Chapter 10 · Designing the study
How many patients do you need?
What decides sample size, how to calculate it for means and proportions, and how to adjust it.
14 TNMGRMU questions answered hereset 17 times, 2009–2025
The problem first
You are a second-year resident. Your thesis protocol says 30 patients per group, because your senior used 30. The research committee sends it back with one line: “Justify the sample size.”
You keep 30. Two years later the result is “no significant difference”, and at the viva the examiner asks one question.
“Was your study ever able to find a difference?”
Start here
Why not study everyone, or just a handful?
You cannot enrol every patient in the world, so you study a sample. With too few patients, chance alone can decide the result. With too many, you use more patients, money and time than the question needs.
Too few: run the same small trial twice and it can favour a different drug each time.
Too many: the answer was already clear at 126 patients.
What decides the number · 1
How large is the difference you want to detect?
Large differences show up in small studies. Small differences need many more patients, in the same way that a displaced fracture is obvious on an X-ray and a hairline crack is easy to miss.
What decides the number · 2
How different are your patients from each other?
Fit 25-year-olds respond to treatment in similar ways. A mix of ages, smokers and diabetics does not. The more results vary between patients, the harder a real difference is to detect.
What decides the number · 3
How sure do you want to be?
There are two ways to be wrong. You decide in advance how often you will allow each.
With a 1-point change and SD 2: 63 per group at α 5% and 80% power. Raise power to 90%: 85. Tighten α to 1%: 94. Lower error rates need more patients.
A one-sided test at α 5% uses Zα = 1.645 instead of 1.96, and needs 50 per group. The choice must be made and justified before the study. ICH E9 prefers a one-sided α of half the two-sided level, 2.5%, which brings the number back to 63. See the hypothesis testing chapter.
Interactive
Change the numbers and watch the sample size change
Example study: Drug A vs Drug B for pain after surgery, pain score (0–10) at 24 hours. Two-sided false-positive rate (α) 5%.
Smaller change → more patients
More spread → more patients
More lost → recruit more
63 needed + 16 extra for dropouts · 158 in total
Interactive
Change the number of patients per group and watch the power change
Try this: start at 30 patients per group, the number in the problem at the top of this chapter. Move it to 63 and watch the power rise from 49% to 80%.
Example study: Drug A vs Drug B for pain after surgery, pain score (0–10) at 24 hours. Two-sided false-positive rate (α) 5%. The difference you set is assumed to be real.
More patients → more power
Larger difference → more power
More spread → less power
With 30 patients per group, studies like this one would detect a real 1 point difference about 49 times in 100. That is less than a 50% chance of detecting it.
80% power needs 63 per group · 90% power needs 85 per group
Decide
Which formula do you need?
Answer two questions about your study.
What is your main outcome?
Which formula: all possible answers
- Two-means formula: n per group = 2(Zα/2 + Zβ)² × SD² ÷ d². Then adjust for dropouts and clustering.
- One-mean formula: n = (Zα/2 × SD ÷ d)², where d is how close to the true average you need to be.
- Two-proportions formula: Uses both expected percentages p₁ and p₂. Rare outcomes need very large groups.
- Single-proportion formula: n = Z² × p(1 − p) ÷ d². Unsure of p? Use 50%.
- Time-to-event calculation: Survival studies are sized by the number of events you need to observe, not only by headcount. Use dedicated software and involve a statistician. Events needed = 4(Zα/2 + Zβ)² ÷ (ln HR)², then divide by the share of patients expected to have the event.
- Paired formula: n patients = (Zα/2 + Zβ)² × SD of the differences² ÷ d². Usually far fewer than two separate groups.
- Diagnostic accuracy formula: People with the disease = Z² × Se(1 − Se) ÷ d². Divide by the prevalence to get the number to test.
- Correlation formula: n = [(Zα/2 + Zβ) ÷ C]² + 3, where C = ½ ln[(1 + r) ÷ (1 − r)].
- Case–control formula: Uses the expected exposure among controls and the smallest odds ratio worth detecting. More controls per case reduce the cases needed, and each extra control saves fewer cases than the one before.
