Confidence Interval Calculator
Calculate confidence intervals for means and proportions with interactive visualization, assumption checking, and detailed statistical interpretation.
Confidence Interval Calculator
Known σ uses Z-distribution, sample s uses t-distribution (unless n ≥ 30)
Confidence Interval Results
95% Confidence Interval for Population Mean
[70.706, 79.294]
Margin of Error: ±4.294
What This Means
We are 95% confident that the true population mean lies between 70.706 and 79.294.
Confidence Level: If we repeated this sampling process many times,95% of the intervals would contain the true population mean.
Interval Width: 8.588(narrower intervals are more precise)
Lower Bound
70.7059
Minimum likely value
Upper Bound
79.2941
Maximum likely value
Practical Implications
• The true population mean is very likely between 70.71 and 79.29
• This interval has a width of 8.59 units
• To get a narrower interval: increase sample size or decrease confidence level
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What is Confidence Interval Calculator?
A confidence interval provides a range of plausible values for a population parameter (mean or proportion) based on sample data. It gives you both an estimate and a measure of the uncertainty in that estimate, helping you understand how precise your sample results are.
Understanding Confidence Intervals
- Confidence Level: The percentage of intervals that would contain the true parameter if sampling were repeated
- Margin of Error: The maximum expected difference between the sample statistic and population parameter
- Interval Width: Affected by sample size, variability, and confidence level
- Point Estimate: The sample statistic at the center of the interval
Types of Confidence Intervals
- Mean (μ): Uses Z-distribution (known σ, large n) or t-distribution (unknown σ, small n)
- Proportion (p): Uses Z-distribution with normal approximation (requires np ≥ 5 and n(1-p) ≥ 5)
- Difference of Means: For comparing two populations
- Difference of Proportions: For comparing two proportions
Common Confidence Levels
- 90% Confidence: Z = 1.645, captures 90% of possible intervals
- 95% Confidence: Z = 1.96, most commonly used in research
- 99% Confidence: Z = 2.576, higher confidence but wider intervals
Real-World Applications
- Medical Research: Treatment effectiveness, drug dosage ranges
- Market Research: Customer satisfaction rates, market share estimates
- Quality Control: Process capability, defect rates
- Political Polling: Election predictions, approval ratings
- Education: Test score populations, intervention effectiveness
- Economics: Unemployment rates, inflation estimates
Factors Affecting Interval Width
- Sample Size (n): Larger samples → narrower intervals
- Confidence Level: Higher confidence → wider intervals
- Population Variability: More variability → wider intervals
- Measurement Precision: Better measurement → narrower intervals
Common Misconceptions
- A 95% CI doesn't mean there's a 95% chance the parameter is in the interval
- It means 95% of such intervals would contain the true parameter
- Wider intervals aren't "worse" - they're more honest about uncertainty
- The interval is about the parameter, not future individual observations
FAQ - Confidence Interval Calculator
A 95% confidence interval means that if you repeated your sampling process many times, 95% of the intervals you calculate would contain the true population parameter. It's about the long-run performance of the method, not the probability for this specific interval.