Margin of Error vs. Confidence Interval: What’s the Difference?


Often in statistics we use confidence intervals to estimate the value of a population parameter with a certain level of confidence.

Every confidence interval takes on the following form:

Confidence Interval = [lower bound, upper bound]

The margin of error is equal to half the width of the entire confidence interval.

For example, suppose we have the following confidence interval for a population mean:

95% confidence interval = [12.5, 18.5]

The width of the confidence interval is 18.5 – 12.5 = 6. The margin of error is equal to half the width, which would be 6/2 = 3.

The following examples show how to calculate a confidence interval along with the margin of error for several different scenarios.

Example 1: Confidence Interval & Margin of Error for Population Mean

We use the following formula to calculate a confidence interval for a population mean:

Confidence Interval = x  +/-  z*(s/√n)

where:

  • xsample mean
  • z: the z-critical value
  • s: sample standard deviation
  • n: sample size

Example: Suppose we collect a random sample of dolphins with the following information:

  • Sample size n = 40
  • Sample mean weight x = 300
  • Sample standard deviation s = 18.5

We can plug these numbers into the Confidence Interval Calculator to find the 95% confidence interval:

The 95% confidence interval for the true population mean weight of turtles is [294.267, 305.733].

The margin of error would be equal to half the width of the confidence interval, which is equal to:

Margin of Error: (305.733 – 294.267) / 2 = 5.733.

Example 2: Confidence Interval & Margin of Error for Population Proportion

We use the following formula to calculate a confidence interval for a population proportion:

Confidence Interval = p  +/-  z*(√p(1-p) / n)

where:

  • p: sample proportion
  • z: the chosen z-value
  • n: sample size

Example: Suppose we want to estimate the proportion of residents in a county that are in favor of a certain law. We select a random sample of 100 residents and ask them about their stance on the law. Here are the results:

  • Sample size n = 100
  • Proportion in favor of law p = 0.56

We can plug these numbers into the Confidence Interval for a Proportion Calculator to find the 95% confidence interval:

The 95% confidence interval for the true population proportion is [.4627, .6573].

The margin of error would be equal to half the width of the confidence interval, which is equal to:

Margin of Error: (.6573 – .4627) / 2 = .0973.

Additional Resources

Margin of Error vs. Standard Error: What’s the Difference?
How to Find Margin of Error in Excel
How to Find Margin of Error on a TI-84 Calculator

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