Margin of Error Calculator: Precise Statistical Confidence Intervals
A margin-of-error tells you how far your sample result can stray from the real population value. Plug in sample size, deviation or proportion, and a confidence level—the calculator returns the ± error and a full confidence interval. Larger samples shrink the error: at n = 1 000 and p = 0.5 the 95 % MOE ≈ ±3.1 % (U.S. Census Bureau, 2022).
Margin of Error Calculator
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How to use the tool
Enter Sample Size (n)
Type the number of observations. Example 1: 750. Example 2: 1 200.
Choose Data Type
• Leave the “Proportion” box unchecked for means (continuous data).
• Check it for proportions (yes/no data).
If Mean Analysis
Standard Deviation (s): e.g., 12 or 22.5
Confidence Level: 80 %, 90 %, 95 %, or 99 %.
Optional Mean (x̄): e.g., 105.4 or 98.0 to build a confidence interval.
If Proportion Analysis
Either Sample Proportion (p) (e.g., 0.42 or 0.68) or Successes (x) (e.g., 315 out of 750).
Press “Calculate” to view Margin of Error, Critical Value and (when possible) Confidence Interval.
Formulas used
Mean data (n < 30): $$ME = t_{α/2,\,df}\,*\,\frac{s}{\sqrt{n}}$$
Mean data (n ≥ 30): $$ME = Z_{α/2}\,*\,\frac{s}{\sqrt{n}}$$
The margin of error is the maximum expected difference between your sample statistic and the true population value (U.S. Census Bureau, 2022).
Why does bigger n shrink the margin?
A larger sample reduces the standard error by 1/√n, directly shrinking the margin of error (Lohr, 2019).
Which critical value does the tool pick?
For n < 30 it uses Student’s t; for n ≥ 30 it switches to the normal Z value (NIST Handbook 151, 2022).
What if I know successes but not p?
The calculator divides successes by n to estimate p, then applies the proportion formula accurately.
Can margin of error be zero?
No. Only a census of the entire population eliminates sampling error; surveys always carry some uncertainty (OECD Glossary, 2017).
How do I report a confidence interval?
State “estimate ± MOE” or present both interval endpoints, citing the confidence level, e.g., “±2.2 % at 90 % CL.”
Why is p = 0.5 the ‘worst case’?
When p = 0.5, p(1-p) peaks at 0.25, generating the largest variance and widest margin (Fowler, 2014).
What standard quote backs these formulas?
“For large samples the distribution of the sample mean approaches normality” (Central Limit Theorem, Casella & Berger 2002).
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