Confidence Interval Calculator

What Is a Confidence Interval?

A confidence interval is a range of plausible values for a population parameter, such as a population mean, based on sample data and a chosen confidence procedure. The steps are: take a sample, compute an estimate (the sample mean), work out a margin of error from the variability and the sample size, and report the estimate ± the margin of error.

Example: a café times 25 randomly chosen orders and finds a mean of 72 seconds with a standard deviation of 10 seconds. The 95% confidence interval for the mean time of all orders is 72 ± 4.13, or 67.87 to 76.13 seconds.

The confidence level describes the method, not one particular interval: if you repeated the sampling many times and built a 95% interval each time, about 95% of those intervals would contain the true mean. It does not mean there is a 95% probability that the true mean lies in the interval you calculated, and it says nothing about where 95% of individual orders fall.

How to Calculate a Confidence Interval

1. Find the sample mean x̄, the standard deviation (s, or σ if known) and the sample size n.

2. Compute the standard error: SE = s / √n.

3. Find the critical value for your confidence level: t* with n − 1 degrees of freedom, or z* if σ is known.

4. Multiply to get the margin of error: E = t* × SE.

5. The interval is x̄ − E to x̄ + E.

For x̄ = 72, s = 10, n = 25 at 95%: SE = 10 / √25 = 2, df = 24, t* ≈ 2.0639, E ≈ 2.0639 × 2 ≈ 4.1278, so the interval is 67.8722 to 76.1278.

Standard Deviation, Standard Error and Margin of Error

These three are easy to confuse. The standard deviation (s or σ) measures how spread out individual observations are. The standard error, SE = s / √n, measures how much the sample mean itself would vary from one sample to another; it shrinks as the sample grows. The margin of error, E = critical value × SE, is the amount added to and subtracted from the sample mean to form the confidence interval.

Z vs T Distribution

Use the t-distribution when the population standard deviation is unknown and you estimate it with the sample standard deviation s. This is the usual situation. The t-distribution has heavier tails than the normal distribution, which allows for the extra uncertainty in s, so t-intervals are a little wider. Its shape depends on the degrees of freedom, df = n − 1: with few observations the estimate of spread is shaky and t* is large (12.71 for df = 1 at 95%), and as df grows t* falls towards the normal value (2.064 for df = 24, 1.984 for df = 99, 1.960 in the limit).

Use the z-distribution when the population standard deviation σ is known, for example from a long-running, well-studied process. A large sample does not by itself turn a t-interval into a z-interval; it only makes the two nearly identical.

Confidence Level Comparison

For the same data, a higher confidence level gives a wider interval, because the method must capture the true mean more often. With x̄ = 72, s = 10 and n = 25:

90%: t* ≈ 1.711, interval 68.58 to 75.42 (narrower).

95%: t* ≈ 2.064, interval 67.87 to 76.13 (the common standard).

99%: t* ≈ 2.797, interval 66.41 to 77.59 (wider).

The choice is a trade-off between confidence and precision; 95% is the usual default in science and business.

Sample Size and the Confidence Interval

With the same method and similar variability, a larger sample gives a smaller standard error and a narrower interval. Because SE = s / √n, quadrupling the sample halves the standard error. With s = 10: n = 25 gives SE = 2 and a 95% margin of error of about 4.13; n = 100 gives SE = 1 and a margin of about 1.98 (t* also drops slightly, from 2.064 to 1.984).

In real data the sample standard deviation also changes from sample to sample, so a bigger sample usually, but not always, gives a narrower interval.

Confidence Interval Assumptions

Whether this interval is appropriate depends on how the data were collected, not on the arithmetic. The standard interval for a mean assumes:

A random sample (or random assignment) from the population of interest.

Independent observations, with no clustering and no repeated measurements of the same unit counted separately.

For small samples, a population that is reasonably close to normal; strong skew or outliers can make a t-interval misleading. With larger samples the interval is more robust, thanks to the central limit theorem.

The correct choice of z or t, depending on whether the population standard deviation is known.

The calculator cannot check these conditions; look at how the sample was taken and plot the data first.

Confidence Interval vs Margin of Error

The margin of error is the distance from the estimate to either end of the interval. The confidence interval is the estimate ± the margin of error. For 72 ± 4.13, the margin of error is 4.13 and the confidence interval is 67.87 to 76.13; the full width of the interval is twice the margin of error.

Confidence Interval vs Prediction Interval

A confidence interval estimates a population parameter, such as the mean order time. A prediction interval estimates where a single future observation, such as the next order, is likely to fall. Individual values vary much more than the mean does, so a prediction interval is much wider. For the café data, the mean is pinned down to about ±4 seconds, but an individual order could easily be 20 seconds from the mean.