Confidence Interval for Mean Calculator
Calculate population mean confidence intervals ($\bar{x} \pm z^* \cdot \frac{s}{\sqrt{n}}$) at 90%, 95%, or 99% confidence levels, margin of error, and SEM.
Calculate Population Mean Confidence Interval
Enter sample mean (x̄), sample SD (s), sample size (n), and confidence level.
Calculation Results
Calculated using confidence interval for mean equations: \text{CI} = \bar{x} \pm z^* \cdot \left(\frac{s}{\sqrt{n}}\right)
Quick Summary
The Confidence Interval for Mean Calculator evaluates population mean confidence bounds ($\bar{x} \pm z^* \cdot \text{SEM}$), margin of error, and interval width.
Formula Explanation
\text{CI} = \bar{x} \pm z^* \cdot \left(\frac{s}{\sqrt{n}}\right)
\text{Margin of Error ME} = z^* \cdot \text{SEM}How It Works
A confidence interval estimates the plausible range for an unknown population mean $\mu$. The Confidence Interval for Mean Calculator evaluates standard error $\text{SEM} = s / \sqrt{n}$ and multiplies by critical value $z^*$.
Step-by-Step Worked Example
Practical Problem: Sample mean $\bar{x} = 100$, SD $s = 15$, sample size $n = 36$. Calculate 95% confidence interval for population mean $\mu$.
- Step 1: Identify critical multiplier ($z^*$ for 95%): $z^* = \mathbf{1.9600}$.
- Step 2: Calculate Standard Error ($\text{SEM}$): $\text{SEM} = 15 / \sqrt{36} = 15 / 6 = \mathbf{2.5000}$.
- Step 3: Calculate Margin of Error ($\text{ME}$): $\text{ME} = 1.960 \times 2.50 = \mathbf{4.9000}$.
- Step 4: Subtract ME from mean for lower bound: $100 - 4.90 = \mathbf{95.1000}$.
- Step 5: Add ME to mean for upper bound: $100 + 4.90 = \mathbf{104.9000\text{ (95\% CI: [95.10, 104.90])}}.$
Real-World Calculation Examples
Scenario 1: 95% CI (x̄ = 100, s = 15, n = 36)
Parameters: x̄ = 100, s = 15, n = 36
Result: [95.10, 104.90] (ME = ±4.90).
Scenario 2: Manufacturing Component Length Audit
Parameters: x̄ = 50.2mm, s = 0.8mm, n = 100
Result: 95% CI: [50.04, 50.36] mm.
Scenario 3: Customer Satisfaction Score (n = 400)
Parameters: Large sample survey mean
Result: High-precision narrow confidence interval.
Scenario 4: Patient Blood Pressure Trial
Parameters: Medical clinical trial mean
Result: 99% confidence interval estimation.
Key Benefits of Using This Calculator
Flexible Confidence Levels
Computes intervals for 90%, 95%, and 99% confidence levels.
Margin of Error & SEM
Outputs standard error $\text{SEM}$ and margin of error $\text{ME}$.
Interval Width Metrics
Displays full interval width ($2 \times \text{ME}$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a confidence interval for a mean?
A confidence interval estimates a range of values likely to contain the true population mean $\mu$ with a specified level of confidence.
What does 95% confidence mean?
It means that if we repeat the sampling process 100 times, approximately 95 of the generated confidence intervals will contain the true population mean.
How do I calculate a confidence interval in Excel?
Use formula =CONFIDENCE.NORM(alpha, standard_dev, size) or =CONFIDENCE.T.
What happens to interval width as sample size n increases?
As $n$ increases, standard error decreases ($\text{SEM} \propto 1/\sqrt{n}$), making the confidence interval NARROWER and more precise.
What happens to interval width as confidence level increases?
Higher confidence levels (e.g. 99% vs 90%) require larger critical multipliers $z^*$, making the interval WIDER.
When should I use t-distribution critical values instead of Z?
Use $t$-distribution critical values ($t^*$) when sample size is small ($n < 30$) and population SD is unknown.
What is margin of error?
Margin of error is the radius of the confidence interval ($\text{ME} = z^* \cdot \text{SEM}$).
Does a 95% CI mean there is a 95% probability the true mean is inside?
In frequentist statistics, the true mean is a fixed unknown value; the 95% probability applies to the long-run capture rate of the sampling procedure.
What is relationship between 95% CI and 2-tailed hypothesis test at α = 0.05?
If hypothesized mean $\mu_0$ falls OUTSIDE the 95% CI, $H_0$ is rejected at $\alpha = 0.05$.
How does sample standard deviation s affect interval width?
Higher variability (larger $s$) produces wider confidence intervals.