What sample size do I need for 95% confidence and 5% margin of error?
For a proportion with p = 0.5 (worst case): n = (1.96² × 0.5 × 0.5) / 0.05² = 384. This is the classic "standard survey" benchmark. For a finite population of 10,000, the FPC reduces it to about 370.
Does population size always affect the sample size I need?
Only when your sample is large relative to the population (n/N > 5%). For populations above 100,000, the standard infinite-population formula works fine — the difference is under 1%. This is why a poll of 330M people uses only ~1,000 respondents.
What proportion should I use if I don't know the true value?
Use p = 0.50. This maximises variance (p × (1−p) is highest at 0.5) and gives the most conservative — therefore safest — estimate. Any other value assumes knowledge you don't have and could leave your study underpowered.
What is statistical power and why should I care?
Power (1 − β) is the probability of detecting a real effect. At 80% power, you have a 20% chance of missing a real difference (Type II error). Studies with power below 60% are often not worth conducting — they're likely to produce inconclusive results even when a real effect exists.
What is Cohen's effect size?
Cohen's d measures the standardised magnitude of a difference: d = (μ₁ − μ₂) / σ. Conventions: Small = 0.2 (like height difference between 15/16-year-olds), Medium = 0.5 (like IQ difference between college and high-school groups), Large = 0.8 (obvious differences). For proportions, Cohen's h is analogous.
Can I use the same formula for means and proportions?
No — they use different formulas. Proportions use n = z²p(1−p)/E². Means use n = (zσ/E)². For means, you must supply an estimate of the population standard deviation (σ), typically from a pilot study or published literature.
Why does 99% CI require so many more samples than 95%?
Because the z-critical value scales non-linearly: z₉₅ = 1.96 but z₉₉ = 2.576. Since n scales with z², the 99% CI needs (2.576/1.96)² ≈ 72% more samples than 95% for the same margin of error. The jump from 95% to 99% is much larger than from 90% to 95%.
What is the minimum sample size for valid statistics?
A common rule of thumb is n ≥ 30, which is roughly where the Central Limit Theorem kicks in and the normal approximation becomes reliable. For proportions, you also want np ≥ 10 and n(1−p) ≥ 10. For very small samples, use t-distribution methods or exact tests.
How do I reduce my required sample size?
Four levers: (1) Widen margin of error — doubling E reduces n by 4×. (2) Lower confidence level from 99% to 95% or 90%. (3) Use a known proportion closer to 0 or 1 instead of 0.5 (if justified). (4) Apply finite population correction if your population is small. Each lever has trade-offs in precision or confidence.
What is Type I vs. Type II error?
A Type I error (α) is a false positive — concluding an effect exists when it doesn't. The confidence level controls this: 95% CI means α = 5%. A Type II error (β) is a false negative — missing a real effect. Power = 1 − β. These are inversely related: increasing one typically increases the other for a fixed sample size.
How do I choose the right margin of error?
Consider the decision at stake: ±10% is sufficient for rough feasibility checks; ±5% is standard for most business surveys; ±3% is typical for political polls; ±1% is needed for high-stakes regulatory studies. Always balance precision against cost — collecting 4× more data to halve the margin of error is rarely justified unless decisions are very sensitive to that precision.
Why do national polls use such small samples?
Because required sample size depends mostly on desired precision, not population size. For a population of 330 million vs. 10,000, the difference in required n (at 95%, ±3%) is only ~0.03%. This counterintuitive result comes directly from the finite population correction formula: when N is huge, n/(n+N−1) ≈ 0 and correction is negligible.