Biology · Research Design, Data & Scientific Practice
Variation, effect sizes, uncertainty and statistical inference
The assigned records four/six/eight/ten and six/seven/seven/eight both average seven. Their spreads differ. A graph that shows only their means hides this difference, while an uncertainty bar without its definition can suggest the wrong comparison.
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.
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Separate observation spread, effect size and uncertainty of an estimate, and interpret a model-conditional test and interval without turning them into certainty.
An average does not describe all variation
A mean summarises location by adding the observations and dividing by their count. An effect size or contrast states the difference, ratio or other comparison relevant to the question. It needs its scale and denominator. A difference of two centimetres and a twenty-percent relative difference can describe the same pair only when their reference is stated.
The first original record lies one and three units on either side of mean seven; the second lies at six, seven, seven and eight. Their sums are both twenty-eight. For the first record, squared deviations total twenty; for the second they total two. When these records are treated as samples for the stated model, sample variance divides those totals by n - 1, where n is four.
Their sample variances are 20/3 and 2/3; sample standard deviations are the square roots. These describe spread among the observed values. Estimated standard error of a mean instead describes the uncertainty associated with estimating a population mean under a sampling model. For independent, identically distributed observations, the usual estimate is s/sqrt(n), where s is sample standard deviation. Nested observations need a model that respects their dependence.
Equal means can conceal different spreads
Read the diagram in words
Both records total twenty-eight and average seven. Squared deviations total twenty versus two; sample variances are 20/3 versus 2/3. Their standard deviations describe spread; division by sqrt(4) gives the estimated mean standard errors under the stipulated independent model.
| Assigned sample record | Individual observed value | Assigned response units |
|---|---|---|
| More dispersed assigned record | 1 | 4 |
| More dispersed assigned record | 2 | 6 |
| More dispersed assigned record | 3 | 8 |
| More dispersed assigned record | 4 | 10 |
| Less dispersed assigned record | 1 | 6 |
| Less dispersed assigned record | 2 | 7 |
| Less dispersed assigned record | 3 | 7 |
| Less dispersed assigned record | 4 | 8 |
| Assigned sample record | Number of independent observations in the stipulated model | Sample mean | Sum of squared deviations from the mean | Sample variance with divisor n minus one | Sample standard deviation | Estimated standard error of the mean |
|---|---|---|---|---|---|---|
| More dispersed assigned record | 4 | 7 | 20 | 6.666666666666667 | 2.581988897471611 | 1.2909944487358056 |
| Less dispersed assigned record | 4 | 7 | 2 | 0.6666666666666666 | 0.816496580927726 | 0.408248290463863 |
Two invented four-value samples. The variance divisor is n-1; estimated mean standard errors use the stated independent, identically distributed sampling assumption. No independence is inferred merely from four rows, and no empirical treatment effect is supplied.
An interval depends on the design and model
For the first four-value sample, mean is seven and estimated standard error is sqrt(5/3), about 1.291. The original interval example stipulates independent draws from one normal population with unknown variance. A two-sided 95-percent t interval uses three degrees of freedom and a supplied multiplier about 3.18245. It gives seven plus or minus about 4.10852, or approximately 2.89148 to 11.10852.
The confidence level describes repeated coverage by that construction under its assumptions. It does not mean that 95 percent of individual plant values lie in this interval, and it does not assign a 95-percent probability to a fixed true mean being inside this particular realised frequentist interval. Those are different quantities.
An interval can become narrow around a biased result if the sample excludes relevant units or the measurements share a systematic offset. Correct arithmetic does not check all design assumptions. The four original values and the normal sampling model are supplied for teaching; no actual biological population or precision requirement is established.
An interval estimates a mean under stated assumptions
Read the diagram in words
Mean seven, sample variance 20/3 and standard error sqrt(5/3) combine with multiplier 3.182446305284263. Half-width is approximately 4.10852, giving endpoints approximately 2.89148 and 11.10852. The interval concerns a population mean under the stated normal independent model.
| Assigned response units |
|---|
| 4 |
| 6 |
| 8 |
| 10 |
| Sample mean | Estimated standard error | Degrees of freedom | Supplied two-sided t multiplier | Calculated half-width | Lower endpoint for the population mean | Upper endpoint for the population mean | Nominal repeated-procedure coverage under the model |
|---|---|---|---|---|---|---|---|
| 7 | 1.2909944487358056 | 3 | 3.182446305284263 | 4.108520513521758 | 2.891479486478242 | 11.108520513521757 | 0.95 |
n = 4; s² = 6.666666666666667
mean ± t(0.975,3) × s/sqrt(n)
Original sample 4/6/8/10, stipulated independent identically distributed draws from a normal population with unknown variance. Classical two-sided t interval for the population mean, not a prediction or tolerance interval. No actual biological population or study-size recommendation.
