Confidence Interval

How a Confidence Interval Is Calculated

A confidence interval is a range of values, built around a sample’s own mean, that likely contains the true population mean. Enter a sample’s mean, standard deviation, and size, choose a confidence level, and this calculator finds the range within which the actual population mean probably falls.

A “95% confidence interval,” for example, means that if you repeated the same sampling process many times, about 95% of the intervals it produced would contain the true population mean. It does not mean there’s a 95% chance the true mean falls inside this one specific interval — a common, technically-incorrect way people phrase it. A wider interval (higher confidence level) casts a wider net and is more likely to contain the true mean; a narrower interval is more precise but less certain.

The Formula

$$\vE{\text{Standard Error}} = \frac{\vA{s}}{\sqrt{\vB{n}}}$$ $$\vF{\text{Margin of Error}} = \vC{z} \times \vE{\text{Standard Error}}$$ $$\text{Confidence Interval} = \vD{\bar{x}} \pm \vF{\text{Margin of Error}}$$

Where $\vA{s}$ is the sample’s standard deviation, $\vB{n}$ is the sample size, $\vC{z}$ is the critical value for the chosen confidence level (looked up from the standard normal distribution), and $\vD{\bar{x}}$ is the sample mean. A bigger sample size shrinks the standard error — and therefore the whole interval — since more data gives a more precise estimate of the true mean.

Worked Example

A sample of 25 items has a mean of 100 and a standard deviation of 15, at a 95% confidence level (a critical value of z = 1.96):

1. Standard error: $\vE{\text{Standard Error}} = \vA{15} \div \sqrt{\vB{25}} = \vE{3}$.
2. Margin of error: $\vF{\text{Margin of Error}} = \vC{1.96} \times \vE{3} = \vF{5.88}$.
3. Confidence interval: $\text{Confidence Interval} = \vD{100} \pm \vF{5.88} = [94.12, 105.88]$.

So the true population mean is likely (with 95% confidence) somewhere between 94.12 and 105.88.

Key Factors to Consider

• A confidence interval says nothing about how the sample was collected. The math here assumes the sample is a genuinely random, representative sample of the population — a biased or non-random sampling method can produce a confidence interval that looks precise but is centered on the wrong value entirely, a problem this formula cannot detect or correct for.
• The interval assumes the underlying data is at least approximately normally distributed, or that the sample is large enough for the Central Limit Theorem to apply. For a small sample from a heavily skewed distribution, both the z-distribution and t-distribution approximations become less reliable.
• A confidence interval and a margin of error reported in a poll or survey describe the same underlying idea. A news report citing “a margin of error of plus or minus 3 percentage points” at a stated confidence level (often 95%) is describing exactly the interval this calculator computes, just phrased differently.
• Narrower isn’t always better if it comes at the cost of confidence. A tighter interval at a lower confidence level (say 80% instead of 95%) is more precise but less likely to actually contain the true value — the right tradeoff between precision and confidence depends on what the estimate is being used for.

Common Mistakes

• Reading “95% confidence” as “95% probability the true mean is in this interval.” Once a sample has actually been collected and the interval computed, the true population mean either is or isn’t inside it — there’s no probability left to talk about. The 95% describes the long-run behavior of the method across many repeated samples, not this one specific result.
• Applying this z-based formula to a small sample without checking the assumptions. The z-distribution assumes either a large sample (conventionally n of about 30 or more) or a known population standard deviation. A small sample with only an estimated standard deviation should technically use the t-distribution instead, which produces a somewhat wider — and more honest — interval.
• Confusing the confidence interval with the range of the raw data. The interval describes uncertainty about the population mean, not where individual data points fall — a dataset can have values spread far wider than its confidence interval while the interval on the mean itself stays narrow, especially with a large sample.

Useful to Know

Because the standard error shrinks with the square root of the sample size rather than the sample size itself, cutting a margin of error in half generally means roughly quadrupling the sample size, not just doubling it — a detail that matters when planning how large a survey or experiment actually needs to be. This is also why polls with only a few hundred respondents can still report a respectably tight margin of error: once a sample reaches a few hundred, the square-root relationship means each additional respondent buys progressively less precision, so pollsters rarely see much value in pushing sample sizes into the tens of thousands just to shave a fraction of a point off the margin.