Confidence Intervals Explained (Without the Jargon)

A confidence interval gives a range of plausible values for something you cannot measure directly, such as the average score of all students, based on a sample. It is the sample mean plus or minus a margin of error: $\bar{x} \pm z^*\frac{\sigma}{\sqrt{n}}$. A 95% interval uses $z^* \approx 1.96$.

The formula

Worked example

A sample of 50 students has mean score 72.5, with $\sigma = 8$.

1. Standard error: $\frac{8}{\sqrt{50}} \approx 1.131$.
2. Margin of error: $1.96 \times 1.131 \approx 2.217$.
3. Interval: $72.5 \pm 2.217$, which is $(70.28,\; 74.72)$.

What "95% confident" means

If you took many samples and built an interval from each, about 95% of those intervals would contain the true mean. It does not mean there is a 95% chance the true mean is in this particular interval; the true mean is fixed, and the interval either contains it or not.

Where 1.96 comes from

95% of a normal distribution lies between $z = -1.96$ and $z = 1.96$, because $\Phi(1.96) \approx 0.975$, leaving 2.5% in each tail.

Sample size matters

The margin of error shrinks with $\sqrt{n}$. To halve it, you need four times as many people.

Common mistakes

• Dividing by $n$ instead of $\sqrt{n}$.
• Saying "95% probability the mean is in this interval".
• Using $z$ for very small samples with an estimated $\sigma$ (a t-interval is better).

Practice questions

1. Find the standard error when $\sigma = 10$ and $n = 25$.
2. What is the 95% margin of error for question 1?
3. How does the interval change if $n$ is multiplied by 4?

Answers: 1) 2 2) 3.92 3) It becomes half as wide

Build intervals with the confidence interval calculator and explore the bell curve with the normal distribution calculator. Background: the normal distribution and z-scores.

Frequently asked questions

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