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Confidence Interval Calculator

Confidence Interval Calculator

Calculate confidence intervals for the population mean when the population standard deviation is unknown. This calculator uses the t-distribution and provides both upper and lower bounds with detailed calculation steps.

Enter your sample statistics and select confidence level to calculate

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain the true population parameter (such as the population mean) with a specified level of confidence. It provides an estimate of the uncertainty associated with a sample statistic and helps quantify how precise our estimate is.

When to Use This Calculator

This calculator is designed for situations where:

  • You have a sample from a population and want to estimate the population mean
  • The population standard deviation is unknown (most common scenario)
  • The sample size is relatively small (n < 30) or the population is normally distributed
  • You want to determine the range within which the true population mean likely falls

Formula and Calculation

When the population standard deviation is unknown, we use the t-distribution to calculate confidence intervals. The formula is:

CI = x̄ ± tα/2,n-1 × (s/√n)

Where:

  • x̄ = sample mean
  • tα/2,n-1 = critical t-value for the given confidence level and degrees of freedom (n-1)
  • s = sample standard deviation
  • n = sample size
  • s/√n = standard error of the mean

Interpreting Confidence Levels

The confidence level represents the percentage of times that the confidence interval would contain the true population parameter if we repeated the sampling process many times. Common confidence levels include:

90% Confidence: We can be 90% confident that the true population mean lies within the calculated interval.
95% Confidence: We can be 95% confident that the true population mean lies within the calculated interval.
99% Confidence: We can be 99% confident that the true population mean lies within the calculated interval.

Example Calculation

Suppose we have a sample of 25 students with a mean test score of 78.5 and a sample standard deviation of 12.3. To find the 95% confidence interval:

Given: x̄ = 78.5, n = 25, s = 12.3, confidence level = 95%
Degrees of freedom: df = n - 1 = 24
Critical t-value: t0.025,24 = 2.064
Standard error: SE = s/√n = 12.3/√25 = 2.46
Margin of error: ME = t × SE = 2.064 × 2.46 = 5.08
Confidence interval: 78.5 ± 5.08 = (73.42, 83.58)

Important Notes

  • Higher confidence levels result in wider intervals
  • Larger sample sizes result in narrower intervals
  • The t-distribution approaches the normal distribution as sample size increases
  • This calculator assumes the underlying population is approximately normally distributed