Comparison
Power or precision: which one is your study about?
Comparing groups is sized by power. Estimating one number is sized by precision. Mixing them up gives the wrong formula.
| Feature | Power | Precision |
|---|---|---|
| Question | Is there a difference between groups? | How big is this one number? |
| Study type | Trials, case-control and cohort comparisons | Prevalence surveys, estimating an average |
| You fix | α, power, smallest difference d, SD or p₁ and p₂ | Confidence level, margin of error d, SD or expected p |
| The answer is | Enough people to detect d if it is real | An estimate within ±d of the truth |
| Example | Drug A vs Drug B on pain score | Anaemia prevalence in a district, ±5 points |
The formula
Comparing two averages
When the outcome is a measurement (pain score, blood pressure, HbA1c), the number per group is:
n per group = 2 × (Zα/2 + Zβ)² × SD² ÷ d²
- Z α/2
- 1.96 for a two-sided α of 5% (2.58 for 1%)
- Z β
- 0.84 for 80% power, 1.28 for 90%
- SD
- standard deviation of the outcome, from an earlier study or a pilot
- d
- smallest difference worth detecting (the MCID)
2 × (1.96 + 0.84)² × 2² ÷ 1² = 62.7 → 63 per group
Always round up. A quick check at α 5% and 80% power: n per group is about 16 × (SD ÷ d)², here 16 × 4 = 64.
Estimating one average
A survey that estimates one mean, with no comparison, is sized by precision. Teaching example: mean haemoglobin in pregnant women, SD 1.4 g/dL, to within ±0.3 g/dL at 95% confidence.
n = (Zα/2 × SD ÷ d)² = (1.96 × 1.4 ÷ 0.3)² = 83.7 → 84 women
The same formula works backwards. If only 30 women can be examined, the margin of error is 1.96 × 1.4 ÷ √30 = 0.50 g/dL. Decide before the study whether that is precise enough.
The formula
Estimating a percentage, or comparing two
Surveys ask "how common is it?" The question becomes: how precise must your answer be?
n = Zα/2² × p × (1 − p) ÷ d²
- p
- expected prevalence, from an earlier survey (as a fraction: 30% = 0.30)
- d
- absolute precision you want, e.g. ±5 percentage points = 0.05
1.96² × 0.30 × 0.70 ÷ 0.05² = 322.7 → 323 people
To compare two percentages (cure rate, infection rate), the formula uses both expected proportions p₁ and p₂:
n per group = [Zα/2√(2pq) + Zβ√(p₁q₁ + p₂q₂)]² ÷ (p₁ − p₂)²
q = 1 − p, and p (p-bar) is the average of p₁ and p₂.
Interactive
Change the two expected percentages and watch the number per group change
Try this: move the standard treatment to 2% and the new treatment to 1%, and watch the number per group rise from 199 to 2,319.
Example study: a complication (yes or no) with a standard treatment and with a new one. You enter the percentage of patients expected to have the complication with each.
Rarer outcome → more patients
Smaller gap between p₁ and p₂ → more patients
398 in total · p̄ = 0.150 · two-sided α 5%, power 80%
Of 199 patients in each group, about 40 with the standard treatment and 20 with the new treatment are expected to have the complication.
Comparison
Is your margin of error absolute or relative?
"±2 percentage points" and "±20% of the prevalence" are different targets. For an uncommon condition, a relative margin needs many more people.
| Feature | Absolute precision | Relative precision |
|---|---|---|
| Margin stated as | A fixed number of percentage points (d) | A fraction ε of the expected prevalence (d = ε × p) |
| Formula | n = Z² × p(1 − p) ÷ d² | n = Z² × (1 − p) ÷ (ε² × p) |
| Expected prevalence 8%, 95% confidence | ±5 points: 114 people | ±20% of 8%, which is ±1.6 points: 1,105 people |
| Suits | Common conditions, where ±5 points is a small share of p | Rare conditions, where ±5 points would be larger than p itself |
Absolute: 1.96² × 0.08 × 0.92 ÷ 0.05² = 113.1 → 114
Relative: 1.96² × 0.92 ÷ (0.20² × 0.08) = 1,104.4 → 1,105
An estimate of 8% ± 5 points could mean anything from 3% to 13%. For a prevalence this low, write the precision in the protocol as a relative margin, or choose a narrower absolute one.
