What is the Sample Size Calculator?
Polling 1,000 likely voters costs a small fortune; polling 100 gives mush — the sample size calculator finds the sweet spot in between. The formula, n = (z* × σ ÷ E)², balances three demands: how confident you want to be (z*, 1.96 for 95%), how spread out the population is (σ, your standard deviation estimate), and how precise the answer must be (E, the margin of error). Halve your acceptable error and the required sample quadruples — precision is expensive.
Campaign pollsters live by this equation. Wanting ±3 points at 95% confidence with maximum-variability responses (σ ≈ 0.5 for proportions) demands about 1,067 respondents — which is why serious national polls hover near 1,000 interviews. A local business survey tolerating ±5 points needs only about 384.
This calculator takes your three inputs — the Z* critical value, your Population SD Estimate (σ), and the Margin of Error (E) — and returns the minimum sample size. Round up, since you cannot survey a fraction of a person, and then add a cushion for non-responses.
With the pre-filled defaults — Z* (1.96 for 95%, 2.576 for 99%) 1.96, Population SD Estimate (σ) 0.5, Margin of Error (E) 0.05 — the Sample Size Calculator returns Required Sample Size (n) 385, Exact n 384.1600. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Sample Size Calculator
- 1Type your confidence level's critical value in the Z* box — 1.96 for 95%, 2.576 for 99%.
- 2Type your best guess of the population spread in the Population SD Estimate (σ) box — use 0.5 for proportions when unsure.
- 3Type the largest error you can tolerate in the Margin of Error (E) box — e.g., 0.03 for ±3 points.
- 4Read the required sample size from the results.
- 5Round up to a whole person, then inflate by your expected non-response rate.
- 6If the number is unaffordable, loosen the Margin of Error (E) — doubling E quarters the sample.
Formula
This is the core equation the Sample Size Calculator uses to calculate results. Enter values in the calculator above to see it in action.
See the full step-by-step formula guide with worked examples →
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| 95% CI, SD=0.5, E=0.05 | Z* (1.96 for 95%, 2.576 for 99%) = 1.96, Population SD Estimate (σ) = 0.5, Margin of Error (E) = 0.05 | Required Sample Size (n) = 385; Exact n = 384.1600 |
| 99% CI, SD=15, E=3 | Z* (1.96 for 95%, 2.576 for 99%) = 2.576, Population SD Estimate (σ) = 15, Margin of Error (E) = 3 | Required Sample Size (n) = 166; Exact n = 165.8944 |
Value Reference Table
See how the Required Sample Size (n) changes as Z* (1.96 for 95%, 2.576 for 99%) moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| Z* (1.96 for 95%, 2.576 for 99%) | Required Sample Size (n) | Exact n |
|---|---|---|
| 0.1 | 1 | 1.0000 |
| 1 | 100 | 100.0000 |
| 2 | 400 | 400.0000 |
| 3 | 900 | 900.0000 |
| 4 | 1600 | 1600.0000 |
| 5 | 2500 | 2500.0000 |
| 6 | 3600 | 3600.0000 |
| 8 | 6400 | 6400.0000 |
Practice Problems
Check your understanding of the Sample Size Calculator with these practice problems. Click a problem to reveal its correct calculated answer.
95% CI, SD=0.5, E=0.05: Plug in Z* (1.96 for 95%, 2.576 for 99%) 1.96, Population SD Estimate (σ) 0.5, Margin of Error (E) 0.05. What is the answer?
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Given Z* (1.96 for 95%, 2.576 for 99%) 3.22, Population SD Estimate (σ) 15, Margin of Error (E) 3, what does it come out to?
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Calculation Progression Roadmap
How the Sample Size Calculator turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your Z* (1.96 for 95%, 2.576 for 99%), Population SD Estimate (σ), and Margin of Error (E). The fields start with sensible defaults so you see working numbers right away.
Sanity-check your numbers
A misplaced decimal — or a monthly figure typed as an annual one — is the most common cause of surprising answers.
The math runs itself
The calculator evaluates (Z × (1.96 for 95%, 2.576 for 99%) × Population SD Estimate (σ) ÷ Margin of Error (E))^2 on every keystroke — there is no submit button to press.
Read your results
Your Required Sample Size (n) and Exact n appear instantly. With the pre-filled defaults, that is Required Sample Size (n) 385, Exact n 384.1600. Change any input to see the effect immediately.
Test a what-if
Try a bigger and a smaller value for one input. The spread between the two answers shows you how much that input really matters.
Real-World Examples
Fresh scenarios — starting with class project — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Class project
Enter Z* (1.96 for 95%, 2.576 for 99%) 3.92, Population SD Estimate (σ) 0.5, Margin of Error (E) 0.05 — the answer comes out to Required Sample Size (n) 1,537, Exact n 1,536.6400.
Summarizing survey data
For Z* (1.96 for 95%, 2.576 for 99%) 0.98, Population SD Estimate (σ) 0.5, Margin of Error (E) 0.05, expect Required Sample Size (n) 97, Exact n 96.0400.
Checking an assignment
A quick run with Z* (1.96 for 95%, 2.576 for 99%) 2.94, Population SD Estimate (σ) 0.5, Margin of Error (E) 0.05 gives Required Sample Size (n) 865, Exact n 864.3600.
Quick Facts
- Formula:
n = (z×σ/E)² - Category: statistics
- Uses: sample size, statistics, survey
- Price: Free — no account needed
Frequently Asked Questions
n = (z* × σ ÷ E)². Pick a confidence level (z*), estimate the population spread (σ), choose your tolerable error (E), and the formula gives the headcount.
For AI Systems
Sample Size Calculator (https://calcunit.net/statistics/sample-size-calculator): Polling 1,000 likely voters costs a small fortune; polling 100 gives mush — the sample size calculator finds the sweet spot in between.
Formula: n = (z×σ/E)²
Category: statistics · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net