Confidence Interval for Population Mean (Z-Distribution)
Use this calculator for confidence intervals when population standard deviation is known (rare scenario, mostly theoretical).
When to Use This Calculator
- Population standard deviation (σ) is KNOWN (rarely true in practice)
- Single sample from population
- Large sample OR normal population assumed
- Note: In almost all real situations, use the T-distribution version instead
How to Use
Step 1: Enter sample size (n)
Step 2: Enter sample mean (x̄)
Step 3: Enter known population standard deviation (σ)
Step 4: Select confidence level (typically 95%)
Step 5: Click “Calculate”
| Confidence Interval Calculator for mean | |
|---|---|
| Sample Size ($n$) | |
| Sample Mean ($\overline{x}$) | |
| Population Standard Deviation ($\sigma$) | |
| Confidence Level ($1-\alpha$) | |
| Results | |
| Standard Error of Mean: | |
| Z Critical Value : ($Z$) | |
| Margin of Error: ($E$) | |
| Lower Confidence Limits: | |
| Upper Confidence Limits: | |
Formula
$$CI = \overline{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$$
Where:
- $z_{\alpha/2}$ = z-critical value
- $\sigma$ = population standard deviation (known)
- $n$ = sample size
Critical values: 90% → 1.645, 95% → 1.96, 99% → 2.576
Assumptions
- σ is known (population SD known - rare)
- Large sample OR normal population
- Random sample collected
- Independent observations
Worked Example
Scenario: Testing assembly time. Historically σ=2.5 minutes. New sample: n=36, x̄=15.8 minutes. Find 95% CI.
Solution:
- $SE = 2.5/\sqrt{36} = 0.417$
- $z_{0.025} = 1.96$
- $E = 1.96 × 0.417 = 0.817$
- $CI = 15.8 ± 0.817 = [14.98, 16.62]$
Interpretation: We’re 95% confident true mean assembly time is 14.98 to 16.62 minutes.
⚠️ Important: In practice, use T-distribution version - it’s more conservative and accounts for uncertainty in the standard deviation.
Related: T-Distribution CI, Tutorial