Sampling

Populations and Samples #

A parameter is fixed; a statistic varies.

A parameter describes the whole population and a statistic describes only your sample

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There are a thousand people in the population, and you can only ask thirty.

The population mean is fixed but unknown. You can compute the sample mean.

So a statistic is an estimate of a parameter, and it changes with each sample.

Sampling Methods #

Chosen to avoid bias, not for convenience.

A sampling method is chosen to avoid bias rather than to be convenient

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In a simple random sample, every member has the same probability of being chosen, and the selections are independent.

In a stratified sample, split the population into groups, then take from each group in proportion to its size.

In a systematic sample, take every tenth name on the list. It is fast, but it relies on the order of the list being fair.

A convenience sample takes whoever is easy to reach. At a gym, the venue decides who can be sampled.

Ask every member instead and it is a census — complete, and often impractical.

In a quota sample, fill a fixed count from each group, choosing freely. It is stratified sampling without the randomness.

Observational Studies and Experiments #

Assigning the treatment is what buys a cause.

Only a study that assigns the treatment at random can support a claim about cause

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In an experiment the researcher assigns the treatment — here by a coin toss.

In an observational study you only record groups people put themselves into.

Self-chosen groups can differ before the treatment, and that alone can explain a gap.

Assigning at random spreads every other difference evenly over the two groups.

The control group is treated identically except for the treatment being tested.

Random assignment supports a cause; observing without it supports only an association.

Random sampling settles who a result covers; random assignment settles whether the result is a cause.

Capture and Recapture #

The tagged share of a catch mirrors the pond.

Matching the tagged fraction of a catch to the pond estimates the whole population

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Tag 20 fish and release them — they scatter through a pond of unknown size.

Recapture 30 fish: 6 carry tags — a fifth of the catch is tagged.

If the catch is representative of the pond, the fractions match — solving gives about 100 fish.

The estimate rests on assumptions: tagged fish mix fully, and every tag stays on.

The Sampling Distribution #

Every possible sample mean, piled up.

Take many samples and their means form a distribution of their own

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The population is ten scores with a mean of 5. One sample of four gives a mean of 6.

Take two more samples of four from the same ten scores, and each one has a different mean.

Keep only the means and they pile up between 4 and 6, where the scores ran 2 to 8.

Mean and Variance of X-bar #

Unbiased, with variance divided by n.

The sample mean is right on average and its variance is divided by the sample size

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On average the sample mean equals the population mean, so it is unbiased.

Independent variances add to $n{\sigma }^2$nσ², and dividing by n divides variance by $n^2$n².

So every extra observation narrows the estimate a little further.

Standard Error #

Halve it and you need four times the data.

The standard error is the standard deviation of the sample mean, so it falls with the root of n

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Take the square root of the variance and the n comes out as a root: $\frac{\sigma }{\sqrt{n}}$σ/√n.

The standard error falls fast at small n and then very slowly as n grows larger.

To halve the standard error you need four times as many readings.

The Central Limit Theorem #

Means go normal whatever the population was.

Sample means become normal as the sample grows, whatever shape the population had

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This population is lopsided and uneven, nothing like a bell.

Take means of samples from it and they pile up symmetrically anyway.

That is the central limit theorem, and it is why the normal curve appears so often.

Applying the Theorem #

Thirty is a rule of thumb, not a sharp edge.

A sample of about thirty is usually enough for the normal approximation to be usable

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Once the sample means are close enough to normal, every method for the normal distribution applies to them.

Thirty is a rule of thumb, not a sharp cutoff.

A badly lopsided population needs more than thirty, not exactly thirty.

Approximating a Binomial #

Match the mean and variance to a normal.

A binomial with enough trials is close enough to a normal to be treated as one

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Binomial bars already look bell-shaped once n is reasonably large.

Give the curve the same mean np and variance np(1 − p) as the bars, and it sits on top of them.

The fit needs room in both tails: np and n(1 − p) must both be greater than 5.

Continuity Correction #

Bars have width; a curve does not.

Swapping bars for a curve needs half a unit of width added back at each edge

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A binomial bar for X = 7 really covers everything from 6.5 to 7.5.

The curve has no bars, so you must say where the bar edges were.

$P\left(X\ge 7\right)$P(X ≥ 7) keeps the whole bar for 7, so the boundary is its left edge, 6.5.

P(X > 7) drops that bar, so the boundary moves out to its right edge, 7.5.