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Confidence Interval

A confidence interval is a range of values, calculated from sample data, that is likely to contain the true value of a population parameter such as the mean. It comes with a confidence level (like 95%) that tells you how sure you can be that the interval captures the true value.

A confidence interval is an estimate of a population parameter expressed as a range, constructed so that, for a given confidence level CC, the interval will contain the true parameter in CC% of all possible samples drawn from the population. The width of the interval depends on the sample size, the variability of the data, and the chosen confidence level. A confidence interval for a population mean takes the form xˉ±z∗⋅σn\bar{x} \pm z^* \cdot \dfrac{\sigma}{\sqrt{n}} when the population standard deviation is known, or uses t∗t^* and the sample standard deviation ss when it is not.

Key Formula

xˉ−z∗⋅σn  ≤  μ  ≤  xˉ+z∗⋅σn\bar{x} - z^* \cdot \frac{\sigma}{\sqrt{n}} \;\leq\; \mu \;\leq\; \bar{x} + z^* \cdot \frac{\sigma}{\sqrt{n}}
Where:
  • xˉ\bar{x} = the sample mean
  • z∗z^* = the critical value from the standard normal distribution for the chosen confidence level
  • σσ = the population standard deviation
  • nn = the sample size
  • μμ = the true population mean (the parameter being estimated)

Worked Example

Problem: A random sample of 64 students has a mean test score of 78. The population standard deviation is known to be 12. Construct a 95% confidence interval for the true mean test score.
Step 1: Identify the known values from the problem.
xˉ=78,σ=12,n=64\bar{x} = 78, \quad \sigma = 12, \quad n = 64
Step 2: Find the critical value. For a 95% confidence level, the z* value is 1.96.
z∗=1.96z^* = 1.96
Step 3: Calculate the standard error by dividing the population standard deviation by the square root of the sample size.
σn=1264=128=1.5\frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{64}} = \frac{12}{8} = 1.5
Step 4: Find the margin of error by multiplying the critical value by the standard error.
z∗⋅σn=1.96×1.5=2.94z^* \cdot \frac{\sigma}{\sqrt{n}} = 1.96 \times 1.5 = 2.94
Step 5: Build the interval by adding and subtracting the margin of error from the sample mean.
78−2.94=75.06and78+2.94=80.9478 - 2.94 = 75.06 \quad \text{and} \quad 78 + 2.94 = 80.94
Answer: The 95% confidence interval for the true mean test score is (75.06, 80.94). We are 95% confident that the true population mean lies within this range.

Visualization

Why It Matters

In practice, you almost never know the exact value of a population parameter — you only have sample data. Confidence intervals give researchers, pollsters, and scientists a principled way to express uncertainty. When a news report says a candidate's approval rating is "52% ± 3%," that margin of error comes directly from a confidence interval calculation.

Common Mistakes

Mistake: Saying "there is a 95% probability that the true mean is in this interval."
Correction: Once calculated, the true mean is either in the interval or it isn't. The correct interpretation is: if you repeated the sampling process many times, about 95% of the resulting intervals would contain the true mean.
Mistake: Thinking a higher confidence level (e.g., 99% vs. 95%) gives a more precise estimate.
Correction: A higher confidence level produces a wider interval, not a narrower one. You gain more confidence at the cost of less precision. To make the interval narrower while keeping the same confidence level, you need a larger sample size.

Related Terms

  • Population — The group whose parameter the interval estimates
  • Mean — Often the parameter being estimated
  • Standard Deviation — Measures spread used in the interval formula