Comparison
Should you analyse the measurement, or turn it into yes or no?
Splitting a measurement at a cut-off throws away information, so the same study needs more patients.
The yes/no version also loses the difference between a patient just below the cut-off and one far below it. Altman and Royston (2006) showed that splitting a continuous variable at its median reduces power by as much as discarding a third of the data. Choose the primary outcome in its measured form when the measurement itself is meaningful, and report the percentage above a cut-off as a secondary result.
The formula
Formulas for other study designs
The same four inputs (α, power, the size of the effect and the variability) appear in every design. What changes is how the effect and the variability are expressed.
1. The same patients before and after (paired design)
n patients = (Zα/2 + Zβ)² × SDdiff² ÷ d²
- SD diff
- SD of the change within each patient, from a pilot or an earlier study
Teaching example: d = 1 point, SD of the change 1.2.
(1.96 + 0.84)² × 1.2² ÷ 1² = 11.3 → 12 patients
Two separate groups with SD 2 need 63 per group (126 patients) for the same 1-point difference. Each patient serves as their own control, so the variation between patients drops out. Designs with several measurements over time are covered in Repeated measures and mixed models.
2. Accuracy of a diagnostic test
With the disease = Z² × Se(1 − Se) ÷ d²
People to test = number with the disease ÷ prevalence
Teaching example: expected sensitivity 90%, specificity 85%, each to within ±5 points, prevalence among the people tested 20%. For sensitivity, 1.96² × 0.90 × 0.10 ÷ 0.05² = 138.3 people with the disease. For specificity (use 1 − prevalence), 1.96² × 0.85 × 0.15 ÷ 0.05² = 195.9 people without it.
Sensitivity: 138.3 ÷ 0.20 = 691.5 → 692 tested
Specificity: 195.9 ÷ 0.80 = 244.9 → 245 tested
Test the larger number, 692. Sensitivity is estimated only from people who have the disease, so a low prevalence raises the number to test (Buderer, 1996). Sensitivity and specificity are explained in the diagnostic tests chapter.
3. Case–control study
Start from the expected exposure among controls (p₀) and the smallest odds ratio worth detecting. The exposure expected among cases is:
p₁ = OR × p₀ ÷ [1 + p₀(OR − 1)]
n cases = [Zα/2√((1 + 1/c) p̄(1 − p̄)) + Zβ√(p₁(1 − p₁) + p₀(1 − p₀)/c)]² ÷ (p₁ − p₀)²
- c
- controls per case
- p̄
- (p₁ + c × p₀) ÷ (1 + c)
This is the Fleiss formula used by the free OpenEpi calculator, written per case. Some textbooks print a simpler Kelsey version, which gives slightly different numbers. State in the protocol which one you used.
4. Cohort study or trial with a risk ratio
Set p₁ = RR × p₀ and use the two-proportion formula above. With a 10% risk among the unexposed and a risk ratio of 2, p₁ = 20%, and the chapter's bar chart already gives 199 per group.
5. Correlation between two measurements
n = [(Zα/2 + Zβ) ÷ C]² + 3
- C
- ½ ln[(1 + r) ÷ (1 − r)], the Fisher transformation of the correlation r you want to detect
Teaching example: r = 0.3, so C = 0.3095, with α 5% and power 80%.
(2.802 ÷ 0.3095)² + 3 = 84.9 → 85 patients
A weaker correlation of r = 0.2 needs 194. Correlation itself is covered in the correlation and regression chapter.
6. Time to an event (survival)
Events needed = 4 × (Zα/2 + Zβ)² ÷ (ln HR)²
Patients = events ÷ share of patients who have the event
Teaching example: hazard ratio 0.75, two equal groups, α 5%, power 80%, and 40% of patients have the event during follow-up.
