Confidence Interval Calculator
Calculate confidence intervals to estimate population parameters with a specified level of confidence.
Understanding Confidence Intervals
Confidence intervals provide a range of values that likely contains an unknown population parameter. They help quantify the uncertainty in sample estimates and make statistical inferences about populations.
What is a Confidence Interval?
A confidence interval is a range of values that has a specified probability of containing the true population parameter. This probability is known as the confidence level, commonly set at 95%.
Formula: CI = x̄ ± (critical value × standard error)
Where:
- CI = Confidence Interval
- x̄ = Sample mean
- Critical value = Z or t value based on confidence level
- Standard error = s/√n (sample standard deviation divided by square root of sample size)
Z-Intervals vs. T-Intervals
Z-Interval:
- Used when population standard deviation (σ) is known
- Also appropriate for large samples (n ≥ 30) even if σ is unknown
- Uses the normal distribution to determine critical values
T-Interval:
- Used when population standard deviation (σ) is unknown and sample size is small (n < 30)
- Uses the t-distribution, which has heavier tails than the normal distribution
- Accounts for the additional uncertainty when estimating σ from a small sample
Interpreting Confidence Intervals
A 95% confidence interval does NOT mean there is a 95% probability that the true parameter lies within the interval. Instead, it means:
- If you were to take many random samples and construct a 95% confidence interval from each sample, about 95% of these intervals would contain the true population parameter.
- The confidence level (95%) refers to the method used to construct the interval, not to any single interval.
- Once calculated, a specific confidence interval either contains the true parameter or it doesn't.
Factors Affecting Confidence Interval Width
- Confidence Level: Higher confidence levels (e.g., 99% vs. 95%) result in wider intervals
- Sample Size (n): Larger samples produce narrower confidence intervals
- Variability (standard deviation): More variable data leads to wider confidence intervals
Common Uses of Confidence Intervals
- Estimating Population Parameters: Mean, proportion, difference between means, etc.
- Evaluating Precision: Narrower intervals indicate more precise estimates
- Hypothesis Testing: If a hypothesized value falls outside the confidence interval, it can be rejected
- Decision Making: Providing ranges for important metrics in research, business, and policy decisions
- Comparing Groups: Non-overlapping confidence intervals suggest significant differences between groups
When to Use Confidence Intervals
- When estimating unknown population parameters from sample data
- When reporting research findings with appropriate margins of error
- When determining the precision of your estimates
- When making inferences about populations based on sample data
- When comparing estimates across different groups or populations
How to Use This Calculator
- Enter your dataset in the text box, separating values with commas, spaces, or new lines
- Select your desired confidence level (90%, 95%, 99%, or 99.9%)
- Choose your preferred decimal precision for the results
- Select the calculation method:
- Z-interval: For known population standard deviation or large samples (n ≥ 30)
- T-interval: For unknown population standard deviation with small samples (n < 30)
- If using Z-interval with known population standard deviation, enter the value
- Click "Calculate" to see the results
- Interpret the confidence interval and statistical summary