A tail proportion is conditional on a reference model
A statistical test compares a chosen statistic with a reference distribution defined under a test hypothesis and other assumptions. A p-value is the probability, under that reference, of a statistic at least as extreme as the observed one according to the specified rule. It is not the probability that the hypothesis is true or that the observed difference arose from chance alone.
The original discrete reference has ten equally likely outcomes. With extremeness defined as absolute statistic size and observed statistic one, four reference outcomes have absolute size at least one. The conditional tail proportion is 4/10 = 0.4. Changing the statistic, the extremeness rule or the reference distribution would change the question.
A large p-value does not prove equality; a small one does not identify a biological mechanism or measure practical importance. Interpretation also depends on measurement quality, the relevant effect size, the design and how analyses were selected. A result chosen from many explored comparisons cannot silently borrow the interpretation of a single prespecified test.
The tail count is conditional on its reference
Read the diagram in words
The ten equally likely outcomes include -2, -1, -0.5, -0.25, zero, zero, 0.25, 0.5, one and two. Four meet absolute value at least one, so the inclusive reference tail probability is 0.4. Both zeros remain separate elementary outcomes, each with probability 0.1.
| Outcome under the stipulated reference | Assigned test statistic | Probability of an elementary reference outcome | Absolute value at least one |
|---|---|---|---|
| Reference outcome 1 | -2 | 0.1 | ✓ |
| Reference outcome 2 | -1 | 0.1 | ✓ |
| Reference outcome 3 | -0.5 | 0.1 | × |
| Reference outcome 4 | -0.25 | 0.1 | × |
| Reference outcome 5 | 0 | 0.1 | × |
| Reference outcome 6 | 0 | 0.1 | × |
| Reference outcome 7 | 0.25 | 0.1 | × |
| Reference outcome 8 | 0.5 | 0.1 | × |
| Reference outcome 9 | 1 | 0.1 | ✓ |
| Reference outcome 10 | 2 | 0.1 | ✓ |
| Observed statistic supplied in the model | Absolute value at least one | Outcome under the stipulated reference | Inclusive tail probability |
|---|---|---|---|
| 1 | 4 | 10 | 0.4 |
An explicitly invented finite reference distribution under a stipulated test hypothesis and other assumptions; ten elementary outcomes equally likely, including two separate zero outcomes. Extremeness is absolute statistic size. This is not a permutation design, source dataset or biological effect estimate.
Worked example
Explain what the uncertainty numbers refer to
For four/six/eight/ten, subtract seven to obtain -3, -1, 1 and 3. Squaring and adding gives 20; divide by three for sample variance 20/3. The sample standard deviation is about 2.582, and its standard error for an independent mean is about 1.291.
The 95-percent t interval uses the stipulated multiplier 3.182446305284263. Multiplying it by sqrt(5/3) gives margin 4.108520513521759; seven minus and plus this margin gives 2.891479486478241 and 11.108520513521759. These endpoints estimate a mean under the specified model, rather than predict an individual plant outcome.
For the separate discrete test, the outcomes -2, -1, 1 and 2 meet the absolute-size-at-least-one rule. Six others do not. Four of ten gives 0.4. No probability for the truth of the test hypothesis has been supplied by this calculation.
Check your understanding
Why can two samples with mean seven have different standard deviations?
Show reasoning
The mean fixes their sum, not their spread. Four/six/eight/ten has squared-deviation total twenty; six/seven/seven/eight has total two. Their sample variances use the same divisor three but differ tenfold.
Does a 95-percent confidence interval contain 95 percent of the population's individual outcomes?
Show reasoning
No. This construction concerns uncertainty in a population mean and repeated coverage under the specified sampling model. An interval for individual outcomes answers a different question.
What does p = 0.4 mean in the ten-outcome reference?
Show reasoning
Four of the ten equally likely reference outcomes are at least as extreme as observed statistic one under the absolute-size rule. It is conditional on that reference model, not a probability that the hypothesis is true.
What to carry forward
Give the comparison, spread or uncertainty measure its name and assumptions; one number cannot establish effect size, cause and certainty at once.
References
Resources supporting this lesson’s explanations and examples.
- e-Handbook of Statistical Methods (Selected sections; web edition inspected 13 October 2026)NIST/SEMATECHStatistical handbook sections
- Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations (European Journal of Epidemiology 31:337–350, 2016; doi:10.1007/s10654-016-0149-3)Sander Greenland, Stephen J. Senn, Kenneth J. Rothman, John B. Carlin, Charles Poole, Steven N. Goodman and Douglas G. AltmanOriginal statistical interpretation paper · PDF