Calculate CI for Mean with Step-by-Step Derivation
Sample Mean (x̄)
Std Dev (σ/s)
Sample Size (n)
Confidence Level
Results
Confidence Interval
Lower Bound
Upper Bound
Step-by-Step Derivation
Confidence Interval Formula
CI = x̄ ± (critical value) × (SE)
SE = σ / √n (standard error)
Z-critical: for σ known or n≥30
T-critical: for σ unknown and n<30
Confidence interval provides a range estimate of the population parameter.
⚠Use z-interval when σ known or n≥30. Use t-interval when σ unknown and n<30.
What is Confidence Interval?
A confidence interval is a range of values that is likely to contain an unknown population parameter. It provides a measure of uncertainty around the sample estimate.
Confidence Level
95% means 95 out of 100 intervals will contain true value
Confidence interval is a range of values likely to contain the population parameter. A 95% CI means we are 95% confident the true value lies within the interval.
What confidence level should I use?▼
95% is standard in most fields. 90% gives narrower interval (less confident), 99% gives wider interval (more confident). Choose based on required certainty.
Z vs T confidence interval?▼
Use z-interval when σ known or n≥30. Use t-interval when σ unknown and n<30. T-distribution accounts for extra uncertainty with small samples.
What affects interval width?▼
Sample size (larger n = narrower), confidence level (higher = wider), standard deviation (higher = wider).
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.