How to Calculate Confidence Intervals

Step-by-step guide to calculating confidence intervals. Learn when to use z-intervals vs. t-intervals, how to choose a confidence level, and how to interpret the results.

What Is a Confidence Interval?

A confidence interval is a range of values, calculated from sample data, that is likely to contain the true population parameter. Rather than reporting a single point estimate (such as a sample mean), a confidence interval provides a range that accounts for sampling variability. For example, a 95% confidence interval for a population mean might be (42.3, 47.7), meaning that based on the sample data, the true population mean is estimated to lie between 42.3 and 47.7. The width of the interval reflects the precision of the estimate: narrower intervals indicate more precise estimates.

Confidence Level Explained

The confidence level, commonly 90%, 95%, or 99%, describes how often the interval construction method would capture the true parameter if you repeated the sampling process many times. A 95% confidence level means that if you took 100 independent random samples and computed a 95% confidence interval from each one, about 95 of those intervals would contain the true population parameter. It does not mean there is a 95% probability that the true value lies in any particular interval. Higher confidence levels produce wider intervals because you need a broader range to be more certain that the parameter is captured.

The Z-Interval Formula

When the population standard deviation (sigma) is known and the sample size is large (typically n >= 30), you use the z-interval formula: CI = x-bar plus or minus z* times (sigma / sqrt(n)). Here x-bar is the sample mean, z* is the critical z-value for the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), sigma is the population standard deviation, and n is the sample size. The term sigma / sqrt(n) is called the standard error and measures how much the sample mean is expected to vary from sample to sample. This formula assumes the data comes from a normally distributed population or the sample size is large enough for the Central Limit Theorem to apply.

The T-Interval Formula

When the population standard deviation is unknown (which is the typical real-world scenario), you replace sigma with the sample standard deviation s and use the t-distribution instead of the z-distribution. The formula becomes: CI = x-bar plus or minus t* times (s / sqrt(n)), where t* is the critical value from the t-distribution with n - 1 degrees of freedom. The t-distribution has heavier tails than the standard normal, which produces slightly wider intervals to account for the additional uncertainty from estimating sigma. As the sample size grows, the t-distribution approaches the z-distribution, and the difference between the two methods becomes negligible.

Choosing the Right Sample Size

The width of a confidence interval depends on three factors: the confidence level, the variability in the data, and the sample size. Since you typically cannot control the first two (the confidence level is chosen by convention and variability is inherent in the data), sample size is the primary lever for controlling precision. The margin of error formula E = z* times (sigma / sqrt(n)) can be solved for n to find the minimum sample size needed for a desired margin of error: n = (z* times sigma / E) squared. For example, to achieve a margin of error of 2 with sigma = 10 at 95% confidence, you need n = (1.96 times 10 / 2) squared = 96.04, so at least 97 observations.

Confidence Intervals for Proportions

When estimating a population proportion (such as the percentage of voters who favor a candidate), the formula adjusts to use the proportion rather than the mean. The confidence interval for a proportion is: p-hat plus or minus z* times sqrt(p-hat times (1 - p-hat) / n), where p-hat is the sample proportion and n is the sample size. This formula requires that both n times p-hat and n times (1 - p-hat) are at least 10 to ensure the sampling distribution is approximately normal. For small samples or proportions near 0 or 1, alternative methods like the Wilson interval or Clopper-Pearson interval provide more accurate coverage.

Common Mistakes and Misinterpretations

The most common misinterpretation of a confidence interval is saying "there is a 95% probability that the true mean lies within this interval." The true parameter is a fixed (though unknown) value, not a random variable; the randomness is in the interval itself. Another common mistake is using the z-interval when sigma is unknown, which underestimates the interval width. Additionally, confidence intervals assume random sampling; intervals computed from biased or convenience samples may not have the stated coverage rate. Finally, do not confuse the confidence interval with a prediction interval, which estimates where a single new observation will fall and is always wider than a confidence interval.

Practical Examples

Suppose a factory samples 50 light bulbs and finds a mean lifetime of 1200 hours with a sample standard deviation of 100 hours. A 95% confidence interval using the t-distribution (t* = 2.009 for 49 degrees of freedom) gives: 1200 plus or minus 2.009 times (100 / sqrt(50)) = 1200 plus or minus 28.4, or (1171.6, 1228.4). This means the factory can be 95% confident that the true mean lifetime of all light bulbs it produces lies between approximately 1172 and 1228 hours. If the factory needs a narrower interval, it can increase the sample size or accept a lower confidence level.

Try These Calculators

Put what you learned into practice with these free calculators.

Confidence Interval Calculator Sample Size Calculator Z-Score Calculator