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There is no single good sample size for a quantitative study. The number comes from what the study has to achieve: either an estimate that lands within a chosen margin of error, or a test that detects a difference of a stated size with a stated probability. Fix those targets, and a formula gives the minimum sample size. Rules of thumb are not a substitute for that calculation.
Short answer: what is a good sample size for a quantitative study?
A good sample size is the smallest one that meets the precision or power target you set before collecting data. Two situations cover most quantitative work:
- Estimating a quantity (a mean or a proportion): the sample size is set by the margin of error you can accept and the variability of the measurement.
- Comparing groups or testing a hypothesis: the sample size is set by the significance level (α), the power (1 − β) and the smallest effect worth detecting. See power analysis and sample size calculation for the full treatment.
Definition in one paragraph
Sample size (N, or n per group) is the number of observations a study analyses. In a priori planning it is the minimum number that makes the study’s primary estimate or test meet a stated precision or power requirement, given assumptions about variability and effect size. It is chosen before data collection, and it is a design decision, not a property of the data. For how participants are drawn, see sampling methods.
Formula for precision: sample size from a margin of error
The NIST/SEMATECH e-Handbook of Statistical Methods (section 7.2.2) gives the minimum sample size for estimating a mean to within a chosen error. For 95 % confidence:
N ≥ (1.96 / δ)² × σ²
- N: the minimum sample size.
- σ: the population standard deviation. NIST’s restriction is that the standard deviation must be known, so in practice you substitute a value from earlier work or a pilot study (see pilot study).
- δ: the margin of error, so the interval is the sample mean ± δ.
- 1.96: the standard normal value z at 0.975, which gives the 95 % confidence level.
The handbook derives this from δ = (σ / √N) × z, which guarantees the error of estimation is less than δ with a probability of 95 %.
Formula for power: sample size from alpha, power and effect size
For a test, the sample size depends on α, power (1 − β) and the size of the difference you want to detect. NIST section 7.2.4 gives the two-sided test of a proportion:
N ≥ ( ( z1−α/2 √(p₀(1 − p₀)) + z1−β √(p₁(1 − p₁)) ) / δ )²
- p₀ is the proportion under the null hypothesis and p₁ the proportion you want to be able to detect.
- δ = |p₁ − p₀|, the difference to detect.
- z1−α/2 and z1−β are standard normal quantiles; power equals 1 − β.
For a difference between two means, most researchers use software rather than a hand formula, because the exact calculation uses the t distribution. R’s power.t.test and G*Power both do this; the worked example below solves one design in each.
Worked example with real numbers
Example 1: precision for a mean
A lab wants to estimate a mean to within δ = 2 units with 95 % confidence, and earlier measurements suggest σ = 10.
N ≥ (1.96 / 2)² × 10² = 0.98² × 100 = 0.9604 × 100 = 96.04. Sample sizes are whole numbers and the requirement is a minimum, so round up: N = 97.
Example 2: power for two independent groups
A researcher wants 80 % power at α = 0.05 (two-sided) to detect a 5-point difference between two group means when the standard deviation is 10, a standardised difference of 0.5. The design is solved three ways:
| Method | Input | Result |
|---|---|---|
R power.t.test |
delta = 5, sd = 10, sig.level = 0.05, power = 0.80 | 63.77, so 64 per group |
| G*Power, a priori, t test for two independent means | d = 0.5, α = 0.05, power = 0.80, allocation ratio 1 | 64 per group, 128 in total |
| Normal approximation in a spreadsheet | 2 × (1.96 + 0.8416)² × (10 / 5)² | 62.8, so 63 per group |
The normal approximation is one participant short because it ignores the extra uncertainty from estimating the standard deviation. At n = 64 per group the exact t-based power is about 0.80; at n = 63 it falls just under, at about 0.795. Use the t-based answer, 64 per group, and round up.
Example 3: power for a proportion
To detect a change from p₀ = 0.50 to p₁ = 0.60 with α = 0.05 (two-sided) and 80 % power, using the NIST formula: z1−α/2 = 1.96 and z1−β = 0.8416. The numerator is 1.96 × √0.25 + 0.8416 × √0.24 = 0.98 + 0.4123 = 1.3923. Divide by δ = 0.10 to get 13.923, and square it: 193.8. Round up to N = 194. NIST also recommends a continuity correction, N = N′ + 1/δ, where N′ is the value from the formula; here that adds 1/0.10 = 10, giving 204 if you apply it.
