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A larger sample can make a poll's estimate more precise, but the gain is slower than the increase in interviews. In a simple model with the same estimated proportion and confidence level, quadrupling the sample size halves the approximate margin of sampling error. Doubling the sample size does not halve it.
That calculation concerns random sampling variation. It does not show that a larger poll has better questions, reaches the right people or predicts an election winner. Start by checking how the sample was selected, then read its size alongside the method.
Define a comparable sample before calculating
Imagine three separate, hypothetical probability samples from the same very large population. Each asks the same binary question, “Would you choose option A rather than option B?”, under the same conditions. Each person gives one yes or no response. There are no missing responses or undecided categories in this exercise.
Assume simple random sampling, equal weights and a sampling fraction small enough to ignore the finite-population correction. Treat the responses as independent for this large-population approximation. There is no clustering, weighting or adjustment for an effective sample size. These assumptions describe an illustrative model, not the design of an actual opinion poll.
We assign the three samples sizes of 400, 1,600 and 6,400, with yes counts of 200, 800 and 3,200. Each observed yes proportion is therefore 0.50, or 50%. The rows are separate examples, not successive updates to one survey or evidence of a stable real opinion.
Calculate the approximate margin in the right unit
Let n be the number of responses and p̂ the observed yes proportion, entered as a fraction. The familiar normal-approximation, or Wald, margin is E = z√[p̂(1 − p̂)/n]. Using the rounded cutoff z = 1.96 gives an approximate two-sided 95% interval of p̂ − E to p̂ + E. NIST lists this common expression while explaining alternatives with better behavior. NIST: confidence intervals for a proportion.
Here, p̂(1 − p̂) = 0.50 × 0.50 = 0.25. For 400 responses, √(0.25/400) = 0.025, and 1.96 × 0.025 = 0.049. A proportion margin of 0.049 is 4.9 percentage points. Subtracting and adding it to 50% gives 45.1% to 54.9%.
| Responses n | Yes / no counts | Observed yes proportion | Estimated standard error, as a proportion | Margin E, as a proportion | Margin in percentage points | Approximate 95% interval |
|---|---|---|---|---|---|---|
| 400 | 200 / 200 | 200/400 = 0.50 | √(0.25/400) = 0.025 | 1.96 × 0.025 = 0.049 | ±4.9 points | 45.1%–54.9% |
| 1,600 | 800 / 800 | 800/1,600 = 0.50 | √(0.25/1,600) = 0.0125 | 1.96 × 0.0125 = 0.0245 | ±2.45 points | 47.55%–52.45% |
| 6,400 | 3,200 / 3,200 | 3,200/6,400 = 0.50 | √(0.25/6,400) = 0.00625 | 1.96 × 0.00625 = 0.01225 | ±1.225 points | 48.775%–51.225% |
The margin is the distance from the center to either endpoint. The full interval width is twice that margin. In the first row, the margin is 4.9 points and the width is 9.8 points. Neither is a relative percentage change in the estimated 50% response.
Use the square-root relationship, not a linear shortcut
With p̂ and z held fixed, the formula makes the margin proportional to 1/√n. Moving from 400 to 1,600 responses multiplies n by four and divides E by two: 4.9 points becomes 2.45 points. Moving from 1,600 to 6,400 repeats the same trade-off.
Doubling n instead divides the margin by √2. For example, with the same observed 0.50 and cutoff, increasing 400 responses to 800 gives 4.9/√2 ≈ 3.465 percentage points, not 2.45 points. Halving the margin requires four times the responses in this calculation.
This comparison deliberately holds the response proportion constant. In real samples the observed proportion can change too. The sample-size ratio alone then does not reproduce the complete change in the estimated margin. Other changes in the sampling design or confidence level also break this simple comparison.
