Sample Size Calculator

Sample size (n) is the number of observations in a study. Larger samples give more accurate results but cost more. Too small → unreliable results, high margin of error. Too large → waste of resources. Example: Poll 50 people about election → +/-14% margin of error (useless). Poll 1,000 → +/-3% (reliable). Sample size affects: statistical power, confidence interval width, ability to detect effects. Most surveys use 1,000-2,000 for +/-3% accuracy at 95% confidence.

Higher confidence requires larger sample. Relationship is squared: For 90% (z=1.645) vs 95% (z=1.96) vs 99% (z=2.576). Example with +/-5% margin: 90% confidence needs n=271, 95% needs n=385 (+42%), 99% needs n=663 (+144%). Each confidence level has critical value: 90%→1.645, 95%→1.96, 99%→2.576. Formula includes z^2, so effect is amplified. Trade-off: higher confidence = more certainty but higher cost.

Margin of Error (MOE) is the +/- range around your estimate. Inverse relationship with sample size: n ∝ 1/MOE^2. To halve MOE, quadruple sample size. Example: MOE +/-5% needs n=385. For +/-2.5% (half), need n=1,540 (4x). For +/-10% (double), need n=96 (¼). This is why most polls use +/-3% (n≈1,000) - good balance. Below +/-2% gets very expensive. Formula: n = (z^2*p*(1-p))/MOE^2.

Population proportion (p) is expected % with characteristic you're measuring. Examples: voting preference (50%), disease prevalence (5%), defect rate (1%). When unknown, use p=0.5 (50%) - gives maximum (conservative) sample size. Example at 95% confidence, +/-5% MOE: p=0.5 needs n=385, p=0.8 needs n=246, p=0.9 needs n=139. Use p=0.5 for polls, elections. Use known rate for quality control, medical studies with historical data.

For two-group comparison (e.g., A/B test, drug vs placebo), need larger samples. Formula: n = 2(z_alpha/2 + z_�^2)^2*p*(1-p)/d^2 where d = minimum detectable difference. Example: Detect 5% difference in 50% baseline conversion, 80% power, 95% confidence → n=1,571 per group (3,142 total). Factors: (1) Baseline rate, (2) Minimum detectable effect (smaller = more samples), (3) Power (usually 80%), (4) Significance level (usually 5%). Online calculators available for this.

Statistical power = probability of detecting real effect when it exists (1-�^2). Standard is 80% power (�^2=0.20). Higher power needs larger samples. Example detecting 10% difference: 80% power needs n=393 per group, 90% power needs n=526 (+34%), 95% power needs n=651 (+66%). Power analysis prevents Type II error (missing real effect). Low power → wasted study, can't detect effects. Used in clinical trials, experiments. Trade-off: higher power = more certainty but higher cost.

Surprisingly, population size barely matters for large populations! Formula: n_finite = n/(1 + n/N). For infinite population at 95% confidence, +/-5% MOE: n=385. If population = 1,000, adjusted n=278. If population = 10,000, adjusted n=370. If population = 1,000,000, adjusted n=384. Only matters when sample is >5% of population. Census (sample everyone) when population <200-300. This is why national polls of 300M people only need ~1,000 samples!

SURVEYS/POLLS: 1,000-2,000 (+/-3% MOE). ACADEMIC RESEARCH: 30-500 depending on effect size. A/B TESTING: 1,000-10,000 per variant (detect 2-10% changes). CLINICAL TRIALS: 100-10,000 (depends on risk/effect). FOCUS GROUPS: 6-10 per group. USABILITY TESTING: 5-8 users (Nielsen). MARKET RESEARCH: 200-500. QUALITY CONTROL: Depends on lot size and AQL. Always do power analysis before expensive studies!

MISTAKE 1: Not accounting for non-response. If 50% response rate expected, double sample size. MISTAKE 2: Using total sample when need per-group sample (comparing groups needs n per group). MISTAKE 3: Ignoring subgroup analysis - if analyzing 5 segments, need larger overall sample. MISTAKE 4: Not considering dropout in longitudinal studies (add 20-30%). MISTAKE 5: Using p=0.5 when you have better estimate (wastes resources). MISTAKE 6: Forgetting finite population correction for small populations.

For continuous data (height, income, test scores), formula: n = (z*rho�/E)^2 where rho�=population SD, E=desired margin of error. Example: Estimate average height +/-2cm, rho�=10cm, 95% confidence: n = (1.96*10/2)^2 = 96. Need estimate of SD from: pilot study, literature, similar studies, or assume SD = range/4. For comparing two means: n = 2(z_alpha/2 + z_�^2)^2*rho�^2/d^2 where d=minimum detectable difference. Larger SD or smaller difference → larger sample needed.

Sample size follows law of diminishing returns. Each doubling gives 30% improvement (sqrt2≈1.41). Cost-benefit analysis: Going from +/-10% to +/-5% MOE: 4x cost for 2x precision. From +/-5% to +/-2.5%: 4x cost for 2x precision. Optimal point: Usually +/-3-5% for surveys (n=400-1,000). Below +/-2% gets very expensive. Factors: (1) Data collection cost, (2) Value of precision, (3) Budget constraints. Example: Phone survey $50/response → n=1,000 costs $50,000 for +/-3% vs $200,000 for +/-1.5%.

Inflate initial sample size by expected loss rate: n_adjusted = n_required/(1 - attrition_rate). Example: Need n=200 completes, expect 30% dropout → recruit n=200/(1-0.30)=286. Common rates: Online surveys 10-30%, Mail surveys 30-50%, Longitudinal studies 20-40%, Clinical trials 10-20%. Calculate at each stage: Initial contact → Agreement → Completion. Example: 40% agree, 80% complete → n=200/(0.40*0.80)=625 initial contacts. Always overrecruit to ensure adequate final sample.