Two terms that students often confuse in statistics are standard error and margin of error.
The standard error measures the preciseness of an estimate of a population mean. It is calculated as:
Standard Error = s / √n
where:
- s: Sample standard deviation
- n: Sample size
The margin of error measures the half-width of a confidence interval for a population mean. It is calculated as:
Margin of Error = z*(s/√n)
where:
- z: Z value that corresponds to a given confidence level
- s: Sample standard deviation
- n: Sample size
Let’s check out an example to illustrate this idea.
Example: Margin of Error vs. Standard Error
Suppose we collect a random sample of turtles with the following information:
- Sample size n = 25
- Sample mean weight x = 300
- Sample standard deviation s = 18.5
Now suppose we’d like to create a 95% confidence interval for the true population mean weight of turtles. The formula to calculate this confidence interval is as follows:
Confidence Interval = x +/- z*(s/√n)
where:
- x: Sample mean
- s: Sample standard deviation
- n: Sample size
- z: Z value that corresponds to a given confidence level
The z-value that you will use is dependent on the confidence level that you choose. The following table shows the z-value that corresponds to popular confidence level choices:
| Confidence Level | z-value |
|---|---|
| 0.90 | 1.645 |
| 0.95 | 1.96 |
| 0.99 | 2.58 |
Notice that higher confidence levels correspond to larger z-values, which leads to wider confidence intervals. This means that, for example, a 99% confidence interval will be wider than a 95% confidence interval for the same set of data.
The standard error would be calculated as:
Standard error = s/√n = 18.5/√25 = 3.7 The margin of error would be calculated as
Margin of error = z*(s/√n) = 1.96*(18.5/√25) = 7.25 And the 95% confidence interval would be calculated as
95% Confidence Interval = x +/- z*(s/√n) = 300 +/- 1.96*(18.5/√25) = [292.75, 307.25] Note that the width of the entire confidence interval is 307.25 – 292.75 = 14.5.
Note that the margin of error is equal to half of this width: 14.5 / 2 = 7.25.
Note also that the margin of error will always be larger than the standard error simply because the margin of error is equal to the standard error multiplied by some critical Z value. In the previous example, we multiplied the standard error by 1.96 to obtain the margin of error.
Additional Resources
What are Confidence Intervals?
Standard Deviation vs. Standard Error: What’s the Difference?