This calculator helps you estimate the true population mean with a specified level of confidence, providing both precision and uncertainty measures for your data. It automatically determines whether to use z-test or t-test methods, handles both raw data uploads and manual statistic entry, and provides comprehensive results including confidence intervals, margins of error, critical values, and detailed statistical interpretation.
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Confidence Interval for a Mean: Definition, Formula, and Interpretation
What is a Confidence Interval for a Mean?
A confidence interval for a mean is a statistical range that estimates where the true population mean likely falls, based on sample data. It consists of a lower bound and upper bound, calculated using the sample mean plus and minus a margin of error.
Unlike a point estimate (which gives just one number), a confidence interval acknowledges the uncertainty inherent in sampling and provides a range of plausible values along with our level of confidence in that range.
Simple Example: Imagine you want to know the average height of all students at a university. You measure 50 randomly selected students and find their average height is 68.5 inches.
Instead of saying "the average height is exactly 68.5 inches," a 95% confidence interval might give you [67.2, 69.8] inches.
This means: "Based on our sample, we're 95% confident the true average height of all students at this university is between 67.2 and 69.8 inches."
See It In Action
Want to see how confidence intervals behave with different sample sizes and confidence levels?
Try our Interactive Confidence Interval SimulationConfidence Interval Formulas for a Mean
The formula depends on whether the population standard deviation is known:
When Population Standard Deviation (σ) is Known (Z-Test):
When Population Standard Deviation is Unknown (T-Test):
Where:
- \ar{x} = sample mean
- = critical z-value for desired confidence level
- = critical t-value with degrees of freedom
- = population standard deviation (known)
- = sample standard deviation (when σ unknown)
- = sample size
Understanding Margin of Error
The margin of error is the value that gets added and subtracted from the sample mean to create the confidence interval bounds:
For Z-Test:
\ ext{Margin of Error} = z_{\alpha/2} \cdot \ rac{\sigma}{\sqrt{n}}For T-Test:
\ ext{Margin of Error} = t_{\alpha/2,\;df} \cdot \ rac{s}{\sqrt{n}}Understanding Confidence Levels
The confidence level (commonly 90%, 95%, or 99%) represents how confident we are in our interval estimation method, not in any single interval.
90% Confidence
Narrower interval, less certainty. Good for preliminary analysis.
95% Confidence
Standard choice. Balances precision with confidence.
99% Confidence
Wider interval, higher certainty. Used for critical decisions.
Important: If we repeated our sampling process 100 times and calculated 95% confidence intervals each time, approximately 95 of those intervals would contain the true population mean.
Z-Test vs T-Test: When to Use Which
Use Z-Test When:
- Population standard deviation (σ) is known
- Sample size is large (n ≥ 30) regardless of population distribution
- Population is normally distributed (any sample size)
Uses standard normal distribution
Use T-Test When:
- Population standard deviation is unknown (most common)
- Small sample size (n < 30)
- Must estimate σ using sample standard deviation (s)
Uses t-distribution with n-1 degrees of freedom
Note: As sample size increases, the t-distribution approaches the standard normal distribution, so the choice becomes less critical for large samples.
Interpreting Your CI Results
Correct Interpretation:
"We are 95% confident that the true population mean lies between [lower bound] and [upper bound]."
Common Misinterpretations to Avoid:
❌ "There's a 95% probability the population mean is in this interval."
❌ "95% of the data falls within this interval."
Key Insights from Your Interval:
- Width: Narrower intervals indicate more precise estimates
- Margin of Error: Half the interval width; shows estimation uncertainty
- Sample Size Effect: Larger samples typically produce narrower intervals
- Confidence Level Trade-off: Higher confidence = wider intervals
Interactive Confidence Interval Simulation
Watch how confidence intervals behave across multiple samples. Each line represents a confidence interval from a different sample.
What you're seeing:
- Red dashed line: True population mean (μ = 50)
- Green intervals: Confidence intervals that capture the true mean
- Red intervals: Confidence intervals that miss the true mean
- Dots: Sample means from each sample
Key insight: About 95% of the confidence intervals should capture the true mean. As you run more samples, the capture rate should approach 95%.
Try This:
- Use "+1" button to add samples one by one
- Increase sample size → intervals get narrower
- Increase confidence level → intervals get wider
- Run 50+ samples to see the capture rate stabilize
Remember:
- Each interval either captures μ or it doesn't
- We can't know which intervals are "correct"
- The confidence level is about the method, not individual intervals