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Calculate the margin of error and confidence interval for surveys, polls, and research studies using sample size, population proportion, and confidence level.
MOE = z × √(p(1 − p) / n)
z-scores: 1.645 (90%), 1.96 (95%), 2.576 (99%) | p = population proportion | n = sample size
The table below shows the margin of error for various sample sizes at a 95% confidence level with a population proportion of 50% (the most conservative estimate). Use this as a quick reference when planning surveys or interpreting poll results.
| Sample Size (n) | Margin of Error (±) | Typical Use Case |
|---|---|---|
| 10 | 31.01% | Very small pilot study |
| 25 | 19.60% | Classroom survey |
| 50 | 13.86% | Small focus group study |
| 75 | 11.32% | Preliminary research |
| 100 | 9.80% | Small internal survey |
| 150 | 8.00% | Department-level poll |
| 200 | 6.93% | Customer feedback survey |
| 250 | 6.20% | Product testing panel |
| 300 | 5.66% | Regional poll |
| 400 | 4.90% | Standard market research |
| 500 | 4.38% | Mid-size opinion poll |
| 750 | 3.58% | Large-scale survey |
| 1,000 | 3.10% | National political poll |
| 1,500 | 2.53% | Major media poll |
| 2,000 | 2.19% | Large national survey |
| 5,000 | 1.39% | Government census sample |
| 10,000 | 0.98% | Large-scale academic study |
| 50,000 | 0.44% | Massive population study |
* All values calculated at 95% confidence level with p = 0.5
Margin of error (MOE) is a fundamental concept in statistics that measures the range of uncertainty around a sample estimate. When researchers conduct surveys or polls, they collect data from a subset (sample) of a larger group (population) because surveying every individual is often impractical or impossible. The margin of error quantifies how much the sample results might differ from the true population values due to random sampling variation.
The concept is most commonly encountered in political polling, where news outlets report results like "Candidate A leads with 52% support, with a margin of error of plus or minus 3 percentage points." This means the true level of support in the entire population is likely between 49% and 55%. The margin of error is always paired with a confidence level, typically 95%, which indicates how often the method produces intervals that capture the true value.
Margin of error depends on three key factors: the sample size (larger samples produce smaller margins), the population proportion (values near 50% produce the largest margins), and the desired confidence level (higher confidence requires wider margins). Understanding margin of error is essential for anyone designing surveys, interpreting research results, reading poll data, conducting quality control testing, or making business decisions based on sample data. It serves as the bridge between sample statistics and population-level conclusions, providing a transparent measure of statistical precision.
MOE = z × √(p(1 − p) / n)
z = z-score for the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%)
p = estimated population proportion (as a decimal, e.g., 0.5 for 50%)
n = sample size (number of respondents or observations)
A polling firm surveys 1,000 voters at a 95% confidence level. What is the margin of error using p = 0.5?
Result: MOE = ±3.10%
If 52% favor Candidate A, the confidence interval is 48.90% to 55.10%.
A company surveys 400 customers at 95% confidence. Prior data suggests 80% satisfaction (p = 0.8).
Result: MOE = ±3.92%
Satisfaction is between 76.08% and 83.92% with 95% confidence.
A researcher surveys 200 respondents at a 99% confidence level with p = 0.5.
Result: MOE = ±9.11%
At 99% confidence with only 200 respondents, the margin is quite wide.
For a quick 95% confidence estimate with p = 0.5, use the formula: MOE ≈ 1 / √n. For example, with n = 400: MOE ≈ 1/√400 = 1/20 = 0.05 = 5%. The exact answer (4.90%) is very close. This shortcut works because 1.96 × √(0.25) = 0.98, which is approximately 1.
Compare how different confidence levels affect the margin of error for the same sample size. Higher confidence levels produce wider margins.
| Sample Size | 90% Confidence | 95% Confidence | 99% Confidence |
|---|---|---|---|
| 100 | ±8.22% | ±9.80% | ±12.88% |
| 250 | ±5.20% | ±6.20% | ±8.15% |
| 500 | ±3.68% | ±4.38% | ±5.76% |
| 750 | ±3.00% | ±3.58% | ±4.70% |
| 1,000 | ±2.60% | ±3.10% | ±4.07% |
| 1,500 | ±2.12% | ±2.53% | ±3.33% |
| 2,000 | ±1.84% | ±2.19% | ±2.88% |
This table demonstrates how the population proportion affects the margin of error. Proportions closer to 50% produce the largest margins, while extreme proportions near 0% or 100% result in smaller margins.
| Proportion (p) | Margin of Error | Confidence Interval |
|---|---|---|
| 10% | ±1.86% | 8.14% to 11.86% |
| 20% | ±2.48% | 17.52% to 22.48% |
| 30% | ±2.84% | 27.16% to 32.84% |
| 40% | ±3.04% | 36.96% to 43.04% |
| 50% | ±3.10% | 46.90% to 53.10% |
| 60% | ±3.04% | 56.96% to 63.04% |
| 70% | ±2.84% | 67.16% to 72.84% |
| 80% | ±2.48% | 77.52% to 82.48% |
| 90% | ±1.86% | 88.14% to 91.86% |
Margin of error is critical for interpreting political polls. A 2-point lead within a 3-point margin of error means the race is statistically too close to call. Understanding MOE prevents overconfidence in poll results and helps media outlets report findings responsibly.
