Sample Size Calculator

Complete Sample Size Guide & Information

1. What is Sample Size?

Sample size is the number of observations or individuals included in a statistical study. Choosing the right sample size is critical: too small and results may be unreliable; too large and the study wastes time and resources. This calculator helps determine the minimum sample size needed to achieve a desired level of statistical precision.

2. Key Statistical Concepts

Confidence Level

The confidence level is the probability that the true population parameter lies within the calculated confidence interval. Common values are 90%, 95%, and 99%. A 95% confidence level means that if the same study were repeated many times, 95% of the resulting intervals would contain the true population value.

Margin of Error

The margin of error describes the range above and below a sample estimate within which the true population value is likely to fall. A smaller margin of error requires a larger sample size.

Population Proportion (p)

The expected percentage of the population that has the characteristic being measured. When unknown, use 50% (0.5) because it produces the maximum variance and thus the most conservative (largest) sample size estimate.

Population Size (N)

The total number of individuals in the population. For very large or unknown populations, this can be left blank (infinite population assumption). For small populations, the finite population correction factor reduces the required sample size.

3. Sample Size Formula

Infinite population (Cochran's formula)

• n = required sample size
• z = z-score corresponding to confidence level
• p = population proportion (0 to 1)
• e = margin of error (0 to 1)

Finite population correction

where N is the population size.

4. Margin of Error Formula

For finite populations, the standard error is adjusted by the finite population correction factor:

5. Z-Scores for Common Confidence Levels

Confidence Level	Z-Score (z)	Common Usage
80%	1.282	Exploratory studies
85%	1.440	Preliminary research
90%	1.645	Business surveys
95%	1.960	Standard scientific research
99%	2.576	Medical and high-stakes research

6. Input & Control Definitions

• Confidence Level dropdown: Select the desired statistical confidence level (80% to 99%).
• Margin of Error: Maximum acceptable error percentage (sample size mode).
• Sample Size: Number of survey respondents (margin of error mode).
• Population Proportion: Expected percentage with the attribute. Use 50% if unknown.
• Population Size: Total population size. Leave blank for infinite population.
• Calculate button: Computes the result based on inputs.
• Clear button: Resets all input fields.
• Help icons (?): Hover for quick definitions.
• Chart: Shows how sample size and margin of error relate to each other.
• Tabs: Switch between calculating sample size and margin of error.

7. Worked Examples

Example 1 — Sample size (default): 95% confidence, 5% margin, 50% proportion, infinite population

• z = 1.96, p = 0.5, e = 0.05
• n = (1.96² × 0.5 × 0.5) / 0.05² = 384.16
• Round up: 385 respondents needed

Example 2 — Margin of error (default): 95% confidence, n = 100, p = 60%

• e = 1.96 × √(0.6 × 0.4 / 100)
• e = 1.96 × 0.0490 = 0.0960
• Margin of error: 9.60%

8. Real-World Applications

• Market research: Designing surveys and opinion polls
• Clinical trials: Determining number of patients needed
• Quality control: Sampling for manufacturing inspection
• Political polling: Designing representative voter surveys
• Social science research: Questionnaire and experiment design
• Auditing: Sample sizes for financial and compliance audits
• Education: Assessment and program evaluation studies
• Public health: Disease prevalence and vaccination surveys

9. Important Notes

• Always round sample size upward to the next whole number — you cannot survey a fraction of a person.
• Using p = 50% gives the largest required sample size and is the most conservative assumption.
• Higher confidence levels require larger sample sizes.
• Smaller margins of error require much larger sample sizes (quadratic relationship).
• Finite population correction matters most when sample size is more than 5% of the population.
• Sample size calculation assumes simple random sampling.
• Response rate should be considered — plan to invite more people than the calculated sample size.
• This calculator assumes normal approximation, which is valid when np ≥ 5 and n(1−p) ≥ 5.

10. Related Statistical Concepts

• Confidence interval: Range of values likely to contain the true parameter
• Statistical power: Probability of detecting an effect when it exists
• Type I error (α): False positive rate (1 − confidence level)
• Type II error (β): False negative rate
• Standard error: Standard deviation of a sample statistic
• Sampling distribution: Distribution of a statistic across repeated samples
• Central Limit Theorem: Justifies normal approximation for large samples
• Stratified sampling: Sampling method that can reduce required sample size

11. References

1. Cochran, William G. "Sampling Techniques." John Wiley & Sons. 1977.
2. Krejcie, Robert V. and Morgan, Daryle W. "Determining Sample Size for Research Activities." Educational and Psychological Measurement. 1970.
3. Israel, Glenn D. "Determining Sample Size." University of Florida Extension. 1992.
4. Bartlett, Joanna E. et al. "What is the best sample size for a survey?" International Journal of Market Research. 2001.
5. Lemeshow, Stanley et al. "Sample Size Determination in Health Studies." John Wiley & Sons. 1990.
6. Moore, David S. and McCabe, George P. "Introduction to the Practice of Statistics." W.H. Freeman. 2012.