4 × 2.802² ÷ (ln 0.75)² = 379.4 → 380 events
380 ÷ 0.40 = 950 patients
The power of a survival comparison depends on the number of events, not on the number of patients enrolled (Schoenfeld, 1983). A longer follow-up or a higher-risk population produces the events with fewer patients. Hazard ratios and the models behind them are covered in Cox regression.
7. Non-inferiority trial with a measured outcome
n per group = 2 × (Z1−α + Z1−β)² × SD² ÷ Δ²
- Z 1−α
- 1.96 for a one-sided α of 2.5%
- Δ
- non-inferiority margin
This version assumes the two treatments truly have the same effect. With SD 2 and a margin of 1 point: 63 per group. Halving the margin to 0.5 points gives 252 per group. The margin plays the part of d, so a narrow margin needs a large trial. If the new treatment may be slightly less effective, assuming no difference underestimates the size needed. The formula for a yes/no outcome is in the clinical trials chapter, and testing against a margin is covered in Equivalence and non-inferiority tests.
8. More than two groups
If three groups will each be compared with each other (three comparisons), a simple approach is to divide α by the number of comparisons: 0.05 ÷ 3 = 0.0167, so Zα/2 = 2.394. For SD 2 and d = 1: 2 × (2.394 + 0.842)² × 4 ÷ 1 = 83.8 → 84 per group instead of 63. Other ways of handling several comparisons are covered in Multiple comparisons.
Interactive
Change the hazard ratio and the share of patients with an event, and watch the events and patients needed
Try this: keep the hazard ratio at 0.75 and move the share of patients who have the event from 40% to 20%. The events needed stay at 380, and the number of patients doubles.
Teaching example: a trial of a new treatment against standard care with death or relapse as the outcome, two equal groups, two-sided α 5%.
Closer to 1 → many more events
Fewer events per patient → more patients
To detect a hazard ratio of 0.75 (a 25% lower hazard) with 80% power, the trial must observe 380 events. If 40% of patients have the event during follow-up, that means 950 patients, about 475 per group.
4 × (1.96 + 0.84)² ÷ (ln 0.75)² → 380 · 380 ÷ 0.40 → 950
After the formula
Three adjustments before you finalise
The formula assumes that every patient completes the study, that each patient's result is independent of the others, and that the population is very large. Real studies often do not meet these assumptions.
1. People drop out
Recruit = n ÷ (1 − dropout rate) → 63 ÷ 0.8 = 79 per group
2. People come in clusters
Design effect = 1 + (m − 1) × ICC
- m
- people sampled per cluster
- ICC
- intracluster correlation: how alike people in one cluster are (0 = not at all, 1 = identical)
m = 10, ICC = 0.05 → 1.45 × 323 = 468.4 → 469 people
3. The population is small
n adjusted = n ÷ [1 + (n − 1) ÷ N]
N is the total population. If only 1,000 people exist, the 323 needed falls to 245. When N is very large, the correction changes almost nothing.
4. The SD has to come from somewhere
Take it from a pilot study or a published study in similar patients. Papers often give a 95% confidence interval for a mean instead of the SD. The Cochrane Handbook (section 6.5.2.2) gives the conversion:
SD = √n × (upper limit − lower limit) ÷ 3.92
Teaching example: mean fall in HbA1c 0.8% (95% CI 0.6 to 1.0) in 120 patients.
√120 × 0.4 ÷ 3.92 = 1.12
The handbook uses 3.92 for large groups (say more than 100) and, for small groups (say fewer than 60), replaces it with 2 × the t value for n − 1 degrees of freedom. For 30 patients with a CI of 0.4 to 1.2: t = 2.045, so SD = √30 × 0.8 ÷ 4.09 = 1.07.
Interactive
Change the survey inputs and watch the number of people to invite change
Try this: set people sampled per cluster to 10 (ICC 0.05) and watch 323 become 469. Then move the expected prevalence to 50%.
Example: a survey to estimate the prevalence of anaemia in a district. The default inputs are the chapter's worked example.
p nearer 50% → more people
Narrower margin → more people
More people per cluster → larger design effect
Has no effect when m is 1
More non-response → invite more
1.96² × 0.30 × 0.70 ÷ 0.05² = 322.7 → 323
If the true prevalence is 30%, about 95 of every 100 surveys of this size give an estimate between 25% and 35%.
