样本量计算器
确定需要多少调查受访者或研究参与者。输入所需置信水平、误差范围和（可选）总体数量，计算统计有效结果所需的最小样本量。
n = Z² × p(1-p) / e²
where Z = z-score, p = proportion, e = margin of error
Z = 1.960 — standard for most research
Use 50% if unknown — gives the largest (most conservative) sample size.
Sample Size Quick Reference
Required sample size at 95% confidence, p = 50%:
| Margin of Error | Infinite Pop. | Pop. 10,000 | Pop. 1,000 |
|---|---|---|---|
| ±1% | 9,604 | 4,900 | 906 |
| ±2% | 2,401 | 1,937 | 707 |
| ±3% | 1,068 | 965 | 517 |
| ±5% | 385 | 370 | 278 |
| ±10% | 97 | 96 | 88 |
* Smaller margin of error requires a larger sample. Use finite population correction when your population is small.
Frequently Asked Questions
样本量计算器是如何工作的？
样本量计算基于三个关键参数：置信水平（通常 95%，对应 Z 值 1.96）、误差边界（可接受的精度，如 ±5%）和预期总体比例（未知时取 50%，给出最保守的最大样本量）。公式为：n = (Z²×p×(1-p)) / E²。
置信水平、误差边界和样本量如何相互关系？
置信水平越高（如从 90% 到 99%），所需样本量越大；误差边界越小（精度要求越高），所需样本量也越大。样本量与误差的平方成反比：误差缩小一半，样本量增大约 4 倍。
什么时候需要有限总体修正？
当调查总体规模较小（通常小于 10,000 人）时，应使用有限总体修正公式：n_adj = n / (1 + (n-1)/N)，其中 N 为总体大小，n 为无限总体样本量。修正后的样本量会小于原始计算值。
为什么 p=50% 给出最大样本量？
在置信度和误差边界固定的情况下，p=50% 时，p×(1-p) 的乘积最大（= 0.25），导致所需样本量最大。如果不知道总体比例的先验估计，使用 50% 是最保守、最安全的假设，确保样本量足够大。
Sample Size Formula
The standard sample size formula for estimating a proportion in a population is derived from the normal distribution. For an infinite (or very large) population:
Infinite Population Formula:
n = Z² × p × (1 - p) / e²
Where:
- n = required sample size
- Z = z-score for the desired confidence level (e.g., 1.960 for 95%, 1.645 for 90%, 2.576 for 99%)
- p = expected proportion (use 0.5 if unknown — maximizes sample size)
- e = margin of error as a decimal (e.g., 0.05 for ±5%)
| Confidence Level | Z-Score | Common Use |
|---|---|---|
| 80% | 1.282 | Exploratory research, quick estimates |
| 85% | 1.440 | Preliminary studies |
| 90% | 1.645 | Market research, less critical decisions |
| 95% | 1.960 | Standard social science and business research |
| 99% | 2.576 | Medical studies, high-stakes decisions |
Finite Population Correction
When your population is not very large relative to the required sample, the standard formula overestimates the needed sample size. The Finite Population Correction (FPC) adjusts for this:
Finite Population Formula:
nₐ = n₀ / (1 + (n₀ - 1) / N)
where n₀ = infinite-population estimate, N = total population size
The FPC becomes significant when the sample represents more than about 5% of the population. For example, surveying 385 people from a population of 1,000 (38.5% sample fraction) yields a corrected sample size of around 278 — a 28% reduction.
n₀ = 385 (95% CI, ±5%, p = 0.5, infinite pop.)
N = 1,000
nₐ = 385 / (1 + (385 - 1) / 1,000)
nₐ ≈ 278 respondents needed
Leave the population field blank if you don't know your population size, or if it is very large (e.g., a national survey). The standard formula will be used in those cases.
Choosing Margin of Error
The margin of error (MoE), also called the confidence interval half-width, defines how close your sample estimate will be to the true population value. A ±5% margin means your results are within 5 percentage points of the actual answer.
Guidelines by Research Type
| Research Type | Recommended MoE | Rationale |
|---|---|---|
| National polls, political surveys | ±2–3% | Close races require precision |
| Customer satisfaction surveys | ±5% | Industry standard for business decisions |
| Market research, internal surveys | ±5–10% | Trend detection, directional insights |
| Exploratory / pilot studies | ±10% | Quick, cheap screening before full study |
| Academic / clinical research | ±1–3% | Rigorous evidence requirements |
Sample size grows quadratically as margin of error decreases — cutting your MoE in half requires four times as many participants. Choose the largest MoE your research objectives can tolerate.
Understanding Confidence Levels
A 95% confidence level means that if you repeated the survey 100 times with different random samples, 95 of the resulting intervals would contain the true population value. It does notmean there is a 95% probability that the true value lies in your specific interval — once you collect data, the true value either is or isn't in the interval.
Higher confidence levels require larger samples:
- Going from 90% to 95% confidence increases sample size by approximately 41%.
- Going from 95% to 99% confidence increases sample size by approximately 73%.
For most surveys and business research, 95% confidence is the standard. Use 99% for high-stakes decisions (medical, legal, financial). Use 90% to reduce cost in exploratory or screening studies.
Sample Size Calculation Examples
Example 1: National Survey (Unknown Proportion)
A polling organization wants to estimate voting intent with ±3% margin of error at 95% confidence. Since the true proportion is unknown, they use p = 0.5 (worst case).
Z = 1.960, e = 0.03, p = 0.5
n = (1.960)² × 0.5 × 0.5 / (0.03)²
n = 3.8416 × 0.25 / 0.0009
n = 1,068 respondents needed
Example 2: Customer Survey (Known Proportion)
A company surveys customers about satisfaction. From past data, ~70% are satisfied. They want ±5% margin of error at 95% confidence.
Z = 1.960, e = 0.05, p = 0.70
n = (1.960)² × 0.70 × 0.30 / (0.05)²
n = 3.8416 × 0.21 / 0.0025
n = 323 respondents needed (vs. 385 with p = 0.5)
Example 3: Small Population (Finite Correction)
An HR team surveys 500 employees. They want ±5% margin at 95% confidence. Without correction: 385 needed. With FPC:
n₀ = 385, N = 500
nₐ = 385 / (1 + 384 / 500)
nₐ ≈ 218 employees needed