Margin of Error Calculator (Population Proportion) for 95% Confidence with 400 Sample Size
Common survey scenario using a 95% confidence level with a sample size of 400 and maximum variability (p = 0.5).
Calculates the margin of error for a population proportion in surveys or polls using the standard statistical formula. Enter your Z-score, Sample Proportion (p), Sample Size (n) to get an instant margin of error. Formula: z * sqrt((p * (1 - p)) / n).
Margin of Error
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How It Works
How It Works
This calculator measures how much uncertainty there is in a survey result. It uses the sample proportion (p), the sample size (n), and the chosen confidence level (z-score) to estimate how far the true population value could be from the sample result.
The formula multiplies the z-score by the square root of (p × (1 − p)) divided by n. This calculation gives the margin of error in decimal form, showing the possible variation above or below the sample proportion.
- The z-score reflects your chosen confidence level (e.g., 1.96 for 95%).
- The sample proportion (p) is the survey result in decimal form.
- The sample size (n) affects accuracy — larger samples reduce error.
- The square root part measures natural variation in the data.
- The final result is a single decimal value.
Understanding the Results
The margin of error tells you how much the survey result may differ from the true population proportion. It shows the range of likely values around your sample result.
To interpret it, add and subtract the margin of error from your sample proportion. This creates a confidence interval, which represents the range where the true population value is likely to fall.
- A smaller margin of error means more precise results.
- Larger sample sizes produce smaller margins of error.
- Higher confidence levels (larger z-scores) increase the margin of error.
- The result is shown in decimal form (e.g., 0.049 = 4.9%).
- Use it to build a confidence interval around your sample proportion.
Frequently Asked Questions
What does the margin of error represent in a survey or poll?
The margin of error indicates how much the sample proportion may differ from the true population proportion. It provides a range around your sample result to estimate the likely true value. For example, if your sample proportion is 0.50 and the margin of error is 0.049, the true population proportion is likely between 0.451 and 0.549 at the chosen confidence level.
When should I use this margin of error calculator?
Use this calculator when you have survey or poll results expressed as a proportion (decimal form) and want to estimate the uncertainty around that result. It is appropriate for large random samples where you know the sample size, sample proportion, and desired confidence level. It is commonly used in opinion polls, market research, and academic studies.
How do I choose the correct Z-score?
The Z-score depends on your desired confidence level. For a 90% confidence level, use 1.645; for 95%, use 1.96; and for 99%, use 2.576. A higher Z-score increases the margin of error because it reflects greater confidence in capturing the true population value.
Why must the sample proportion (p) be entered as a decimal?
The formula requires the sample proportion in decimal form because it performs mathematical operations directly on that value. For example, 50% should be entered as 0.5, and 25% should be entered as 0.25. Entering percentages instead of decimals will produce incorrect results.
How does sample size affect the margin of error?
A larger sample size reduces the margin of error because it provides more information about the population. For instance, increasing the sample size from 400 to 1,000 (while keeping the same Z-score and proportion) will produce a smaller margin of error. This is why larger surveys generally yield more precise estimates.
Why is the margin of error largest when p is 0.5?
The formula includes the term p × (1 − p), which reaches its maximum value when p equals 0.5. This means variability is highest at 50%, producing the largest possible margin of error for a given sample size and confidence level. Researchers sometimes use p = 0.5 when planning studies to estimate the most conservative (largest) margin of error.
Disclaimer
This calculator provides estimates for informational purposes only. It is not professional advice. Verify results with a qualified professional. Disclaimer.