more reliable estimate you need _________

Time to learn why.

AnimationNot sure if this will work.http://onlinestatbook.com/stat_si

m/sampling_dist/index.htmlWe will now look at some key

ideas in sampling.

Distribution of MeansWhen sampling a population that

follows a normal distribution the means of all possible samples (of size n) will create another data set.

This dataset also follows a normal distribution.

If the original distribution is X~N(μ,σ2) then the means of these samples follow

2,~n

NX

Distribution of MeansThe distribution of means is

So the mean of the sample means is (the population mean).

The standard deviation (called the standard error of the sample means) is

2,~n

NX

n

Using (and hints)

None Normal Distributions

None Normal Distributions

Central Limit TheoremIn very simple terms the central

limit theorem states that regardless of the population distribution the sample means will still follow

Provided the sample size is large enough (n ≥ 30) though we can usually go ahead whatever.

2,~n

NX

Worth a Mark or Two

Unbiased EstimatorsBecause the distribution of sample

means has a mean that is the same as the population being sampled then the mean of a sample is already an unbiased estimator.

What about variances and standard deviations of samples how do they compare to the variance and standard deviations of the population?

Unbiased Estimators of VarianceSince the Variance/Standard

Deviation of samples is naturally less than that of the population being sampled we amend it to be more representative (often called s2/s).

This is done by dividing the variance by n-1 rather than n

Unbiased Estimators of Variance

Note the normal standard deviation can be converted to s by multiplying by

1

12

2

2

N

xxs

N

xxs

1122

n

ns

n

ns

Example

316002xx

Round-UpThe means of samples of

populations with a mean μ and standard deviation σ will follow

If the population is already normally distributed then this is always the case.

If not normally distributed it will still be the case if n≥30 this is called Central Limit Theorem.

2,~n

NX

Round-UpThe mean of samples is, on

average, the mean of the population. This means that the mean of a sample is an unbiased estimate for the mean of the population.

The variance/standard deviation of a sample is less varied than the population it is from. This is corrected using

1

2

2

N

xxs