Question

Why does increasing the number of trials increase confidence in the results of the experiment?

Answer 1

Because repeated trials reduce the effect of random variation, giving a more precise estimate of the true value.

Key points:
- Random error averages out. Each trial is subject to random fluctuations; when you take many independent trials their random ups and downs cancel out and the sample average (or proportion) gets closer to the true value.
- Law of large numbers: as the number of trials n grows, the sample mean converges to the expected (true) mean.
- Standard error shrinks with n. For a sample mean the standard error is σ/√n (σ = population standard deviation). That 1/√n factor means doubling n reduces the standard error by about 29% (not by half). For a proportion p the standard error is √(p(1−p)/n).
- Narrower confidence intervals and greater statistical power. Smaller standard error gives narrower confidence intervals around estimates and increases the test statistic (often proportional to √n), so you have more power to detect real effects.

Simple numeric examples:
- If σ = 10, n = 4 → SE = 10/2 = 5; n = 100 → SE = 10/10 = 1.
- For p = 0.5, n = 100 → SE = 0.05; n = 10,000 → SE = 0.005.

Caveats:
- Increasing n reduces random (statistical) uncertainty but does not fix systematic errors (bias), poor experimental design, or nonindependent samples. Those must be addressed separately.
- There are diminishing returns because error decreases like 1/√n, so very large increases in n are needed to get modest further precision.

In short: more trials reduce random noise, tighten uncertainty estimates, and make you more confident that your measured effect reflects the true effect rather than chance.