Last updated: 2026-05-18

The confidence interval

A confidence interval is a range of plausible values for an unknown population parameter — typically the mean — computed from sample data together with a stated confidence level. It pairs a point estimate (the sample mean) with a margin of error that reflects how much the estimate could vary from one sample to the next.

Formulas

With sample mean $\bar{x}$, standard deviation $\sigma$, sample size $n$, and critical Z-value $z^*$:

Standard error:

Margin of error:

Confidence interval:

Critical Z-values (standard normal distribution):

Interpretation

A confidence interval does not state the probability that the true mean falls inside a particular interval. The true population mean $\mu$ is a fixed (unknown) constant — it is not random. The interval is the random object, because it is built from a random sample.

The precise meaning is a statement about the procedure: if the sampling process were repeated many times and a 95% confidence interval were built from each sample, approximately 95% of those intervals would contain the true mean. Any single interval either contains $\mu$ or it does not.

In practice, a 95% confidence interval is a range that covers the true value for 95% of samples drawn the same way; on average 1 interval in 20 misses.

When to use Z vs. t

This calculator uses the Z-distribution (standard normal). This is appropriate when:

• The population standard deviation $\sigma$ is known, OR
• The sample size is large ($n \geq 30$), so the Central Limit Theorem guarantees the sampling distribution is approximately normal

When $\sigma$ is unknown and $n < 30$, use the t-distribution with $n - 1$ degrees of freedom. The t-distribution has heavier tails, producing wider (more conservative) intervals. For $n \geq 30$, the difference between Z and t is negligible.

How sample size affects the interval

The margin of error shrinks as $\sqrt{n}$ grows. Halving the margin of error requires four times the sample size — a key constraint in survey design.

Worked example: exam score analysis

A teacher randomly selects 35 exam papers from a class. The sample mean score is 47.3 points with a standard deviation of 11.8.

Standard error: $SE = 11.8 / \sqrt{35} \approx 1.994$

95% confidence interval: $ME = 1.96 \times 1.994 \approx 3.91$

Interpretation: "Based on this sample of 35, we estimate the class average lies between 43.4 and 51.2, with 95% confidence."

What changes with 99% confidence? The interval widens: $ME = 2.576 \times 1.994 \approx 5.14$, giving $[42.2, 52.4]$. More confidence costs width.