By specialty
Sample size problems by specialty
Two drugs compared on HbA1c at 6 months. A continuous outcome: you need the SD from an earlier trial and the smallest HbA1c difference worth acting on.
The smaller the difference you aim to detect, the larger the trial must be.
Surgical site infection with a new skin preparation. Infection is a yes/no outcome that most patients never have.
Low event rates need hundreds per group. Consider a multicentre study or a more frequent outcome.
A thesis measures pain scores, PONV, rescue analgesia and patient satisfaction. The calculation must be based on one of them.
Name one primary outcome. Secondary outcomes are usually underpowered.
A neonatal unit sees a few eligible babies a month. The calculation says 80 per group.
Check feasibility: months needed = total n ÷ eligible babies per month.
A resident wants to show a drug reduces maternal deaths. Deaths are far too rare for a thesis-sized study.
Choose an outcome that is frequent and still clinically meaningful.
Depression rating scales vary widely between patients, so the SD is large.
Large SD means large n. Trained raters and a reliable scale reduce the spread.
Anaemia prevalence in a district, surveyed village by village.
Use p and precision, then multiply by the design effect for clusters.
Outcomes after bilateral knee replacement, counting 60 knees from 30 patients.
Two knees from one person are not independent. Count patients, or adjust.
Common myths
Common mistaken beliefs
Myth: 30 per group is enough for any study.
30 is not a rule. The number depends on the difference, the spread, α and power.
Myth: A bigger sample is always better.
Beyond the calculated number you add cost and expose more patients, and very small differences become statistically significant.
Myth: "Not significant" means there is no difference.
In a small study it often means "could not tell". Look at the confidence interval.
Myth: You can work out the sample size after collecting the data.
It is fixed before the study. Changing n after looking at results raises the false-positive rate. A planned re-estimation is different: ICH E9 (section 4.4) allows a revised sample size after an interim check on blinded data, justified in a protocol amendment.
Myth: Take as many patients as the unit has.
Calculate first, then check feasibility. If you cannot reach n, change the outcome or add centres.
Myth: Post-hoc power explains why my study was negative.
Observed power is a one-to-one function of the p-value, so it adds nothing (Hoenig and Heisey, 2001). Judge a negative study by its confidence interval.
Why this matters to you
Five consequences of getting sample size wrong
Protocol sent back
Research and ethics committees ask you to justify the number before approval.
No conclusion after two years
An underpowered thesis cannot show a real difference even when one exists.
A wrong conclusion
You write "no difference" when the study was simply too small to tell.
Questions you cannot answer at the viva
"My guide said 30" does not explain how you reached n.
Misreading papers
Small negative trials get quoted as proof that a treatment does not work.
History
History of power and sample size calculation
- 1933
Jerzy Neyman and Egon Pearson
Introduced the idea of choosing the test most likely to reject a false hypothesis: the concept we now call power.
- 1962
Jacob Cohen
Reviewed 70 psychology studies. Their average power was 0.18 for small effects, 0.48 for medium and 0.83 for large. For a medium effect, the typical study had less than a 50% chance of detecting it.
- 1978
Freiman, Chalmers, Smith and Kuebler
Examined 71 "negative" randomised trials in the medical literature. 67 had more than a 10% risk of missing a true 25% improvement; 50 could have missed a 50% improvement.
- 1983
D.A. Schoenfeld
Published a sample size formula for the proportional hazards regression model, based on the number of events needed.
- 1988
Jacob Cohen
Proposed 0.80 as a conventional power, treating a type I error as about four times as serious as a type II (β 0.20 against α 0.05).
- 2001
John Hoenig and Dennis Heisey
Showed that "observed power" after a study is determined completely by the p-value and adds nothing to interpretation.
Guidelines
Sample size in guidelines
“The number of subjects in a clinical trial should always be large enough to provide a reliable answer to the questions addressed.” The protocol must give the calculation method, the estimates used, and the basis for those estimates. Type II error is conventionally set at 10–20%.
