One plant adds more grams while another multiplies its starting mass more quickly. Both statements can be correct. Before calling one plant the faster grower, decide whether you mean absolute mass gain or growth relative to existing mass.
Relative growth rate expresses proportional biomass change over time. This guide compares unequal starting masses, explains the logarithmic calculation, and shows why averaging plants before taking logarithms can change the answer. All data are invented teaching examples; they are not results from a named plant trial.
THE GOLD STANDARD OF BOTANY
Choose the question you need to answer
Define and calculate growth · Chapters 1–3. Separate grams, mass ratios and logarithmic rates.
Respect the measurement design · Chapters 4–5. Handle harvest samples and the limits of normalisation.
Practise and continue · Chapters 6–8. Try changed masses and follow the related reading.
Open the chapter contents · Try the transfer checks · The Gold Standard of Botany
Chapter contents
Define and calculate growth
1. Choose the biological quantity before ranking plants
2. Derive the interval relative rate from a mass ratio
3. Distinguish a logarithmic rate from a percentage increase
Respect the measurement design
4. Keep destructive harvests separate from individual tracking
Practise and continue
6. Test the comparison when the starting point changes
A plant beginning at 10 g and reaching 15 g adds 5 g. Another beginning at 2 g and reaching 4 g adds only 2 g, but doubles. A nursery interested in harvestable output and a researcher interested in proportional growth may therefore ask different questions of the same measurements.
Absolute growth rate over an interval is (final mass − initial mass) ÷ elapsed time. Its units might be grams per day. A relative rate describes change in relation to the mass already present and is commonly expressed per day.
Neither quantity automatically measures health, suitability or treatment benefit. A plant can allocate more biomass to roots while showing little height gain. A taller plant need not have a larger dry mass. Define the plant part and mass basis consistently.
For dry-biomass investigations, drying and weighing generally involve destructive harvests, so initial and final samples usually contain different individuals. For now, treat each numerical pair as an idealised mass trajectory. Chapter 4 returns to what actual harvesting permits you to claim.
If mass follows W(t) = W₀ × exp(r × t) with a constant proportional rate r, dividing final mass W₂ by initial mass W₁ gives W₂/W₁ = exp(r × Δt). Taking the natural logarithm gives r = ln(W₂/W₁) ÷ Δt, equivalently [ln W₂ − ln W₁] ÷ Δt.
The ratio inside the logarithm is dimensionless when both masses use the same unit. Time supplies the rate unit. If time is in days, r has units day⁻¹. Use natural logarithms, usually labelled ln on a calculator; a base-10 logarithm without conversion produces another numerical scale.
For 2 g becoming 4 g over ten days, r = ln(2) ÷ 10 ≈ 0.0693 day⁻¹. For 10 g becoming 15 g, r = ln(1.5) ÷ 10 ≈ 0.0405 day⁻¹. The smaller starting plant has the larger interval relative rate, although its absolute gain is smaller.
An endpoint log ratio remains a defined interval summary for positive masses even when the rate changes during the interval. Interpreting it as a constant process rate or using it to forecast future mass adds assumptions that two endpoints cannot verify.
| Idealised trajectory | Absolute rate over 10 days | Interval relative rate |
|---|---|---|
| 2 g → 4 g | 0.20 g/day | ln(2)/10 = 0.0693 day⁻¹ |
| 10 g → 15 g | 0.50 g/day | ln(1.5)/10 = 0.0405 day⁻¹ |
The first trajectory increases by 100% over ten days; the second increases by 50%. Dividing those percentages by ten gives simple changes per starting mass per day, but those are not the same calculation as the logarithmic interval rate.
For a constant r of 0.0693 day⁻¹, the one-day multiplication factor is e⁰·⁰⁶⁹³, about 1.0718. That corresponds to about 7.18% discrete growth in one day. Calling r exactly 6.93% ordinary daily growth mixes continuous and discrete descriptions.
Over short intervals and for small changes, the numerical difference may be modest. It becomes more visible over longer intervals or larger growth factors. State which definition you use rather than deciding that one calculator answer must be an error.
A calculated doubling time ln(2)/r is a conditional projection under constant exponential growth. It does not promise that a plant will continue doubling. Changes in light, crowding, development and resource availability can alter the trajectory.
You cannot dry a plant to measure its initial dry mass and then continue growing that same plant. A destructive-harvest design instead allocates comparable plants to harvest dates. Its inference concerns the sampled population under the design, not a directly observed before-and-after dry mass for each individual.
Hoffmann and Poorter examine calculation bias in classical growth analysis. Their result makes an important distinction: averaging logarithms of individual masses is not generally interchangeable with taking the logarithm of an arithmetic mean mass.
