Margin of Error Calculator

What is Margin of Error?

The margin of error is a statistical measure of the amount of random sampling error in a survey's results. It represents the range of values above and below the sample statistic in a confidence interval.

Formula and Its Components

The formula for margin of error is:

\[MoE = z \times \sqrt{\frac{p(1-p)}{n}} \times \sqrt{\frac{N-n}{N-1}}\]

Where:

• \(z\) is the z-score (typically 1.96 for a 95% confidence level)
• \(p\) is the sample proportion (if unknown, use 0.5 for maximum margin of error)
• \(n\) is the sample size
• \(N\) is the population size

Calculation Steps

1. Determine the z-score based on the desired confidence level (usually 1.96 for 95% confidence).
2. Calculate \(\sqrt{\frac{p(1-p)}{n}}\), which represents the standard error of the sample proportion.
3. Calculate \(\sqrt{\frac{N-n}{N-1}}\), which is the finite population correction factor.
4. Multiply these components together to get the margin of error.
5. Convert to a percentage by multiplying by 100 if desired.

Example Calculation

Let's calculate the margin of error for a survey with:

• Sample size (n) = 1000
• Probability (p) = 0.5 (maximum variability)
• Population size (N) = 100,000
• Confidence level = 95% (z = 1.96)

Plugging into our formula:

\[MoE = 1.96 \times \sqrt{\frac{0.5(1-0.5)}{1000}} \times \sqrt{\frac{100000-1000}{100000-1}}\]

\[MoE = 1.96 \times 0.0158 \times 0.9950 = 0.0309 = 3.09\%\]

Visual Representation

This graph illustrates the concept of margin of error. The blue curve represents the sampling distribution, and the red lines show the margin of error around the sample statistic.