Trial reports must state “how sample size was determined, including all assumptions supporting the sample size calculation”. This replaced item 7a of CONSORT 2010.
Steps
Calculating sample size, step by step
- Write one primary outcome and the study design.
- Decide the smallest difference that matters, or for a survey, the expected prevalence and the precision you need.
- Take the SD or expected proportions from an earlier study or a pilot.
- Fix α (usually 5%, two-sided) and power (80% or 90%).
- Apply the formula, or free software such as G*Power or OpenEpi, and round up.
- Adjust for dropouts, clustering and a small population.
- Write every number, and where it came from, in the protocol.
Reading a paper
Five checks on any study's sample size
- Is the calculation based on the primary outcome?
- Are α, power, the difference to detect and the SD (or expected proportions) all stated?
- Do they say where those numbers came from?
- Did they recruit the calculated number, with room for dropouts?
- If the result is negative, is the confidence interval narrow enough to rule out a difference that matters?
Why is it called that?
Word surgery
“Power”
Strength, force, authority. A "powerful" study sounds like a big one.
A probability: the chance your study detects a real difference of the size you planned. A large study can still have low power for a tiny difference.
“Sample”
A free trial pack, a taste of the product.
The part of a population actually studied, chosen to stand for the whole.
More spread, higher power, a stricter α and more dropouts increase n. A larger difference decreases n.
Sample
Latin exemplum: "that which is taken out"
The part of a population that is studied to draw conclusions about the whole.
Power
everyday meaning: ability
The probability that a study detects a real difference of the planned size. With 80% power, a real difference is detected 8 times in 10.
α and β
the first two Greek letters: the first and second kind of error
α: a false positive (type I error). β: a missed difference (type II error). Power = 1 − β.
Precision
Latin praecisio: "a cutting off"
How narrow the range around an estimate is. ±2.5 points is more precise than ±10, and needs more people.
MCID
Minimal · Clinically · Important · Difference
The smallest change a patient would notice and value. It sets d in the formula.
Exam answers
Model answer outlines
Parameters required for estimating sample size, and their role
MD Community Medicine 2019, 2022 · Ph.D. RM 2021| Parameter | What it means | If it increases, n… |
|---|---|---|
| Effect size (d) | Smallest difference worth detecting (MCID) | falls (halve d → n × 4) |
| Variability (SD) | How much the outcome varies between people | rises (double SD → n × 4) |
| Level of significance (α) | Accepted risk of a false positive; usually 5% | stricter α (5% → 1%) → n rises |
| Power (1 − β) | Chance of detecting a real difference; 80–90% | rises |
| Expected prevalence (p) | For surveys; p(1 − p) is largest at 50% | rises as p nears 50% |
| Precision (d, surveys) | Acceptable margin of error | tighter margin → n rises |
| Dropout / non-response | Expected losses to follow-up | recruit n ÷ (1 − rate) |
| Design effect | Cluster sampling, paired or repeated measures | rises with cluster size and ICC |
| One- or two-sided test | Decided and justified before the study | two-sided at the same α → n rises |
| Type of precision (surveys) | Absolute (± points) or relative (± a fraction of p) | relative margin for an uncommon condition → n rises steeply |
| Allocation ratio | Controls per case, or unequal group sizes | more controls per case → fewer cases, more people in total |
| Expected number of events (survival) | Power depends on events, not headcount | HR nearer 1 → more events |
| Prevalence (diagnostic accuracy) | Only people with the disease inform sensitivity | lower prevalence → more people tested |
Role in quality: the right n gives adequate power, a narrow confidence interval and an ethical use of participants. Too small a sample risks a type II error. Too large a sample wastes resources and exposes extra people to risk.
Importance of sample size (in clinical trials)
MD Pharmacology 2014, 2022 · Ph.D. RM 2024- A sample studies a part to learn about the whole. Its size decides how far the result can be trusted.
- Too small: low power, real effects missed (type II error), wide confidence intervals, inconclusive results, and patients exposed without an answer.
- Too large: wasted time and money, extra patients given the inferior treatment, and trivial differences reaching statistical significance.
