Sample size

1. What Is Sample Size?

Sample size (n) is

the number of observations used to estimate a population parameter or test a hypothesis.

Sample size directly affects:

• Precision of estimates
• Statistical power
• Stability and reliability of conclusions

2. Why Sample Size Matters

Sample size determines how much information the data contain.

Small n:

• High uncertainty
• Low power
• Large random fluctuations

Large n:

• More precise estimates
• Higher power
• Greater sensitivity to small effects

Sample size controls the resolution of statistical inference.

3. Sample Size and Variability

A central relationship:$$\text{Standard Error} \propto \frac{1}{\sqrt{n}}$$​

Implications:

• Doubling n does not halve uncertainty
• To reduce standard error by half, n must increase fourfold

4. Sample Size and Estimation Precision

In estimation problems (e.g., confidence intervals):

• Larger n → narrower confidence intervals
• Smaller n → wider confidence intervals

Interpretation:

Sample size determines how precisely we can estimate population parameters.

5. Sample Size and Hypothesis Testing

In hypothesis testing, sample size affects:

(1) Test Statistic

$$\text{Test statistic} = \frac{\text{Effect}}{\text{Standard Error}}$$

Since standard error ↓ as n ↑:

• Larger n → larger test statistic (for the same effect)
• Easier to exceed the critical value

(2) Statistical Power

• Larger n → higher power
• Small effects become detectable only with large samples

6. Sample Size vs Effect Size

Sample size and effect size trade off:

• Large effect → small n may suffice
• Small effect → large n required

Key idea:

Sample size compensates for weak signals.

7. Sample Size Does NOT Do Everything

Important limitations:

• Large n cannot fix:
  • Bias
  • Poor measurement
  • Wrong model
• Large n can make trivial effects statistically significant

Precision does not equal correctness or importance.

8. Sample Size Planning (Conceptual)

Before data collection, sample size is chosen based on:

• Desired power (e.g., 0.80)
• Expected effect size
• Acceptable Type I error (α)
• Variability of the data

This process is called power analysis.

9. After-the-Fact Interpretation

After analysis:

• Non-significant result + small n → inconclusive
• Significant result + very large n → check practical relevance

Sample size affects interpretation, not just design.

10. Sample Size and Reproducibility

Small samples:

• High variance across studies
• Poor reproducibility

Large samples:

• More stable estimates
• Higher replication success

But only if bias and design issues are controlled.

11. Common Misconceptions

“Larger sample size guarantees truth”

False — it guarantees precision, not validity

“Sample size only matters before the study”

False — it affects interpretation afterward as well

“Non-significant results mean no effect”

Often false when n is small

12. Sample Size in Different Contexts

• Estimation: precision-focused
• Hypothesis testing: power-focused
• Machine learning: generalization-focused
• Observational studies: confounding-focused

Same concept, different emphasis.

13. Key Takeaway Statements

• Sample size controls uncertainty
• Standard error decreases as 1/√n
• Larger samples increase power
• Large n does not fix bias
• Statistical significance must be interpreted with context

14. Concept Map

Sample Size (n)
      ↓
Standard Error ↓
      ↓
Precision ↑   Power ↑

Sample Size (n)
      ↓
Standard Error ↓
      ↓
Precision ↑   Power ↑