Confidence Interval Calculator

Confidence Interval Formulas

Confidence Interval for Mean (known population standard deviation)

Where:

• x̄ is the sample mean
• z_α/2 is the critical value from the standard normal distribution
• σ is the known population standard deviation
• n is the sample size

Confidence Interval for Mean (unknown population standard deviation)

Where:

• x̄ is the sample mean
• t_{α/2, n-1} is the critical value from the t-distribution with n-1 degrees of freedom
• s is the sample standard deviation
• n is the sample size

Confidence Interval for Proportion

Where:

• p̂ is the sample proportion (x/n)
• z_α/2 is the critical value from the standard normal distribution
• n is the sample size

Confidence Interval for Difference Between Means (unknown but equal variances)

Where:

• x̄₁ and x̄₂ are the sample means
• sₚ is the pooled standard deviation
• t_{α/2, n₁+n₂-2} is the critical value from the t-distribution
• n₁ and n₂ are the sample sizes

Confidence Interval for Difference Between Means (unequal variances)

Where df is calculated using the Welch-Satterthwaite equation.

Confidence Interval for Difference Between Proportions

Confidence Interval for Variance

Where χ² is the critical value from the chi-square distribution.

Sample Size for Estimating Mean

Where E is the desired margin of error.

Sample Size for Estimating Proportion

Where p is the estimated proportion and E is the desired margin of error.

Understanding Confidence Intervals

A confidence interval provides a range of values that is likely to contain the true population parameter with a certain level of confidence. For example, a 95% confidence interval means that if you were to repeat your sampling process many times, about 95% of the resulting intervals would contain the true parameter.

Key points:

• The width depends on the confidence level, sample size, and data variability.
• Higher confidence levels yield wider intervals.
• Larger sample sizes yield narrower intervals.
• More variable data results in wider intervals.

When to Use Different Methods

Z-interval for mean: Use when the population standard deviation is known, or for large samples (n ≥ 30).

T-interval for mean: Use when the population standard deviation is unknown and the sample size is small (n < 30).

Interval for proportion: Use when estimating a population proportion. For small samples or extreme proportions, consider Wilson score or exact methods.

Variance/SD interval: Use when estimating the variability of a population. These intervals are often asymmetric.