Confidence Interval Calculator

Determine confidence intervals for means and proportions. Reliable Confidence Interval Calculator for statistical estimation.

What is a Confidence Interval?

A confidence interval (CI) is a range of values that is likely to contain the true population parameter (such as the mean or proportion) with a certain level of confidence. It provides an estimated range of values which is likely to include an unknown population parameter.

Confidence intervals are crucial in statistics because they account for sampling error. Instead of providing a single estimate (point estimate), they provide a range where the true value likely lies.

Interpretation

A 95% confidence interval means that if we were to take 100 different samples from the same population and compute a confidence interval for each sample, approximately 95 of those intervals would contain the true population parameter. It does not mean there is a 95% probability that the specific interval contains the parameter (once calculated, the interval either contains the parameter or it doesn't).

Common Confidence Levels and Z-Scores

The confidence level determines how sure you can be. Higher confidence levels result in wider intervals.

  • 90% CI: Z = 1.645
  • 95% CI: Z = 1.96 (Most common)
  • 99% CI: Z = 2.576

Calculating Confidence Intervals

For Population Mean (Known Standard Deviation)

CI=xˉ±Z×σnCI = \bar{x} \pm Z \times \frac{\sigma}{\sqrt{n}}

Where:

  • xˉ\bar{x} = Sample mean
  • ZZ = Z-score corresponding to the confidence level
  • σ\sigma = Population standard deviation
  • nn = Sample size

For Population Mean (Unknown Standard Deviation)

When the population standard deviation is unknown and the sample size is small (n<30n < 30), use the t-distribution:

CI=xˉ±t×snCI = \bar{x} \pm t \times \frac{s}{\sqrt{n}}

Where:

  • tt = t-score (based on degrees of freedom df=n−1df = n-1)
  • ss = Sample standard deviation

Example Calculation

Scenario: We want to estimate the average height of students in a university. We take a random sample of 100 students.

  • Sample Mean (xˉ\bar{x}) = 170 cm
  • Population Standard Deviation (σ\sigma) = 10 cm
  • Sample Size (nn) = 100
  • Confidence Level = 95% (Z≈1.96Z \approx 1.96)

Step 1: Calculate Standard Error SE=10100=1010=1SE = \frac{10}{\sqrt{100}} = \frac{10}{10} = 1

Step 2: Calculate Margin of Error MOE=1.96×1=1.96MOE = 1.96 \times 1 = 1.96

Step 3: Construct Interval Lower=170−1.96=168.04Lower = 170 - 1.96 = 168.04 Upper=170+1.96=171.96Upper = 170 + 1.96 = 171.96

Result: We are 95% confident that the true average height is between 168.04 cm and 171.96 cm.

Frequently Asked Questions

Q: Why does a higher confidence level widen the interval? A: To be more certain that your interval captures the true parameter, you need to cast a wider net. A 99% CI must cover more range than a 90% CI.

Q: How does sample size affect the confidence interval? A: Increasing the sample size decreases the standard error, making the confidence interval narrower (more precise).

Q: What is the difference between a Z-test and a t-test interval? A: Use Z-scores when the population standard deviation is known or n≥30n \ge 30. Use t-scores when the sample size is small (n<30n < 30) and the population standard deviation is unknown.