How this calculator works
For independent, identically distributed observations, averaging leaves the centre unchanged and divides the population SD by sqrt(n). Adding multiplies the centre by n and the SD by sqrt(n). These moment formulas are exact under independence; the claim of an approximately normal shape is an additional large-sample result, not a guarantee at n = 30.
Formula
Mean: E[X̄] = μ, SD[X̄] = σ/√n. Sum: E[S] = nμ, SD[S] = σ√n. Required n = ceil((σ/target SE)²).
Worked example
For μ = 100, σ = 15 and n = 25, sample means have centre 100 and SE 3. Sums have centre 2,500 and SD 75.
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Frequently asked questions
Does the theorem make the original data normal?
No. It concerns the sampling distribution of averages or sums.
Is n = 30 always enough?
No. Strong skew or heavy tails can require much larger samples, and infinite variance breaks this form of the theorem.
Does this include finite population correction?
No. Use independent sampling or a population large enough that sampling without replacement is effectively independent.