Interpretation: how the result changes with your assumptions
The sample size is only as credible as the inputs. These values come from the arithmetic above and show how sensitive N is.
| If you change this | With σ = 10, 95 % confidence, N from the precision formula |
|---|---|
| Margin of error δ = 1 | (1.96 / 1)² × 100 = 384.16, so N = 385 |
| Margin of error δ = 2 | 96.04, so N = 97 |
| Margin of error δ = 4 | (1.96 / 4)² × 100 = 24.01, so N = 25 |
Halving the margin of error roughly quadruples the required sample. For power, the same pattern holds for the effect you hope to detect. For a two-sided, two-group t test at α = 0.05 and 80 % power, exact t-based calculation gives:
| Standardised difference (d) | n per group |
|---|---|
| 0.5 | 64 |
| 0.75 | 29 |
| 1.0 | 17 |
| 1.5 | 9 |
For d = 1 the R documentation’s own example, power.t.test(power = .90, delta = 1), targets 90 % power instead, which needs more participants than the 80 % figure above. Choosing a realistic effect size is the hardest input; the guide to sensitivity power analysis covers working backwards from a fixed sample.
Assumptions and common mistakes
- Treating σ as known. The NIST formula requires the standard deviation, so an optimistic guess gives a sample that is too small.
- Mixing up per-group and total n. In R’s
power.t.test, n is the number of observations per group. A two-group design needs twice that in total. - Forgetting to round up. The result is a minimum. 96.04 means 97, not 96.
- Counting enrolled instead of analysable participants. If you expect dropouts or exclusions, plan for them separately from the calculated N.
- Powering for one test and analysing another. The calculation applies to the primary statistical test; a different analysis needs its own calculation. Regression has its own considerations, covered in events per variable and minimum sample size for regression.
- Trading off error rates silently. Changing α or power changes N. See Type I and Type II errors.
How to report it (APA example)
State the test, α, power, the effect size and where it came from, the software, and the resulting number:
An a priori power analysis for an independent-samples t test (two-tailed, α = .05, power = .80, d = 0.50) indicated that 64 participants per group (N = 128) were required. The analysis was run in G*Power 3.1 (Faul et al., 2007).
For a precision-based plan: “To estimate the mean within ±2 units with 95 % confidence, assuming σ = 10 from earlier measurements, at least 97 observations were required.” For how this fits a grant or protocol, see sample size justification for a grant or protocol.
How to calculate it in Excel, R or Python
Excel (precision example)
=ROUNDUP((NORM.S.INV(0.975)/2)^2*10^2, 0)Returns 97 for δ = 2 and σ = 10. For the two-group normal approximation: =ROUNDUP(2*(NORM.S.INV(0.975)+NORM.S.INV(0.8))^2*(10/5)^2, 0) returns 63, which is the approximation noted above; use R or G*Power for the t-based 64.
R (two-group power)
power.t.test(delta = 5, sd = 10, sig.level = 0.05, power = 0.80,
type = "two.sample", alternative = "two.sided")
# n = 63.77 -> use 64 per groupExactly one of n, delta, power, sd and sig.level must be left as NULL; that is the one R solves for.
Python (precision example, standard library only)
import math
from statistics import NormalDist
z = NormalDist().inv_cdf(0.975) # 1.96
sigma, delta = 10, 2
n = math.ceil((z / delta) ** 2 * sigma ** 2)
print(n) # 97G*Power is a free program for power analyses of t tests, F tests, chi-square tests, z tests and some exact tests, and it graphs the results.
FAQ
What is a good sample size for a quantitative study?
It is the minimum that meets your precision or power target. For a mean with σ = 10 and a margin of error of 2 at 95 % confidence, that is 97; for a two-group comparison of a medium difference (d = 0.5) at 80 % power it is 64 per group.
Is there a minimum number that always works?
No. The required number moves with variability, margin of error, α, power and effect size, as the tables above show.
Should I round up or down?
Always up. The formula returns a minimum.
Where do I get the standard deviation before I have data?
From earlier studies of the same measure or from a pilot study; the NIST formula assumes it is known.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.2, Sample sizes required to estimate a mean (checked 2026-10-29).
- NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.4, Sample sizes required for a test of a proportion (checked 2026-10-29).
- R Documentation, power.t.test: Power calculations for one and two sample t tests (checked 2026-10-29).
- G*Power, Heinrich Heine University Düsseldorf, G*Power software page (checked 2026-10-29). Faul, F., Erdfelder, E., Lang, A.-G., and Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program. Behavior Research Methods, 39, 175-191.
