Most cross-sectional studies estimate a proportion: how many households have water within 30 minutes, how many mothers attended antenatal care. The sample size answers one question: how many people do you need for that estimate to be precise enough to be useful?
1. Three numbers to decide first
Confidence level. Usually 95%, which gives z = 1.96.
Expected proportion (p). Take it from a similar earlier study in a similar population. If there is none, use 50%: it gives the largest sample, so you are safe.
Margin of error (d). How close you need the estimate to be, often 5 percentage points (0.05). A smaller margin needs a much larger sample.
| Confidence | z |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
2. The formula
For a single proportion (Cochran’s formula):
n = z² × p × (1 − p) / d²
3. A worked example
A household survey of access to water, with no earlier estimate (p = 0.5), 95% confidence and a 5-point margin:
- n = 1.96² × 0.5 × 0.5 / 0.05² = 384.2, rounded up to 385
- Cluster sampling, design effect 1.5: 385 × 1.5 = 577.5, so 578
- Expected non-response 10%: 578 / 0.9 = 642.2, so 643 households
Always round up at each step: a sample of 384.2 people means 385.
4. Cluster sampling: the design effect
People in the same village or kebele tend to be alike, so a cluster sample tells you less than a simple random sample of the same size. The design effect (DEFF) corrects for this:
DEFF = 1 + (m − 1) × ICC
where m is the number interviewed per cluster and ICC is the intracluster correlation. Without an ICC from earlier work, many studies use a design effect of 1.5 or 2; say which, and why, in your methods.
5. Small populations
When the whole population (N) is small, for example 800 health workers in a zone, the finite population correction lowers the sample:
n′ = n / (1 + (n − 1) / N)
With n = 385 and N = 800: 385 / (1 + 384 / 800) = 260.1, so 261 health workers.
6. What to write in the methods section
“The sample size was calculated with the single population proportion formula, assuming a proportion of 50%, 95% confidence and a 5% margin of error (385). Allowing for a design effect of 1.5 and 10% non-response, the final sample was 643 households.”
Do this in Researchite
The sample size calculator works this through for you, with 20 methods and the reference for each formula ready for your methods section. No account needed.
References
- Cochran WG. Sampling Techniques. 3rd ed. New York: John Wiley & Sons; 1977.
- Kish L. Survey Sampling. New York: John Wiley & Sons; 1965.
- Lwanga SK, Lemeshow S. Sample Size Determination in Health Studies: A Practical Manual. Geneva: World Health Organization; 1991.
Tesfahun Mulatu
Epidemiologist and biostatistician with more than 10 years in academia and research, including multi-country and mixed-methods studies with local and international research institutions, and monitoring, evaluation and learning across Ethiopia’s development and humanitarian sectors on projects funded by UN agencies and international donors. Tesfahun’s work spans maternal and child health, sexual and reproductive health, gender-based violence and mental health, with over 19 publications.
Epidemiologist and biostatistician, founder of Researchite · LinkedIn