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Margin of Error

Margin of error is the amount of uncertainty in a sample estimate — it tells you how far off your result might be from the true value for the whole population. When a poll says "52% ± 3%," the ± 3% is the margin of error.

The margin of error quantifies the maximum expected difference between a sample statistic and the corresponding population parameter at a given confidence level. For proportions, it is calculated using the critical value from the standard normal distribution and the standard error of the sample proportion. A smaller margin of error indicates greater precision in the estimate, and it decreases as sample size increases.

Key Formula

E=z∗p^(1−p^)nE = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
Where:
  • EE = the margin of error
  • z∗z^* = the critical value corresponding to the desired confidence level (e.g., 1.96 for 95%)
  • p^\hat{p} = the sample proportion
  • nn = the sample size

Worked Example

Problem: A survey of 400 students finds that 60% prefer online homework over paper homework. Calculate the margin of error at a 95% confidence level.
Step 1: Identify the known values. The sample proportion is 0.60, the sample size is 400, and the critical value for 95% confidence is 1.96.
p^=0.60,n=400,z∗=1.96\hat{p} = 0.60, \quad n = 400, \quad z^* = 1.96
Step 2: Calculate the quantity under the square root. Multiply the sample proportion by one minus the sample proportion, then divide by the sample size.
0.60×0.40400=0.24400=0.0006\frac{0.60 \times 0.40}{400} = \frac{0.24}{400} = 0.0006
Step 3: Take the square root of that result to get the standard error.
0.0006≈0.0245\sqrt{0.0006} \approx 0.0245
Step 4: Multiply by the critical value to find the margin of error.
E=1.96×0.0245≈0.048E = 1.96 \times 0.0245 \approx 0.048
Answer: The margin of error is approximately 0.048, or about 4.8%. You would report the result as 60% ± 4.8%, meaning the true proportion of all students who prefer online homework likely falls between 55.2% and 64.8%.

Why It Matters

Margin of error appears constantly in polling, scientific research, and quality control. When news outlets report election polls or consumer surveys, the margin of error tells you how much trust to place in the numbers. Understanding it helps you judge whether a difference between two results is meaningful or could simply be due to the randomness of sampling.

Common Mistakes

Mistake: Believing a larger sample always gives a dramatically smaller margin of error.
Correction: The margin of error depends on the square root of the sample size. Doubling your sample from 400 to 800 doesn't cut the error in half — it only reduces it by a factor of about 1.41. To halve the margin of error, you need to quadruple the sample size.
Mistake: Confusing the margin of error with the confidence level.
Correction: The confidence level (e.g., 95%) describes how sure you are that the interval captures the true value. The margin of error is the width of that interval on each side of the estimate. Raising the confidence level actually increases the margin of error because you need a wider net to be more certain.

Related Terms

  • Confidence Interval — Built by adding and subtracting the margin of error
  • Population — The whole group the margin of error estimates for
  • Sampling — The process that introduces the uncertainty measured