Biology · Research Design, Data & Scientific Practice
Sampling, independent replication and avoiding pseudoreplication
An imaginary population contains eight plants with assigned values two, two, four, four, six, six, eight and eight. Its complete mean is five. Selecting only the four high-valued plants gives seven, even though every selected value is measured perfectly.
Life, Evidence & Biological Scale introduces evidence and biological organisation. Organisms, Environments & Ecological Niches and Population Ecology & Life-History Patterns develop the importance of boundaries and denominators. Their English chapters are available in this working manuscript; their publication remains pending. The examples here require arithmetic, averages and ordinary graph reading. The sample-variance and confidence-interval example explains its notation as it is introduced; it can be revisited after the simpler comparisons.
Back to the Biology mapBy the end, you can
Distinguish target population, sampling frame, allocation unit and measurement unit, and identify replication at the level of the question.
Coverage comes before a sample average
The target population is the set about which the question seeks information. The sampling frame is the list or area system used to reach eligible members. If the frame omits part of the target, a random selection from that frame still cannot include the omitted part. A large sample does not by itself repair this coverage difference.
The original finite example supplies all eight values so that its true mean can be checked. A convenience frame containing only six, six, eight and eight has mean seven, two above the full-population mean five. More repeated readings of those same four plants would not reveal the omitted low-valued plants. This is selection, not measurement noise.
Probability sampling specifies calculable inclusion probabilities under a selection design. They need not all be equal in every design. The simple random two-plant example here gives each distinct unordered pair the same probability, without replacement. It leaves sampling variation: different selected pairs can have means below, equal to or above five.
A selection rule changes which population is seen
Read the diagram in words
All eight values total forty and average five. The restricted six/six/eight/eight frame averages seven. Twenty-eight distinct unordered pairs are equally likely in the complete-frame design; their means span two to eight and average five.
| Complete assigned population | Assigned plant value |
|---|---|
| Plant 1 | 2 |
| Plant 2 | 2 |
| Plant 3 | 4 |
| Plant 4 | 4 |
| Plant 5 | 6 |
| Plant 6 | 6 |
| Plant 7 | 8 |
| Plant 8 | 8 |
| Restricted high-value sampling frame | Assigned plant value |
|---|---|
| Plant 5 | 6 |
| Plant 6 | 6 |
| Plant 7 | 8 |
| Plant 8 | 8 |
| Complete assigned population | Mean of one unordered two-plant subset | Selection probability for each subset |
|---|---|---|
| Plant 1 / Plant 2 | 2 | 0.03571428571428571 |
| Plant 1 / Plant 3 | 3 | 0.03571428571428571 |
| Plant 1 / Plant 4 | 3 | 0.03571428571428571 |
| Plant 1 / Plant 5 | 4 | 0.03571428571428571 |
| Plant 1 / Plant 6 | 4 | 0.03571428571428571 |
| Plant 1 / Plant 7 | 5 | 0.03571428571428571 |
| Plant 1 / Plant 8 | 5 | 0.03571428571428571 |
| Plant 2 / Plant 3 | 3 | 0.03571428571428571 |
| Plant 2 / Plant 4 | 3 | 0.03571428571428571 |
| Plant 2 / Plant 5 | 4 | 0.03571428571428571 |
| Plant 2 / Plant 6 | 4 | 0.03571428571428571 |
| Plant 2 / Plant 7 | 5 | 0.03571428571428571 |
| Plant 2 / Plant 8 | 5 | 0.03571428571428571 |
| Plant 3 / Plant 4 | 4 | 0.03571428571428571 |
| Plant 3 / Plant 5 | 5 | 0.03571428571428571 |
| Plant 3 / Plant 6 | 5 | 0.03571428571428571 |
| Plant 3 / Plant 7 | 6 | 0.03571428571428571 |
| Plant 3 / Plant 8 | 6 | 0.03571428571428571 |
| Plant 4 / Plant 5 | 5 | 0.03571428571428571 |
| Plant 4 / Plant 6 | 5 | 0.03571428571428571 |
| Plant 4 / Plant 7 | 6 | 0.03571428571428571 |
| Plant 4 / Plant 8 | 6 | 0.03571428571428571 |
| Plant 5 / Plant 6 | 6 | 0.03571428571428571 |
| Plant 5 / Plant 7 | 7 | 0.03571428571428571 |
| Plant 5 / Plant 8 | 7 | 0.03571428571428571 |
| Plant 6 / Plant 7 | 7 | 0.03571428571428571 |
| Plant 6 / Plant 8 | 7 | 0.03571428571428571 |
| Plant 7 / Plant 8 | 8 | 0.03571428571428571 |
| Mean of all eight plants | Mean among the four reachable plants | Average over all equally likely subset means |
|---|---|---|
| 5 | 7 | 5 |
28 × 1/28 = 1; 7 − 5 = 2
Eight entirely invented finite plant values, with complete population information supplied for teaching. Simple random sampling is without replacement; each unordered pair has probability 1/28. The complete enumeration is not an empirical sample or biological sampling recommendation.
