The two most common Margin of Error formulas

“Margin of error” is a family name for several closely related formulas. Which one you use depends on what you measured. In everyday polling, you’re usually estimating a proportion (a percent), so the proportion formula dominates headlines. In experiments and measurements, you may be estimating a mean (an average), which uses a different standard error.

Suppose you surveyed n people and x of them answered “yes”. Your sample proportion is p = x / n (often reported as a percent). The standard error (the typical sampling fluctuation) for a proportion is:

SE(p) = √( p(1 − p) / n )

To convert that into a “confidence interval wiggle room”, you multiply by a critical value from the normal distribution. For 95% confidence, that critical value is approximately z = 1.96. So the margin of error becomes:

MOE = z · √( p(1 − p) / n )

If you enter p as a percent (like 48), the calculator internally converts it to 0.48, computes MOE, then converts the MOE back into percentage points (like ±3.1%).

If you measured a continuous variable—height, weight, response time, temperature—you often summarize results with a mean. The sampling variability of a mean depends on how spread out the data are. That spread is captured by the standard deviation (σ). The standard error for a mean is:

SE(x̄) = σ / √n

And the margin of error at confidence level based on z is:

MOE = z · ( σ / √n )

If you also supply your sample mean x̄, the calculator displays the confidence interval [x̄ − MOE, x̄ + MOE]. If you don’t supply x̄, you still get MOE, which is the “±” uncertainty around whatever mean you plan to report.

Step-by-step (what the calculator does)

The calculator follows a simple pipeline. This makes it easy to audit, explain in class, or copy into a report. Here’s the exact flow.

• Proportion: assumes a Bernoulli “yes/no” style result and uses p(1−p).
• Mean: assumes a continuous outcome and uses σ/√n.

For most quick work, z-scores are standard: 90% → 1.645, 95% → 1.96, 99% → 2.576. If you choose “Custom z-score,” you can use anything—like 1.28 for ~80% confidence.

• Proportion: SE = √( p(1−p)/n )
• Mean: SE = σ/√n

Final step is just MOE = z · SE. The confidence interval is “estimate ± MOE.”

If you enter a target MOE, the calculator estimates how large your sample should be to hit that precision, holding z (confidence) fixed.

• Proportion sample size: n ≥ z² · p(1−p) / MOE²
• Mean sample size: n ≥ ( z·σ / MOE )²

If you don’t know p yet for a proportion, the most conservative choice is p = 0.5, because p(1−p) is largest there, producing the largest required n.

Realistic examples you can copy/paste

Examples are where margin of error becomes intuitive. Here are three scenarios—one classic poll, one small sample, and one mean measurement. Try them in the calculator and compare.

A poll surveys n = 1,000 people. 48% say they support Candidate A. At 95% confidence, use z = 1.96.

• Proportion: p = 0.48
• SE: √(0.48·0.52/1000) ≈ 0.0158
• MOE: 1.96·0.0158 ≈ 0.031 → ±3.1%
• Confidence interval: 44.9% to 51.1%

Interpretation: “48% ± 3.1%” means that, under the sampling model, the true support is likely in that interval. Notice how easy it is for “48” to be basically “a tie” after you apply uncertainty.

Now keep p = 0.48 but reduce to n = 100 (like a quick classroom poll).

• SE: √(0.48·0.52/100) ≈ 0.050
• MOE: 1.96·0.050 ≈ 0.098 → ±9.8%
• Confidence interval: 38.2% to 57.8%

That wide interval explains why small samples often fail to “settle” an argument. The result is not useless, but it’s more like a rough signal than a precise estimate.

You measure the time (in seconds) for users to complete a task. You have n = 64 users, a sample mean x̄ = 42.0, and estimated standard deviation σ ≈ 8.0. At 95% confidence:

• SE: σ/√n = 8/8 = 1
• MOE: 1.96·1 = 1.96
• Confidence interval: 42.0 ± 1.96 → [40.04, 43.96]

Interpretation: the mean task time is estimated pretty tightly because n is decent and σ is moderate.

Frequently Asked Questions

• Is “margin of error” the same as “confidence interval”?

  Margin of error is the “±” part. A confidence interval is the full range: estimate − MOE to estimate + MOE. People often say “MOE” because it’s easy to quote, but the interval is what you should interpret.

• Why do most polls use 95% confidence?

  It’s a common convention that balances caution and practicality. Higher confidence (like 99%) increases z and makes MOE bigger. Lower confidence makes MOE smaller but increases the chance the interval misses the true value. 95% is a “good middle ground” in many fields.

• Does MOE include “bias” or question wording issues?

  No. The classic MOE formula mainly captures sampling variability. If the sample is nonrandom, response rates are low, or the question is leading, the true error can be much larger than the MOE.

• What if I don’t know the true proportion yet?

  If you’re planning a survey, use p = 0.5 for a conservative sample size calculation. It produces the largest MOE (worst-case), so your sample won’t be “too small” for any real p.

• Can I use this for very small samples?

  You can, but be cautious. The proportion formula uses a normal approximation that can be rough for small n or extreme p values (near 0% or 100%). The output is still a useful estimate, but if you need high accuracy, consider exact or adjusted intervals (like Wilson) in a stats package.

• What’s a “good” margin of error?

  It depends on the stakes. For national political polls, ±3% is common. For high-stakes decisions (medical, safety, finance), you may want much tighter intervals—and you might need larger samples or better measurement, not just a formula.