Margin of Error in Polls

What is margin of error?

The margin of error (MOE) is the "±" in poll results: "52% ± 3.1% support the proposal." It defines the width of the confidence interval.

$MOE = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Step-by-step

Given: $\hat{p} = 0.52$, $n = 1000$, 95% confidence ($z^* = 1.96$).

Step 1 — Standard error of proportion:

$SE = \sqrt{\frac{0.52 \times 0.48}{1000}} = \sqrt{\frac{0.2496}{1000}} = \sqrt{0.0002496} = 0.01580$

Step 2 — Margin of error:

$MOE = 1.96 \times 0.01580 = 0.0310 = 3.1\%$

Step 3 — Confidence interval: $52\% \pm 3.1\% = (48.9\%, 55.1\%)$

Interpretation

We're 95% confident that the true population support is between 48.9% and 55.1%. Since this range includes 50%, we cannot conclude the proposal has majority support with 95% confidence.

How to reduce MOE

• Increase sample size: $n = 4000$ would cut MOE in half.
• Accept lower confidence: 90% CI has smaller MOE than 95%.

Try it in the visualization

Enter poll results. The MOE is computed and shown as an interval. Adjust $n$ to see the MOE shrink. The 50% line shows whether the result is statistically significant.