S2 Chapter 6: Sampling and Sampling distribution

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A population is: the full collection of people or things.

A sample is: some subset of the population intended to represent the population.

Population

Sample

Data obtained from all members of the population is known as a census.

Advantages of sampling

• Cheaper/quicker than taking a census.
• Useful when testing of items results in their destruction (e.g. life-time of light bulb)

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Disadvantages of sampling

• Potential for bias.
• Natural variation between any two samples due to variation in data.

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Sample

🖉 Each individual thing in the population that can be sampled is known as a sampling unit.

🖉 The list of all those within the population that can be sampled is known as the sampling frame.

Bro Helping Hand: This might conceptually seem confusing as a population is a list of things. The population can be represented as a distribution where the outcomes are possible samples.

For example, if a population is all possible lottery tickets, then the distribution representing it is a uniform distribution whose outcomes are all the possible tickets.

How could we represent the possible choice of 1^st member of our sample?

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This means for example that if the first person chosen for our sample is Indian, that doesn’t make it any less or more likely our second choice will be Indian, i.e. our second choice is independent of the first.

This will all become a lot clearer once we do an example…

We might wish to calculate some numerical property of a population or a sample, e.g. mean, variance, mode, range.

🖉 A population parameter is a quantity calculated from the population.

🖉 A statistic is a quantity calculated (solely) from the observations in a sample.

🖉 The sampling distribution of a statistic gives all the values of a statistic and the probability that each would happen by chance alone.

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Statistics for this sample could be the mode number of children, median, maximum, mean, …

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Possible Samples?

Note: Because each thing in the sample is independently drawn from the population, we technically have sampling with replacement, and hence the same item could be in the sample twice. In practice however (and in exams) you won’t have to worry about this, as the population in exams is assumed to be infinitely large.

Sampling distribution for sample maximum.

Suppose we had 10 families which form the population of an island (The Isle of Bob), for which we know the number of children in each family. Suppose we took a (very small!) sample of 2 families.

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Let’s reflect on what we did.

#1: We considered all possible samples, and the probability of each sample occurring.

#2: We’re interested in some statistic for each sample (let’s say the sample maximum)

#3: Thus we now have a distribution over possible values of the statistics across all possible samples we could have had, i.e. the ‘sampling distribution’.

Edexcel S2 May 2013 Q1

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Key Points:

a) Ensure you don’t forget possibilities through other possible orderings, etc.

b) If we know all the possible values of the statistic, we can find the probability of the last by just subtracting from 1 (as it’s a probability distribution!)

Edexcel S2 June 2007 Q4

Step 1: List possible samples (and statistic for each if possible).

Step 2: Use this to work out the probability of obtaining each value of the statistic.

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Sometimes it is not practical to list out all the possible samples, but we can tell what the sampling distribution is by thinking about what the statistic represents.

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Bro Note: The mark schemes likes the idea of a ‘list’ and the idea that things in the sampling frame can be clearly identified.

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Continue onto Exercise 6C if you’re done.

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