Confidence Interval Calculator

What a Confidence Interval Means?

A 95% confidence interval does not mean there is a 95% chance the true value lies within the range. It means that if you repeated this sampling procedure 100 times, approximately 95 of those intervals would contain the true population value. The interval is a statement about the method, not about any single result. This is a subtle but important distinction that even scientists frequently misstate.

What's the key thing to understand about Choosing Your Confidence Level?

90% CI uses Z equals 1.645 — narrower interval, less certainty. Used when a wider margin of error is acceptable and a smaller sample size is necessary. 95% CI uses Z equals 1.96 — the standard for most research and polling. 99% CI uses Z equals 2.576 — wider interval, very high certainty. Used when false positives are costly, such as pharmaceutical trials.

What should I know about Margin of Error in Practice?

Election polls quoted as X% plus or minus 3% use a 95% confidence interval with a sample of approximately 1,000 respondents. Doubling precision (halving the margin of error) requires quadrupling the sample size — the main reason polls rarely use samples larger than 1,000–2,000. A margin of error of plus or minus 3% at 95% confidence with n equals 1,000 is the standard in UK political polling.

What's the difference between Sample Size and Margin of Error?

For proportion-based CIs at 95% confidence: n=100 gives plus or minus 10%, n=400 gives plus or minus 5%, n=1,000 gives plus or minus 3.1%, n=2,500 gives plus or minus 2%, n=10,000 gives plus or minus 1%. The diminishing returns of larger samples are clear — going from 1,000 to 10,000 only reduces margin from 3% to 1%. This is why practical research balances cost against precision.

What's the key thing to understand about Common Misinterpretations of Confidence Intervals?

A 95% confidence interval is one of the most frequently misunderstood results in statistics. It does not mean there is a 95% probability that the true population value lies within this specific interval you calculated — once an interval is computed, the true value either is or isn't in it, with no remaining probability involved. The correct frequentist interpretation is about the method, not the single result: if you repeated the same sampling process many times and built a confidence interval each time, approximately 95% of those intervals would contain the true population parameter. It's also a mistake to interpret a 95% confidence interval as meaning 95% of individual data points fall within it — that describes a prediction interval or the empirical rule for standard deviation, a different concept entirely. Finally, a wide confidence interval isn't a 'more confident' result — it reflects more uncertainty (often from a small sample), while a narrow interval at the same confidence level reflects a more precise estimate.

What's the difference between the t-Distribution and the Normal (Z) Distribution?

The choice between the z-distribution and the t-distribution depends on what you know about the data. The z-distribution (standard normal) is appropriate when the population standard deviation is known and/or the sample size is large (conventionally n ≥ 30, where the Central Limit Theorem makes the sampling distribution closely normal regardless of the underlying data). The t-distribution is used when the population standard deviation is unknown and must be estimated from the sample itself — the usual real-world situation — especially with smaller samples. The t-distribution looks like the normal distribution but has heavier tails, reflecting the extra uncertainty introduced by estimating the standard deviation from limited data; its exact shape depends on the degrees of freedom, calculated as n − 1. As sample size grows, the t-distribution converges toward the normal distribution, and for samples larger than about 30 the two give very similar results. Using z when t is appropriate tends to make confidence intervals artificially narrow, overstating precision.

What's the key thing to understand about Confidence Intervals for Proportions vs Means?

A confidence interval for a population mean uses the sample mean ± (critical value × standard error of the mean), where the standard error is s/√n. A confidence interval for a population proportion follows the same structure but uses a different standard error: p̂ ± (critical value × √[p̂(1-p̂)/n]), where p̂ is the sample proportion. The two share the same underlying logic — both widen with a smaller sample and narrow with a larger one — but proportion intervals are typically built from the normal approximation and require a "success-failure condition" (np̂ and n(1-p̂) both at least 10) to be valid, since proportions near 0% or 100% behave differently from those near 50%. This is why national election polls, which estimate a proportion, report a margin of error calculated this way rather than using a standard-deviation-based formula.

What should I know about One-Sided vs Two-Sided Confidence Intervals?

The confidence intervals calculated by most tools are two-sided, meaning they bound the true population parameter on both the upper and lower side symmetrically, such as the familiar form estimate ± margin of error. A one-sided confidence interval instead provides a bound in only one direction — either an upper bound (claiming the true value is below some limit, with the stated confidence) or a lower bound (claiming the true value is above some limit) — leaving the other side completely unbounded. One-sided intervals are appropriate whenever only one direction of deviation matters practically: a quality control process might only care about an upper bound on a contaminant's concentration, with no interest in how low it might be, while a manufacturing specification might only need a lower bound on a component's minimum strength. The calculation differs subtly from the two-sided case: a 95% one-sided interval uses the full 5% significance allowance on just one tail of the distribution, using a smaller critical z or t value than the two-tailed 95% interval would use for that same side, which makes a one-sided 95% bound noticeably tighter than the corresponding side of a two-sided 95% interval. Choosing between one-sided and two-sided framing should always be decided by the practical question being asked, not after looking at the data, since choosing afterward to get a more favourable result is a form of statistical malpractice.

Critical Value Reference Table (z and t, df = 10)

The multiplier used to build a confidence interval depends on your chosen confidence level and whether you're using the normal (z) or t-distribution:

Notice the t-values are always larger than the corresponding z-values for the same confidence level — the t-distribution has fatter tails to account for the extra uncertainty of estimating the population standard deviation from a small sample, and that gap shrinks as degrees of freedom (sample size) increase, eventually converging to the z-values once df exceeds roughly 30.

Worked Example: 95% Confidence Interval for a Sample Mean

A sample of n = 36 measurements has a mean of 52.4 and a sample standard deviation of s = 8.1. Construct a 95% confidence interval for the true population mean (n is large enough to use the z critical value, 1.96). Standard error: SE = s/√n = 8.1/√36 = 8.1/6 = 1.35. Margin of error: ME = z × SE = 1.96 × 1.35 = 2.646. Confidence interval: 52.4 ± 2.646, which is (49.75, 55.05). This means we're 95% confident the true population mean lies between 49.75 and 55.05. If the sample size had instead been only n = 9 (requiring the t-distribution with df = 8, t ≈ 2.306), the standard error would balloon to 8.1/3 = 2.7, and the margin of error to 2.306 × 2.7 = 6.23 — more than double the margin from the larger sample. This is the direct, quantified reason researchers push for larger samples: quadrupling the sample size (from 9 to 36) roughly halves the margin of error, since SE scales with 1/√n.

Worked Example: Confidence Interval for a Proportion

A poll surveys n = 150 voters, and 42% (p̂ = 0.42) say they support a particular candidate. Construct a 95% confidence interval for the true population proportion. Standard error for a proportion: SE = √[p̂(1−p̂)/n] = √[0.42 × 0.58 / 150] = √[0.2436/150] = √0.001624 = 0.0403. Margin of error: ME = z × SE = 1.96 × 0.0403 = 0.0790 (7.9 percentage points). Confidence interval: 0.42 ± 0.079, which is (0.341, 0.499), or 34.1% to 49.9%. Because the upper bound (49.9%) sits just under 50%, this poll cannot confidently conclude the candidate has majority support, even though the point estimate (42%) might look clearly below half — the margin of error is wide enough that the true population proportion could plausibly reach nearly 50%. A larger sample (say n = 1,000 instead of 150) would shrink the standard error by a factor of √(1000/150) ≈ 2.58, tightening the margin of error down to roughly 3.1 percentage points and giving a much more decisive interval.