Researchers use margin of error to determine how many participants their study needs. A study with too-wide margins may fail to detect meaningful effects, while understanding MOE helps reviewers assess the reliability and generalizability of published findings.
Manufacturers use margin of error when testing product batches. Rather than inspecting every item, they test a sample and use MOE to estimate the defect rate for the entire batch. This approach saves time and money while maintaining quality standards.
Companies conduct market research surveys to gauge customer preferences and buying intent. Knowing the margin of error helps decision-makers understand how much they can trust survey results before committing budgets to product launches or advertising campaigns.
If you don't know the expected population proportion before your survey, use p = 0.5 (50%). This gives the largest possible margin of error, ensuring your estimate is conservative. Once you have preliminary results, you can recalculate with the actual observed proportion for a tighter interval.
Margin of error only measures random sampling error. It does not account for non-response bias, question wording effects, coverage gaps, or interviewer influence. A survey with a small MOE can still be highly inaccurate if these other sources of error are present. Always consider total survey error, not just MOE.
Work backwards from your desired margin of error to determine the needed sample size using n = (z² × p(1−p)) / MOE². For example, to achieve ±3% at 95% confidence, you need n = (1.96² × 0.25) / 0.03² = 1,067 respondents. Planning ahead avoids collecting too little or too much data.
Going from 100 to 400 respondents cuts the MOE roughly in half (from 9.8% to 4.9%), but going from 1,000 to 4,000 only reduces it from 3.1% to 1.5%. Each additional respondent provides less improvement. Consider the cost-benefit ratio before pursuing extremely large sample sizes.
If your overall survey has 1,000 respondents with ±3.1% MOE, a subgroup of 200 women in that sample has a much larger MOE of ±6.93%. Always recalculate margin of error for each subgroup you analyze, as the effective sample size shrinks and the margin widens accordingly.
When your sample represents more than 5% of the total population, apply the finite population correction factor: multiply MOE by √((N−n)/(N−1)), where N is the population size. For example, surveying 200 out of 500 employees gives a smaller MOE than the standard formula suggests because you've sampled a large portion of the population.
Margin of error (MOE) is a statistic that quantifies the amount of random sampling error in a survey or poll result. It tells you how much the results from your sample might differ from the true population value. For example, if a poll shows 60% support with a margin of error of plus or minus 3%, the true population support likely falls between 57% and 63% at the stated confidence level.
Margin of error is calculated using the formula MOE = z times the square root of p(1-p)/n, where z is the z-score corresponding to your confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), p is the population proportion (use 0.5 if unknown for maximum MOE), and n is the sample size. Multiply the result by 100 to express it as a percentage.
A margin of error of plus or minus 5% or less is generally considered acceptable for most surveys and polls. Political polls typically aim for plus or minus 3%, while academic research may require plus or minus 2% or less. The acceptable margin depends on the stakes involved and the precision needed. Market research might tolerate plus or minus 5-10%, while clinical trials demand much tighter margins.
The 95% confidence level has become the standard in most fields because it strikes a practical balance between precision and reliability. It means that if you repeated the survey 100 times, approximately 95 of those surveys would produce confidence intervals that contain the true population parameter. This convention originated in the early 20th century with statistician Ronald Fisher and has since become the default in social sciences, business research, and polling.
Sample size and margin of error have an inverse square root relationship. As sample size increases, margin of error decreases, but with diminishing returns. Doubling your sample size does not halve the margin of error. You need to quadruple the sample size to cut the margin of error in half. For example, increasing from 100 to 400 respondents reduces the MOE from about 9.8% to about 4.9% at 95% confidence.
Margin of error is the plus-or-minus value added to and subtracted from a sample statistic, while the confidence interval is the resulting range. For example, if 52% of respondents favor a candidate with a margin of error of plus or minus 3%, the margin of error is 3% and the confidence interval is 49% to 55%. The margin of error is half the width of the confidence interval.
Using p = 0.5 produces the maximum possible margin of error for any given sample size and confidence level. This is because the expression p(1-p) reaches its maximum value of 0.25 when p = 0.5. Pollsters use this conservative assumption when the true proportion is unknown before the survey, ensuring the margin of error is never underestimated regardless of the actual result.
Margin of error can only be zero if you survey the entire population (a census) rather than a sample. In practice, any sample-based estimate will have some margin of error. However, very large sample sizes can reduce the margin of error to negligibly small values. For example, a sample of 100,000 at 95% confidence yields a margin of error of about plus or minus 0.31%, which is close to zero for practical purposes.
No, margin of error only accounts for random sampling error, which is the natural variation that occurs when surveying a subset of the population. It does not account for non-sampling errors such as biased question wording, non-response bias, coverage error (not reaching certain groups), social desirability bias, or measurement error. These non-sampling errors can often be larger than the stated margin of error.