01 · Lesson 09
Statistics
Probability begins with a model and studies possible data. Statistics begins with observed data and uses them to describe a population, estimate unknown quantities, compare explanations and quantify uncertainty.
Scenario: estimating a recovery time
A study records recovery times from a sample of infected patients. The observed times differ because people differ biologically, measurements are imperfect and the sample contains only part of the wider population. We want to summarise the data and learn about the population without pretending the sample is exact.
Population, sample and observation
| Term | Meaning |
|---|---|
| Target population | The individuals or biological systems about which conclusions are intended |
| Sample | The subset actually observed |
| Variable | The measured characteristic, such as recovery time |
| Observation | One recorded value |
| Parameter | An unknown population quantity |
| Statistic | A quantity calculated from the sample |
A large sample does not repair systematic selection bias. How observations were obtained matters as much as how many were obtained.
Describing the centre
For observations \(x_1,\ldots,x_n\), the sample mean is
The median is the middle ordered value. The mean uses every magnitude but can be strongly affected by extreme observations. The median is resistant to extremes but does not replace the mean for every statistical model.
Describing variation
The sample variance is
and the sample standard deviation is \(s\). Standard deviation has the same units as the data. The range and interquartile range give other descriptions of spread.
Variation among patients is not the same as uncertainty in the estimated mean. The first describes individual spread; the second describes how precisely the population mean has been estimated.
Estimation and standard error
If observations are independent with finite standard deviation, the estimated standard error of the sample mean is
Increasing sample size reduces sampling uncertainty approximately with \(1/\sqrt n\), not with \(1/n\). Four times as many independent observations are therefore required to halve this standard error.
Confidence intervals
A confidence interval combines an estimate with its sampling uncertainty. In a simple large-sample setting, an approximate 95% interval for a mean is
The repeated-sampling interpretation is that the procedure captures the true parameter in about 95% of comparable repeated samples under its assumptions. It is not automatically a 95% probability statement about a fixed parameter after the data have been observed.
Association is not causation
Covariance and correlation measure joint variation. A correlation may arise from causation, reverse causation, confounding, selection or chance. Establishing a biological mechanism requires design and subject knowledge in addition to a numerical association.
Regression can describe how an expected response changes with predictors while controlling for declared variables, but conclusions still depend on model form, measurement quality and sampling design.
Connecting data to a mathematical model
Model parameters may be estimated by making model predictions agree with observations according to a defined objective or likelihood. Validation then asks how predictions perform on information not used for fitting.
- Residual checks examine disagreements between data and fitted predictions.
- Sensitivity analysis examines how results respond to parameter changes.
- Identifiability asks whether the available data can distinguish parameter values.
- Uncertainty propagation examines how uncertain inputs affect model outcomes.
These topics are developed in the Data and Parameter Estimation section.
Conditions and limitations
- Define the target population and sampling process.
- Plot and inspect data before relying on summaries.
- Report sample size, missing values and exclusions.
- Distinguish biological variation, measurement error and sampling uncertainty.
- Check independence and distributional assumptions used by an inferential method.
- A statistically significant difference need not be biologically important.
What this lesson adds
You can now distinguish populations, samples, parameters and statistics; interpret centre and spread; separate standard deviation from standard error; read confidence intervals carefully; and explain how statistical evidence connects data to biological models.