- Must be calculated before the study starts, stated in the protocol (ICH E9) and reported with its assumptions (CONSORT 2025, item 16a).
- Changing n after seeing the data inflates the false-positive rate.
How is sample size calculated? (Determination in drug trials)
Ph.D. RM 2015, 2016, 2018, 2023 · MD Pharmacology 2015 · DM Neonatology 2013, 2025 · DM Medical Oncology 2009- State the research question, design and primary outcome.
- Choose α (5%, two-sided) and power (80–90%).
- Specify the effect size: MCID for a continuous outcome, or expected proportions p₁ and p₂ for a binary one.
- Obtain SD or baseline proportion from the literature or a pilot study.
- Apply the formula: two means, n = 2(Zα/2 + Zβ)²SD²/d²; one proportion, n = Z²p(1 − p)/d²; two proportions, the two-proportion formula.
- Worked example: SD 2, d 1, α 5%, power 80% → 63 per group.
- Other designs: paired (SD of the within-patient differences, 12 patients in the example), diagnostic accuracy (divide by the prevalence), case–control (exposure among controls, odds ratio and controls per case), correlation (Fisher's C), survival (number of events, then divide by the share with the event), non-inferiority (margin and one-sided α).
- Adjust for dropouts (÷ 1 − rate), design effect and finite population, then round up.
- Report all assumptions and their sources in the protocol and the final paper.
The one thing to remember
Decide the smallest difference that matters before you start. Half that difference needs four times the patients.
Worked examples use made-up numbers for teaching. Formulas assume a two-sided test and normal approximation. Always round up.
Further reading: Lwanga SK, Lemeshow S. Sample size determination in health studies: a practical manual. Geneva: World Health Organization; 1991. ISBN 92 4 154405 8.
Sources: Neyman J, Pearson ES. Phil Trans R Soc Lond A 1933;231:289–337. Cohen J. J Abnorm Soc Psychol 1962;65:145–53 (doi:10.1037/h0045186). Freiman JA et al. N Engl J Med 1978;299:690–4 (doi:10.1056/NEJM197809282991304). Cohen J. Statistical Power Analysis for the Behavioral Sciences, 2nd ed. 1988. Hoenig JM, Heisey DM. Am Stat 2001;55:19–24 (doi:10.1198/000313001300339897). ICH E9 Statistical Principles for Clinical Trials, 1998. Hopewell S et al. CONSORT 2025 statement. BMJ 2025;389:e081123.
Formulas for other designs: Schoenfeld DA. Biometrics 1983;39:499–503 (PMID 6354290). Buderer NM. Acad Emerg Med 1996;3:895–900 (doi:10.1111/j.1553-2712.1996.tb03538.x; PMID 8870764). Malhotra RK, Indrayan A. Indian J Ophthalmol 2010;58:519–22 (PMID 20952837). Altman DG, Royston P. BMJ 2006;332:1080 (doi:10.1136/bmj.332.7549.1080; PMID 16675816). Dean AG, Sullivan KM, Soe MM. OpenEpi: sample size for unmatched case-control studies (openepi.com). Charan J, Biswas T. Indian J Psychol Med 2013;35:121–6 (PMID 24049221). Higgins JPT et al., eds. Cochrane Handbook for Systematic Reviews of Interventions, version 6.5, 2024, section 6.5.2.2.
Past papers
Every TNMGRMU question on sample size
Each one is answered in this chapter. Tap a sitting to open the university's own question paper.
- ×2Calculation of sample size in Medical research?
- ×2Sample size
- ×2Sample size - Calculation
- ×1Explain the classification of data with suitable examples. Explain the basic concept of sampling. How do you determine the sample size of a research study?
- ×1Importance of sample size
- ×1Estimation of sample size
- ×1Significance of sample size in clinical trials
- ×1What factors determine the sample size for a research?
- ×1Principles in sample size calculation
- ×1Enlist the parameters required for estimating sample size and justify their role in the quality of research
- ×1Explain how a sample size is calculated
- ×1Determining sample size in drug trials
- ×1Importance of sample size calculation in clinical trials
- ×1Determination of Sample Size for a clinical trial
Further reading