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Bias in statistics is a systematic error: a deviation from the truth that points in a consistent direction instead of scattering randomly. The phrase covers two related ideas, the bias of a number that estimates a parameter, and bias built into how a study is designed, sampled, measured or reported. This guide separates the two, then maps the main types of bias in statistics onto four families so you can tell which one you are facing.
Definition: what a type of bias in statistics means
The Cochrane Handbook (chapter 7) defines bias as “a systematic error, or deviation from the truth, in results”, and describes risk of bias as the risk that results will over-estimate or under-estimate the true effect. Every type of bias in statistics fits that wording. What differs is where the systematic error enters.
- Bias of an estimator: a property of a formula. Across all possible samples, the estimator’s average value differs from the parameter it targets.
- Bias in a study: a property of the data-generating process. Who gets sampled, how outcomes are measured, which results get reported, or which other variable drives both exposure and outcome.
Random error is a different thing: it adds scatter that averages out as the sample grows, whereas bias does not. A closely related page is what an unbiased estimator is.
Formula: bias of an estimator
In mathematical statistics (for example Casella and Berger, Statistical Inference, 2nd ed.), the bias of a point estimator W of a parameter θ is the difference between its expected value and θ:
Bias(W) = E(W) − θ
- W is the estimator, a rule that turns a sample into a number (for example the sample mean).
- E(W) is the expected value of W: its average over all samples the sampling process could produce.
- θ is the true parameter value in the population.
An estimator whose bias is zero for every θ is called unbiased. A positive bias means the estimator over-estimates on average; a negative bias means it under-estimates.
Worked example with real numbers
Illustrative example, constructed for this page; the numbers are not from a real study.
Example 1: a biased way of reaching the population
Take a population of ten people whose scores are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The true mean is θ = (1 + 2 + … + 10) / 10 = 55 / 10 = 5.5.
A survey can only reach people who score 5 or more (for example because only those people answer the phone). Everyone it can reach has the scores 5, 6, 7, 8, 9, 10, and the sample mean of that group is (5 + 6 + 7 + 8 + 9 + 10) / 6 = 45 / 6 = 7.5. Sampling more of the reachable group does not move the average away from 7.5, so
Bias = E(W) − θ = 7.5 − 5.5 = +2.0.
A bigger sample does not fix this, because the error comes from who is reachable, not from how many are sampled. This is selection bias; see sampling bias and non-response bias.
Example 2: a biased estimator formula
Estimating a population variance by dividing the sum of squared deviations by n gives E(σ̂²) = ((n − 1) / n) × σ². With n = 5 and a true variance σ² = 4, E(σ̂²) = (4 / 5) × 4 = 3.2, so Bias = 3.2 − 4 = −0.8. Dividing by n − 1 instead gives an expected value of 4, so its bias is zero. The full n versus n − 1 comparison is on the unbiased estimator guide.
The main types of bias in statistics: four families
The grouping below is an organising device used on this page, not an official classification. Cochrane’s own tool lists domains such as random sequence generation, allocation concealment, blinding of participants and personnel, blinding of outcome assessment, incomplete outcome data and selective outcome reporting, and the Catalogue of Bias (Oxford Centre for Evidence-Based Medicine) lists many more named biases, including attrition, ascertainment and availability bias.
| Family | Where it enters | Examples (CASRAI pages) |
|---|---|---|
| Selection | Who ends up in the sample or the analysis | Sampling bias, non-response bias, survivorship bias, collider bias |
| Information / measurement | How exposures or outcomes are recorded | Information bias, measurement bias, recall bias |
| Confounding | A third variable linked to both exposure and outcome | Confounding variable, lurking variable |
| Reporting | Which results reach the literature | Publication bias |
Reporting biases named by Cochrane
The Cochrane Handbook (Table 7.2.a) lists these non-reporting biases: publication bias, time-lag bias, language bias, citation bias, multiple (duplicate) publication bias, location bias, and selective (non-)reporting bias. For how risk of bias is assessed within a single study, see bias in experiments.