Separate sampling precision from other sources of error
Pew Research Center explains that sample design, weighting and subgroup sizes matter alongside the overall respondent count. A whole-sample margin cannot simply be assigned to a smaller subgroup. Pew: understanding polling margins.
| Issue | Effect of increasing n alone | What to check separately |
|---|---|---|
| Random sampling precision | Reduces the displayed margin under the fixed assumptions | Confidence level, response proportion and actual sampling design |
| Coverage and nonresponse bias | Does not automatically remove systematic error | Who can be selected and who actually responds |
| Question wording | Does not correct a misleading question | Exact wording and response choices |
| Weighting or clustering | Raw n alone does not determine precision | The survey's design-aware uncertainty calculation |
| Turnout or an election win | Does not turn a response share into a winning chance | The target population, future event and forecast assumptions |
Sampling error bars leave out errors such as nonresponse bias and differences in how questions are understood. More responses can produce a tightly estimated answer to a poorly measured question. Pew's explanation of error bars distinguishes these limitations from the precision shown by the bars. Pew: understanding error bars.
Keep the confidence level and approximation in view
The nominal 95% level concerns repeated use of a procedure under its assumptions. It does not assign a 95% posterior probability to the fixed population proportion after an interval is calculated, and it is not a 95% chance that a candidate wins. Our normal approximation also does not deliver exact 95% coverage for every population proportion and sample size.
The Wald interval is used here to expose the square-root relationship, not as a universal polling calculator or NIST's preferred general method. NIST discusses Wilson intervals and notes problems with the common symmetric expression, including impossible bounds and inadequate accuracy for small samples or sparse counts. Near-zero or near-one proportions need particular care. The tidy 50% examples do not remove those limitations. NIST: proportion-interval methods.
A margin for one proportion is also not automatically the uncertainty for a difference between two estimates. For chart comparisons, overlapping error bars do not supply a universal significant-versus-insignificant rule. Read the relevant analysis rather than deciding from overlap alone. Pew: comparing error bars.
When reading an actual poll, look for its question, population, fieldwork dates, sample selection, weights, subgroup count and stated uncertainty method. Use that published method rather than inserting raw n into this teaching formula. For a different question—why a survey response share differs from a winning chance—read polls and election odds. To distinguish an average amount from an event probability, see expected value and probability.
Common questions
Does doubling the sample size halve the margin of error?
No. With the same observed proportion, confidence cutoff and simple design, doubling n divides the approximate margin by √2. Quadrupling n halves it. This is why 400 to 1,600 responses changes the example's margin from 4.9 to 2.45 percentage points.
Is a 4.9-point margin the same as a 4.9% relative change?
No. The first example adds or subtracts 4.9 percentage points from 50%, giving 45.1%–54.9%. The margin is a distance on the percentage scale, not a relative change calculated by multiplying 50% by 4.9%.
Can I use the full poll's margin for a small subgroup?
Not automatically. The subgroup has its own respondent base and may have different weighting or design effects. Find the uncertainty calculation for that subgroup; the total number interviewed does not by itself describe its precision.
Does a smaller margin mean the poll is free of bias?
No. A sampling margin does not measure every source of error. Missing parts of the population, different response patterns or misleading wording can affect an estimate even when a large sample gives it a small sampling margin.
Can I apply this formula to any online poll?
No. The example assumes a particular probability-sampling model with equal weights and negligible sampling fraction. An opt-in response count alone does not establish those conditions. Consult the poll's recruitment and estimation methods; do not attach a conventional sampling margin solely because many people answered.
Explore HypaForecast
Optional navigation to the general market page and the BTC page descriptions. These links are separate from the fictional polling examples and do not confirm availability or eligibility.
Sources and method
The three samples and their yes/no counts are original hypothetical examples, not poll data. The teaching calculation uses the common symmetric normal-approximation (Wald) interval listed by NIST, with the rounded cutoff 1.96, observed proportion 0.50, independent equal-weight binary responses from a very large population and negligible sampling fraction. It is not presented as NIST’s preferred general interval or an exact-coverage polling calculator. NIST’s Wilson and small-count cautions, Pew’s 2016 discussion of design and subgroups, and Pew’s 2025 discussion of error bars inform the limits. The examples omit finite-population, weighting, clustering and effective-sample-size calculations. No current poll, political candidate, market price or election forecast is assessed.
HypaForecast publishes educational guides and operates a prediction market. This guide explains a simplified sampling calculation; it does not recommend a transaction or establish a winning chance, current offering or personal eligibility.
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