# 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.