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Calculate the initial scale
A fictional independent measurement model has known standard deviation 8 units. With n = 64, the mean standard error is 8/√64 = 1 unit. The exercise holds the measurement distribution and independence assumption fixed.
Solve for the desired uncertainty
To reach SE = 0.5 units, solve 8/√n = 0.5. Then √n = 16 and n = 256. Increasing 64 to 128 would give approximately 0.7071 units, so doubling the sample does not halve SE.
| n | SE in units | Relative to baseline |
|---|---|---|
| 64 | 1 | 1 |
| 128 | 0.7071068 | 0.7071068 |
| 256 | 0.5 | 0.5 |
Keep collection costs separate
The calculation concerns uncertainty, not the optimal cost of a study. Collecting 256 measurements may change the sampling period, respondent population or measurement process. Those changes can undermine the fixed-model comparison.
Do not use precision as a bias correction
A larger sample can make a biased selection process produce a very precise estimate of the wrong population. Check coverage and data quality alongside the intended sample size. The numeric result is not a universal minimum sample-size rule.