For an illustrative first harvest, let individual masses be 1 g and 9 g. For the second, let masses be 5 g and 5 g. The arithmetic mean is 5 g at both harvests, so a log ratio of the two arithmetic means is zero. Yet the mean log mass changes from [ln(1) + ln(9)]/2 = ln(3) to ln(5). Over ten days that difference gives ln(5/3)/10 ≈ 0.0511 day⁻¹.
These calculations describe different summaries. They do not show that either plant at the first harvest became either plant at the second. For an actual study, choose the estimator and sampling model before analysis, retain individual measurements, and report uncertainty. The paper provides the methodological treatment; this small example only exposes why the order of operations matters.
Relative growth rate changes the comparison scale. It does not automatically make different starting populations biologically equivalent. Plants of different sizes may also differ in developmental stage, leaf area allocation or growing conditions. A rate per unit mass is not a substitute for a suitable experimental comparison.
Keep fresh and dry mass separate. A fresh-mass change includes water as well as other material. If water status changes, fresh mass and dry biomass need not move together. A measurement record should name the quantity rather than use the vague label weight.
Zero and negative masses do not fit the logarithmic formula. Do not add an arbitrary small number merely to obtain a calculator result. A dead plant, a value below reliable measurement and an empty pot are different observations requiring an explicitly defined outcome and analysis.
Two plants are also not an adequate basis for a precise general claim about a species or growing method. Replication, sampling and uncertainty remain necessary. Our arithmetic can be exact while its biological generalisation remains deliberately limited.
Transfer check: equal multiplication, unequal grams
Plants grow from 3 to 6 g and from 12 to 24 g over ten days. Their absolute rates are 0.3 and 1.2 g/day. Their interval relative rates are both ln(2)/10. Explain why equal proportional growth produces unequal gram gains.
Transfer check: equal gain, unequal relative rates
Plants grow from 2 to 4 g and from 10 to 12 g in ten days. Both add 2 g. The relative rates are approximately 0.0693 and 0.0182 day⁻¹. A claim about more grams and a claim about a larger proportional rate now produce different comparisons.
Transfer check: reverse the interval
An idealised mass falls from 4 to 2 g over ten days. The log interval rate is ln(0.5)/10 ≈ −0.0693 day⁻¹. The sign reports a proportional decline in the defined mass quantity. It does not identify the mechanism responsible for that decline.
Practical evidence check
Ask a partner to calculate both rates from an unfamiliar pair of positive masses, state the units, and say whether the values represent repeated individuals or separate harvest samples. Then change grams to milligrams consistently. The relative rate should stay the same because the ratio is unchanged; the absolute rate’s numerical value and unit change together.
Is relative growth rate always the best way to compare plants?
No. Choose the summary that answers the biological or production question. Report absolute gain alongside a relative measure when both meanings matter.
Does a bigger final plant necessarily have a higher relative rate?
No. It may simply have started larger. Compare initial and final values across the same elapsed time.
Can height replace mass in the formula?
You can define a logarithmic height-change summary for positive heights, but it measures a different quantity. Do not label it dry-biomass relative growth rate or assume plant shape stays constant.
Can separate harvest groups be treated as paired plants?
No. Preserve their sampling design. Arbitrary pairing creates individual trajectories that were never observed.
Why retain every mass if I already have the average?
Individual values permit checking variation and calculating the intended transformed summary. An arithmetic mean alone cannot reconstruct a mean logarithm.
A useful report gives the organism and sampled part, fresh or dry mass basis, harvest times, experimental units, individual values, chosen summary and uncertainty. Its conclusion names the result precisely: greater absolute gain, greater interval relative rate, or a difference that remains unresolved.
Use the connected guides for botanical mechanisms, fair growing comparisons and the distinction between a time course and an endpoint. The germination guide answers a separate question about the timing of an event; mass growth needs its own defined quantity and comparison scale.
Subject and learning map
Education: learning that transfers into life
How X Works: connected subject hub
Plant mechanisms and fair comparisons
Botany: plant structure, growth and function
Plants, soil and fair growing experiments
Germination timing: preserve the denominator and observation window
Measurement and biological evidence
Biology: time courses, endpoints and what change means
Biology: what physical pooling preserves and erases
Baseline balance: compare starting conditions
Missing outcomes: distinguish unavailable measurements from valid results
Authoritative sources
Sources checked on 21 October 2026. Numerical cases and practice tasks are original teaching examples, not observations from a published experiment or official statistics.
Hoffmann and Poorter: Avoiding Bias in Calculations of Relative Growth Rate — primary growth-analysis research, Annals of Botany 90 (2002), 37–42; discusses destructive harvests and the order of averaging and logarithmic transformation.