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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:

**x:**sample 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