Confidence Interval Calculator

Calculate confidence intervals for population mean and proportion with step-by-step solutions

Input Values

Results

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain the true population parameter with a specified level of confidence. It provides a way to estimate population parameters from sample data and quantify the uncertainty in our estimates.

Types of Confidence Intervals

Population Mean:

Used when estimating the mean of a population from sample data

CI = x̄ ± (z × SE)

Population Proportion:

Used when estimating the proportion of a population with a certain characteristic

CI = p̂ ± (z × SE)

Confidence Interval Formulas

For Population Mean:

CI = x̄ ± (z × σ/√n)

Where:

  • x̄ = sample mean
  • z = critical value
  • σ = population standard deviation
  • n = sample size

For Population Proportion:

CI = p̂ ± (z × √(p̂(1-p̂)/n))

Where:

  • p̂ = sample proportion
  • z = critical value
  • n = sample size

Critical Values (z-scores)

Confidence Levelz-scoreα/2
90%1.6450.05
95%1.9600.025
99%2.5760.005

How to Use the Calculator

  1. Choose the type of confidence interval (mean or proportion)
  2. Enter the sample data or statistics
  3. Select the confidence level
  4. Click "Calculate" to get the confidence interval
  5. View the step-by-step solution and interpretation

Example Calculations

Population Mean Example:

Sample mean = 50

Population SD = 10

Sample size = 100

Confidence level = 95%

CI = 50 ± (1.96 × 10/√100)

CI = 50 ± 1.96

CI = [48.04, 51.96]

Population Proportion Example:

Sample proportion = 0.6

Sample size = 200

Confidence level = 90%

SE = √(0.6 × 0.4 / 200) = 0.035

CI = 0.6 ± (1.645 × 0.035)

CI = [0.542, 0.658]

Interpretation

  • 95% Confidence Level: If we repeated the sampling process many times, 95% of the intervals would contain the true population parameter
  • Margin of Error: The half-width of the confidence interval
  • Precision: Smaller intervals indicate more precise estimates
  • Sample Size Effect: Larger samples generally produce narrower intervals

Assumptions

  • Random Sampling: Data should come from a random sample
  • Normal Distribution: For means, the sampling distribution should be approximately normal
  • Independence: Observations should be independent
  • Known Population SD: For means, population standard deviation should be known (or use t-distribution)

Applications

  • Market Research: Estimating customer satisfaction rates
  • Medical Studies: Estimating treatment effectiveness
  • Quality Control: Estimating product defect rates
  • Political Polling: Estimating voting preferences
  • Scientific Research: Estimating population parameters

Common Mistakes

  • Confusing confidence level with probability
  • Using the wrong critical value
  • Ignoring the assumptions
  • Interpreting the interval incorrectly
  • Using the wrong formula for the data type