How the Sampling Calculator works (with formulas)

This calculator uses classic “intro stats” formulas that show up in survey methodology, research papers, and quality control. There are two common sampling situations: (1) proportions (a percentage like conversion rate) and (2) means (an average like average score). The inputs you choose are really about one trade-off: precision vs effort.

If you want to estimate a proportion (for example, “What percent of customers prefer Plan A?”), the baseline sample size formula is:

n₀ = (Z² · p · (1 − p)) / E²

• Z = z-score for your confidence level (90% ≈ 1.645, 95% ≈ 1.96, 99% ≈ 2.576)
• p = expected proportion (0.5 is “worst-case” if you don’t know)
• E = margin of error as a decimal (±5% → 0.05)
• n₀ = initial sample size for a large population

If your population is not huge (say you only have 600 students in a school), you can adjust using finite population correction (FPC):

n = n₀ / (1 + (n₀ − 1)/N)

Where N is the population size. This usually makes the recommended sample size smaller. If N is unknown or very large, the correction barely changes anything — which is why survey tools often ignore it.

If you’re estimating a mean (for example, “What’s the average time to complete a task?”), the formula looks similar, but uses an estimated standard deviation (σ):

n₀ = (Z² · σ²) / E²

Here E is the allowable error in the same units as your measurement (e.g., ±2 minutes, ±5 points). If σ is unknown, you can use pilot data (a small initial sample) or a reasonable guess based on past measurements.

If you already have a sample size and want to estimate the margin of error for a proportion, you can rearrange the formula:

E ≈ Z · √(p(1−p)/n)

In practice, many people plug in p = 0.5 when p is unknown. That yields a conservative (slightly larger) margin of error.

Real examples (copy-paste friendly)

You want a 95% confidence estimate with ±5% margin of error for a proportion (p unknown). Use p = 0.5 (worst-case). The calculator returns about 385 responses. That’s why you see “385” in so many survey guides — it’s the default for 95% and ±5%.

You’re surveying a class of N = 120 students, 95% confidence, ±5% error, p = 0.5. Without correction you’d still get ~385, which is impossible (bigger than the class). With finite population correction, the required sample size drops to around 92. That’s a huge difference — and it’s the reason “population size” matters when N is small.

You’re measuring average delivery time. You think σ ≈ 12 minutes from historical data. You want 95% confidence and ±3 minutes error. The calculator returns roughly 62 observations (deliveries).

With 95% confidence and unknown p (use 0.5), a sample size of 200 gives a margin of error around ±6.9%. That’s why small samples can feel “noisy”.

How to pick the “right” inputs (without overthinking)

The “best” sample size isn’t one magic number — it’s a decision based on effort, time, and what you need the result for. Here’s a practical way to choose inputs:

• Proportion: yes/no, conversion rate, approval, defect rate, churn rate.
• Mean: average rating, average completion time, average value in dollars.

• 90%: quick internal signals, early product discovery, rough estimates.
• 95%: standard choice for most surveys and research assignments.
• 99%: high-stakes estimates where you want stronger certainty.

• ±10%: very fast, very rough. Great for early direction.
• ±5%: common, balanced. Good for public-facing survey claims.
• ±3%: more precise, often used in serious polling. Requires much larger samples.
• ±1–2%: highly precise, typically expensive in time or money.

Population size is helpful when the total group is limited and known (a roster, a customer list, a batch). If you’re sampling “future visitors” or “the internet”, leave it blank.

For proportions, if you have no clue, use p = 0.5. If you have a prior estimate (say your conversion rate is usually ~8%), use p = 0.08. This can reduce the required sample size, but be careful: if your guess is wrong, you might under-sample.

For means, σ matters a lot. If you don’t know σ, take a small pilot sample (like 20), compute the sample standard deviation, then use that to plan the full study.

Sampling Calculator FAQs

• Why do I keep seeing “385” as the sample size?

  Because for a proportion with unknown p (use 0.5), 95% confidence, and ±5% margin of error, the formula gives n ≈ 384.16, which rounds up to 385. It’s the standard “default survey” number.

• Does population size change the sample size a lot?

  Only when the population is small. If N is huge, the correction barely moves. If N is a few hundred or a few thousand, finite population correction can noticeably reduce n.

• What if I don’t know the expected proportion (p)?

  Use p = 0.5. It’s “worst-case”, meaning it produces the largest recommended sample size for a given margin of error and confidence level. That’s the safe option.

• Is this the same as power analysis for A/B tests?

  Not exactly. A/B test planning usually depends on statistical power and the minimum effect you want to detect. This tool is perfect for rough planning and survey-style estimation, but power analysis is more specific.

• Can I use this for quality control sampling?

  Yes for approximate planning. If you’re doing regulated QC or acceptance sampling, your industry may require specific sampling plans (like ANSI/ASQ standards).

• What’s the difference between “percent points” and “percent”?

  Margin of error here is in percent points. If you estimate 40% with ±5%, your interval is 35% to 45%. That’s ±5 percentage points (not “±5% of 40”).