Identify where the condition was independently assigned
The biological unit of interest is what the conclusion concerns. The experimental unit is the entity assigned a condition under the design. The observational unit is what is measured. These may coincide, or measurements may be nested inside larger assigned units. The hierarchy matters when choosing an analysis and describing how much independent evidence was collected.
In the original tray design, four trays are independently allocated, two per condition, and the condition is applied at tray level. Each contains three measured plants. Their twelve outcomes retain four tray identities. Treating the twelve plant measurements as twelve independent assignments would ignore the shared tray condition.
Additional measurements inside a tray can describe within-tray variation and help measure that tray more carefully. They do not create additional independently assigned trays. Whether units influence each other, share an application or are allocated together must be considered explicitly. Merely naming a measurement biological rather than technical does not settle independence.
Twelve measurements retain four assignment units
Read the diagram in words
Tray means are four, six, eight and ten. Condition A has two trays and mean five; condition B has two trays and mean nine. The observed difference is four. Twelve plant rows belong to four independently allocated tray units.
| Independently assigned tray | Condition assigned to the tray | Plant measured inside its tray | Assigned response units |
|---|---|---|---|
| Tray one | Condition A | 1 | 3 |
| Tray one | Condition A | 2 | 4 |
| Tray one | Condition A | 3 | 5 |
| Tray two | Condition A | 1 | 5 |
| Tray two | Condition A | 2 | 6 |
| Tray two | Condition A | 3 | 7 |
| Tray three | Condition B | 1 | 7 |
| Tray three | Condition B | 2 | 8 |
| Tray three | Condition B | 3 | 9 |
| Tray four | Condition B | 1 | 9 |
| Tray four | Condition B | 2 | 10 |
| Tray four | Condition B | 3 | 11 |
| Independently assigned tray | Condition assigned to the tray | Mean of the three plant observations |
|---|---|---|
| Tray one | Condition A | 4 |
| Tray two | Condition A | 6 |
| Tray three | Condition B | 8 |
| Tray four | Condition B | 10 |
| Condition assigned to the tray | Number of independent allocation units | Number of observed plants | Mean of the two tray means |
|---|---|---|---|
| Condition A | 2 | 6 | 5 |
| Condition B | 2 | 6 | 9 |
| Number of independent allocation units | Number of observed plants | B minus A mean tray outcome |
|---|---|---|
| 4 | 12 | 4 |
Original four-tray teaching design: each tray independently assigned and receiving its condition at tray level; three nested plants measured once per tray. The model stipulates no cross-tray interference and supplies no real intervention, significance result or animal procedure.
Do not trade one kind of repetition for another
Sampling variation comes from which units the selection design includes. Measurement variation comes from how a quantity is recorded. Variation among independently assigned units and variation among repeated readings within one unit are also distinct. A useful report preserves the relevant identity at every stage rather than treating each table row as a new replicate.
For the eight-plant population there are twenty-eight unordered two-plant subsets. Their means range from two to eight. Averaging the means of all equally likely subsets gives five. This property is about the specified selection distribution, not a promise that one selected pair equals the population mean.
A cluster design or repeated-measures design may legitimately collect many related observations, provided its design and analysis account for the relations. The problem is pretending they are independent when the question and allocation do not support that assumption. No universal count of plants, cells or time points can replace a description of how the data were obtained.
Worked example
Count units before counting rows
The original tray outcomes are 3/4/5 and 5/6/7 in condition A, then 7/8/9 and 9/10/11 in B. Their tray means are four, six, eight and ten. There are four independently allocated trays and twelve plant measurements.
Condition A has mean tray outcome five and B has nine, giving an observed difference four. Both groups contain two assigned trays. The four-unit allocation, rather than the twelve raw rows, is the starting point for evaluating the condition comparison in this stipulated design.
For the separate finite sampling example, the pair of lowest-valued plants has mean two and the highest pair has mean eight. Both are possible simple random samples. The complete distribution's expected sample mean five does not remove the uncertainty of a particular two-plant selection.
Check your understanding
Why can a random sample from an incomplete frame remain poorly suited to the target population?
Show reasoning
The selection cannot include units absent from that frame. Random choice among reachable units does not supply information about the omitted target population.
Do three plants in each of four independently assigned trays provide twelve independently allocated trays?
Show reasoning
No. They provide twelve measured plants nested in four trays. The condition is assigned at tray level, so those four tray identities must remain in the comparison.
Does the expected two-plant sample mean five mean every selected pair averages five?
Show reasoning
No. The twenty-eight equally likely pair means vary from two to eight. Their average is five; any one realised pair can differ from it.
What to carry forward
Count the units that the design selects, assigns and measures separately; preserve their relationships before estimating uncertainty.
References
Resources supporting this lesson’s explanations and examples.
- Statistics: Power from Data! (Selected sections; web edition inspected 13 October 2026)Statistics CanadaAgency educational sections
- What exactly is ‘N’ in cell culture and animal experiments? (PLOS Biology 16(4): e2005282, 2018; doi:10.1371/journal.pbio.2005282)Stanley E. Lazic, Charlie J. Clarke-Williams and Marcus R. MunafòOriginal methodological perspective