What a confidence interval is
A confidence interval is a range of plausible values for an unknown population parameter -- most often a population mean or proportion -- calculated from a sample of data. It is built around a point estimate, such as a sample mean, and extends outward in both directions by a margin of error, so that the interval as a whole is designed to capture the true, unknown population value a specified percentage of the time if the same sampling and calculation procedure were repeated many times.
The chosen percentage, called the confidence level, is most commonly 90%, 95% or 99%. A wider interval provides more confidence that it captures the true value, but at the cost of precision; a narrower interval is more precise but comes with a lower confidence level, or requires a larger sample to maintain the same confidence level. This trade-off between precision and confidence is unavoidable and is central to interpreting any reported interval correctly.
The common misinterpretation of '95% confidence'
It is a common and well-documented misinterpretation to read a 95% confidence interval as meaning there is a 95% probability that the true population value lies within that one specific, already-calculated interval. This reading is not technically correct: once a particular interval has been calculated from a particular sample, the true population value either does or does not lie within it -- there is no remaining randomness to attach a probability to for that single interval. The 95% describes the long-run success rate of the method used to build the interval, across many hypothetical repeated samples, not a probability statement about any one interval after the fact.
Hoekstra and colleagues (2014), publishing in Psychonomic Bulletin and Review, documented that this distinction is widely misunderstood, including among researchers and statistics students, many of whom interpreted a 95% confidence interval as directly describing the probability that the true value fell within that specific interval. The technically correct interpretation concerns the reliability of the method across repeated sampling, not a probability statement about one already-observed result.
Margin of error: the building block of a confidence interval
The margin of error is the distance the confidence interval extends on either side of the point estimate, and it is calculated as a critical value (z, for a normal-based interval) multiplied by the standard error of the estimate. For a mean, the standard error is the standard deviation divided by the square root of the sample size: standard error = standard deviation / sqrt(n). A larger sample shrinks the standard error, and therefore the margin of error and the overall interval width, in proportion to the square root of n.
The critical value z depends on the chosen confidence level: 1.6449 for 90% confidence, 1.96 for 95% confidence, and 2.5758 for 99% confidence, each representing the point on the standard normal distribution that leaves the corresponding total probability in the two tails. A higher confidence level requires a larger critical value, which widens the margin of error and therefore the resulting interval, all else being equal.
Worked example: building a 95% confidence interval for a mean
Suppose a sample of n = 100 observations has a mean of 50 and a standard deviation of 8, and a 95% confidence interval for the population mean is required. Step 1: calculate the standard error, 8 / sqrt(100) = 8 / 10 = 0.8. Step 2: calculate the margin of error using the 95% critical value, 1.96 x 0.8 = 1.568. Step 3: add and subtract the margin of error from the sample mean: 50 - 1.568 = 48.432, and 50 + 1.568 = 51.568.
The resulting 95% confidence interval is 48.432 to 51.568. The correct interpretation is that if this same sampling and calculation procedure were repeated many times on new samples of 100 from the same population, approximately 95% of the intervals constructed this way would contain the true population mean -- not that there is a 95% probability the true mean lies specifically between 48.432 and 51.568 in this one instance.
How confidence level and sample size change the interval
Holding the sample fixed (mean 50, standard deviation 8, n = 100), the table below shows how the same underlying data produce different interval widths depending only on the chosen confidence level. Higher confidence always widens the interval, because a more reliable method (one that succeeds more often across repeated samples) must, by construction, cover a broader range of plausible values.
| Confidence level | z critical value | Margin of error | Resulting interval |
|---|---|---|---|
| 90% | 1.6449 | 1.316 | 48.684 to 51.316 |
| 95% | 1.96 | 1.568 | 48.432 to 51.568 |
| 99% | 2.5758 | 2.061 | 47.939 to 52.061 |
Часто задаваемые вопросы
What does a 95% confidence interval actually mean?
It means that if the same sampling and calculation procedure were repeated many times, approximately 95% of the resulting intervals would contain the true population value. It does not mean there is a 95% probability that the true value falls within this one specific, already-calculated interval -- a distinction documented as a widespread misunderstanding by Hoekstra and colleagues (2014) in Psychonomic Bulletin and Review.
How do you calculate a 95% confidence interval for a mean?
Calculate the standard error (standard deviation divided by the square root of the sample size), multiply it by the 95% critical value of 1.96 to get the margin of error, then add and subtract that margin from the sample mean. For a sample with mean 50, standard deviation 8 and n = 100: standard error = 0.8, margin of error = 1.568, and the interval is 48.432 to 51.568.
What is a margin of error?
The margin of error is the amount added to and subtracted from a point estimate, such as a sample mean, to form a confidence interval. It is calculated as a critical value (which depends on the chosen confidence level) multiplied by the standard error of the estimate. A larger margin of error produces a wider, less precise interval; a smaller margin of error produces a narrower, more precise one.
Why is it wrong to say there's a 95% probability the true value is in this interval?
Because once a specific interval has been calculated from a specific sample, the true population value either is or is not inside it -- there is no remaining randomness to assign a probability to. The 95% is a property of the method used to construct the interval, describing how often it succeeds across many hypothetical repeated samples, not a probability statement about any single interval after it has already been computed.
Does a larger sample size make a confidence interval narrower?
Yes. The standard error, and therefore the margin of error and overall interval width, is calculated by dividing the standard deviation by the square root of the sample size. Because sample size enters under a square root, quadrupling the sample size halves the width of the interval, while a smaller sample produces a proportionally wider, less precise interval for the same confidence level.
Why does a higher confidence level produce a wider interval?
A higher confidence level, such as 99% instead of 95%, requires a larger critical value (2.5758 versus 1.96), which increases the margin of error and widens the resulting interval. This reflects an unavoidable trade-off: a method that needs to succeed more often across repeated samples must cover a broader range of plausible values to do so.
Источники
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods, Section 7.1: Confidence intervals. nist.gov.
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. W. H. Freeman (confidence intervals for a mean; z vs t procedures).
- Hoekstra R, Morey RD, Rouder JN, Wagenmakers EJ. "Robust misinterpretation of confidence intervals." Psychonomic Bulletin and Review, 2014;21(5):1157-1164.