Interpretation: reading a type of bias in statistics
| Result | What it means |
|---|---|
| Bias > 0 | The estimate is too high on average (Example 1: +2.0) |
| Bias < 0 | The estimate is too low on average (Example 2: −0.8) |
| Bias = 0 | The estimator is unbiased; any single estimate can still be off through random error |
| Bias unchanged as n grows | The cause is in the design or measurement, not the sample size |
A true bias can only be computed when the true value is known, as in a simulation or a constructed example. In real studies it is assessed through risk-of-bias judgements, not calculated directly.
Assumptions and common mistakes
- Treating a bigger sample as a cure. Increasing n shrinks random error but leaves selection and measurement bias where it was.
- Confusing “biased” with “wrong in this sample”. An unbiased estimator can still give a poor estimate from one sample.
- Merging randomisation and concealment. CONSORT 2010 lists sequence generation and allocation concealment as separate items; concealment concerns keeping the allocation sequence hidden until participants are assigned.
- Confusing blinding with randomisation. Blinding addresses what happens after allocation (performance and detection effects); randomisation addresses who is allocated to which group.
- Searching only indexed journals. The PRISMA 2020 statement (Page et al., BMJ 2021;372:n71) asks reviewers to report every database, register, website and other source they searched or consulted (item 6), and to state when each was last searched or consulted.
How to report bias (APA example)
Report the estimate, the true or reference value, and the difference, with symbols in italics and consistent decimals. For a simulation or constructed example:
The mean of the reachable group (M = 7.50) exceeded the population mean (μ = 5.50), a bias of +2.00.
For real studies, name the bias assessed and the tool or checklist used, rather than giving a single number.
How to calculate bias in Excel, R or Python
Each snippet computes bias as the average of the estimates minus the true value, using the numbers from Example 1.
Excel (estimates in B2:B7, true mean 5.5): =AVERAGE(B2:B7)-5.5 returns 2 when B2:B7 holds 5, 6, 7, 8, 9, 10.
R:
est <- c(5, 6, 7, 8, 9, 10)
mean(est) - 5.5 # 2Python:
from statistics import mean
est = [5, 6, 7, 8, 9, 10]
print(mean(est) - 5.5) # 2.0FAQ
What are the main types of bias in statistics?
Selection, information (measurement), confounding and reporting bias, plus the bias of an estimator, which concerns the formula itself rather than the study.
What is the difference between bias and random error?
Bias is systematic and has a direction. Random error scatters estimates around their expected value and shrinks as the sample grows.
Does a larger sample remove bias?
Not when the bias comes from who is sampled or how data are measured. In Example 1 the sample mean stays at 7.5 however many reachable people are sampled.
What counts as a type of bias in statistics?
Any distinct mechanism that pushes results consistently in one direction. The Catalogue of Bias lists many named variants; this guide groups them into four families instead of repeating the long tail.
Sources
Web sources checked on 2026-10-29.
- Cochrane Handbook for Systematic Reviews of Interventions, chapter 7, Considering bias and conflicts of interest among the included studies: https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-07
- Catalogue of Bias, Oxford Centre for Evidence-Based Medicine: https://catalogofbias.org/biases/
- Page, M. J., McKenzie, J. E., Bossuyt, P. M., et al. The PRISMA 2020 statement: an updated guideline for reporting systematic reviews. BMJ. 2021;372:n71: https://pmc.ncbi.nlm.nih.gov/articles/PMC8005924/
- Schulz, K. F., Altman, D. G., Moher, D. CONSORT 2010 Statement: updated guidelines for reporting parallel group randomised trials. BMJ. 2010;340:c332: https://pmc.ncbi.nlm.nih.gov/articles/PMC2844940/
- Casella, G. and Berger, R. L., Statistical Inference, 2nd ed., Duxbury, 2002 (definition of the bias of a point estimator).
