Confidence Interval Calculator

A confidence interval (CI) is a range of values, calculated from sample data, that is likely to contain the true population parameter — usually the population mean — with a stated level of confidence. This calculator takes a sample mean, a standard deviation and a sample size, lets you pick a confidence level of 90%, 95%, 98% or 99%, and returns the interval’s lower and upper bounds along with the margin of error and the standard error. It is built for students of statistics, researchers and analysts, and anyone reporting survey or experimental results who needs to express not just an estimate but how precise that estimate is.

The confidence level is widely misread, so it is worth stating carefully. A 95% confidence interval means that if you repeated the same sampling procedure many times and built an interval each time, about 95% of those intervals would contain the true population mean. It is not a 95% probability that the true mean lies inside this one particular interval — the true mean is fixed; it is the interval that varies from sample to sample. The confidence level therefore describes the long-run reliability of the method, not the odds for any single result.

Formula

CI = x̄ ± z × (σ / √n)

Here x̄ is the sample mean, z is the critical value for the chosen confidence level, σ is the standard deviation, and n is the sample size. The piece σ / √n is the standard error (SE) — how much the sample mean is expected to vary around the true mean — and z × SE is the margin of error (ME), which is simply half the total width of the interval. So the interval runs from x̄ − ME up to x̄ + ME.

Critical values (z)

90% → 1.645 · 95% → 1.96 · 98% → 2.326 · 99% → 2.576

The most commonly used value is z = 1.96 for a 95% confidence interval, which is the default standard in most scientific reporting. Higher confidence levels use larger z-values, which widen the interval — the price of more certainty is less precision.

Worked example

Suppose a sample of n = 30 readings has a mean x̄ = 100 and a standard deviation σ = 15, and you want a 95% confidence interval. First the standard error: SE = σ / √n = 15 / √30 ≈ 15 / 5.477 ≈ 2.739. Next the margin of error at 95% (z = 1.96): ME = 1.96 × 2.739 ≈ 5.37. The interval is therefore x̄ ± ME = 100 ± 5.37, giving roughly 94.63 to 105.37. You can report this as: “the true mean is estimated at 100, with a 95% confidence interval of about 94.6 to 105.4.”

Tips for using this calculator

• Larger samples shrink the interval: because the standard error divides by √n, quadrupling the sample size roughly halves the margin of error. Use the what-if slider to see this effect live.
• Pick the confidence level to match the stakes — 95% is the everyday default, while 99% suits high-consequence work such as medical, safety or financial decisions.
• Report the full interval, not just the mean. A point estimate alone hides how precise (or imprecise) your data actually is.
• A narrower interval is not automatically “better” — it may simply reflect a lower confidence level rather than more reliable data.

Common mistakes to avoid

• Misreading the confidence level as the probability the true mean lies in this specific interval. It describes the long-run success rate of the method across many samples.
• Using the z-distribution for small samples with an unknown population standard deviation. For n below about 30, the t-distribution (Student’s t) is the correct choice.
• Confusing standard error with standard deviation. The standard deviation describes the spread of individual data points; the standard error describes the spread of the sample mean and equals σ / √n.
• Forgetting that a wider interval is more, not less, cautious. Raising the confidence level always widens the interval, because you are demanding a higher chance of capturing the true value.