MEDD8815 Chap.3 Producing Data, Probability and Random Variables
Producing Data, Probability and Random Variables
Where data come from
The third session changes the question from how to describe data to how data should be produced. Anecdotes are vivid and unrepresentative, available data were collected for other purposes, and only designed studies, sample surveys and experiments, are built to answer a stated question.
The key distinction is between observing and intervening: an observational study can describe an association, while an experiment that imposes treatments can support a claim about cause.
Designing experiments
The chapter sets out the language of experiments, units, subjects, factors and treatments, and the three principles that make a comparison trustworthy: control, randomisation and replication.
It covers completely randomised designs, matched pairs and block designs, and the protection that double-blinding offers. A worked example counts the factors, treatments and responses of a two-factor design before any data exist, which is exactly the skill the session's homework opens with.
Designing samples
Samples chosen for convenience or by voluntary response are biased.
A simple random sample gives every possible set of the chosen size an equal chance, and a stratified sample does this separately within groups of similar individuals. The chapter then separates parameters, which describe populations, from statistics, which describe samples, and introduces the sampling distribution.
Random sampling removes bias; larger samples reduce variability; and the size of the population hardly matters once it is at least a hundred times the sample. The handout's Excel procedure for drawing a random sample is included.
Randomness and random variables
Probability is defined as a long-run proportion.
A random variable assigns a number to each outcome of a chance process, discrete when it counts and continuous when it measures. The law of large numbers says the mean of many independent observations settles near the population mean, and the chapter shows it with a computed simulation that echoes the session's SPSS laboratory on thousands of simulated sample means.
What this chapter covers
- 01
Anecdotal, available and designed data
- 02
Observational studies against experiments
- 03
Factors, treatments and experimental units
- 04
Control, randomisation and replication
- 05
Matched pairs and block designs
- 06
Simple random and stratified samples
- 07
Bias, variability and sampling distributions
- 08
Random variables and the law of large numbers
Worked example · free
Allocate treatments at random, step by step
- 2Label the pupils and use chance to pick the first group.
- 1Name the design and its comparison.
- 2Describe the change that turns it into a block design, and why it might help.
Key terms
- Observational Study
- A study that records variables without attempting to influence the responses, so it can describe associations but not by itself establish cause.
- Experiment
- A study that deliberately imposes treatments on units in order to observe their responses, the only fully convincing source of evidence about cause.
- Treatment
- A specific experimental condition applied to units; when there are several factors, a treatment is a combination of their levels.
- Simple Random Sample
- A sample of size n chosen so that every set of n individuals in the population has an equal chance of being the sample.
- Stratified Random Sample
- A sample formed by dividing the population into groups of similar individuals and taking a separate simple random sample within each.
- Parameter
- A number describing a population, usually unknown, such as a population mean or proportion.
- Sampling Distribution
- The pattern of values a statistic would take across every possible sample of a given size drawn from one population.
- Random Variable
- A variable whose value is a numerical outcome of a chance process; discrete when it counts, continuous when it measures.
Producing Data, Probability and Random Variables FAQ
Why is random assignment better than letting people choose their group?
People who choose a treatment tend to differ from those who do not, in motivation, background or prior attainment, and those differences become confounded with the treatment. Random assignment spreads such differences evenly by chance, so a large difference in responses can be credited to the treatment.
Does a bigger population need a bigger sample?
Hardly at all. The spread of a statistic depends on the sample size and design, not on the population size, provided the population is at least a hundred times as large as the sample. A well-drawn sample of a thousand describes a district and a country almost equally well.
What is the difference between a parameter and a statistic?
A parameter is a fixed but usually unknown number describing a whole population. A statistic is computed from a sample and changes from one sample to the next; it is used to estimate the parameter, and its sampling distribution describes how much it varies.
Assessment move
Collect three education claims from news or reports and, for each, write two lines: was the treatment imposed or merely observed, and how were the cases chosen? Then state what the study can and cannot conclude. This session's quiz and homework are about exactly that judgement, and practising it on real claims is faster than rereading definitions.
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