Confidence Interval Calculator

Confidence Interval Calculator Mean (Known σ) Mean (Unknown σ) Proportion Difference of Means ...

Confidence Interval Calculator

Confidence Interval for Mean (Known σ)

Sample Mean (x̄)

Population Standard Deviation (σ)

Sample Size (n)

Confidence Level (%)

90% 95% 99% 99.9%

💡 Formula:
CI = x̄ ± Z × (σ/√n)

Confidence Interval for Mean (Unknown σ)

Sample Mean (x̄)

Sample Standard Deviation (s)

Sample Size (n)

Confidence Level (%)

90% 95% 99%

💡 Formula:
CI = x̄ ± t × (s/√n)

Confidence Interval for Proportion

Number of Successes (x)

Sample Size (n)

Confidence Level (%)

90% 95% 99%

💡 Formula:
CI = p̂ ± Z × √(p̂(1-p̂)/n)

Confidence Interval for Difference of Means

Sample 1 Mean (x̄₁)

Sample 1 Standard Deviation (s₁)

Sample 1 Size (n₁)

Sample 2 Mean (x̄₂)

Sample 2 Standard Deviation (s₂)

Sample 2 Size (n₂)

<Confidence Level (%)

90% 95% 99%

💡 Formula:
CI = (x̄₁ - x̄₂) ± t × √(s₁²/n₁ + s₂²/n₂)

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Results

Sample Mean (x̄): 75.00

Population Std Dev (σ): 10.00

Sample Size (n): 100

Confidence Level: 95%

Standard Error: 1.00

Z-Score: 1.96

Margin of Error: ±1.96

Confidence Interval: (73.04, 76.96)

Confidence Interval Visualization

95% CI

x̄ = 75

73.04

76.96

Sample Mean (x̄): 75.00

Sample Std Dev (s): 12.00

Sample Size (n): 25

Confidence Level: 95%

Standard Error: 2.40

T-Score (df=24): 2.064

Margin of Error: ±4.95

Confidence Interval: (70.05, 79.95)

Confidence Interval Visualization

95% CI

x̄ = 75

70.05

79.95

Number of Successes (x): 60

Sample Size (n): 100

Sample Proportion (p̂): 0.60

Confidence Level: 95%

Standard Error: 0.049

Z-Score: 1.96

Margin of Error: ±0.096

Confidence Interval: (0.504, 0.696)

Confidence Interval Visualization

95% CI

p̂ = 0.60

0.504

0.696

Sample 1 Mean (x̄₁): 78.00

Sample 2 Mean (x̄₂): 72.00

Difference of Means: 6.00

Sample 1 Size (n₁): 50

Sample 2 Size (n₂): 45

Confidence Level: 95%

Standard Error: 2.24

T-Score (df≈93): 1.986

Margin of Error: ±4.45

Confidence Interval: (1.55, 10.45)

Confidence Interval Visualization

95% CI

Diff = 6

1.55

10.45

Understanding Confidence Intervals

Confidence intervals are a fundamental concept in statistics that provide a range of values within which we can be reasonably confident that the true population parameter lies. They are essential for making inferences about populations based on sample data.

Confidence level represents the probability that the interval contains the true parameter. A 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each, approximately 95 of those intervals would contain the true population parameter.

Margin of error is half the width of the confidence interval and depends on the sample size, variability in the data, and the chosen confidence level. Larger sample sizes produce narrower (more precise) confidence intervals, while higher confidence levels produce wider intervals.

This calculator helps researchers, students, and analysts compute confidence intervals for means, proportions, and differences between groups, making statistical inference accessible and practical for real-world applications.

Frequently Asked Questions

Q: What's the difference between Z and t distributions?
A: Use the Z-distribution when the population standard deviation is known or when the sample size is large (n ≥ 30). Use the t-distribution when the population standard deviation is unknown and the sample size is small (n < 30). The t-distribution has heavier tails, accounting for additional uncertainty in small samples.

Q: How do I choose the right confidence level?
A: Common confidence levels are 90%, 95%, and 99%. Higher confidence levels provide more certainty but wider intervals. Lower confidence levels give narrower intervals but less certainty. The 95% level is standard in most scientific research, but choose based on your tolerance for error and the consequences of being wrong.

Q: What assumptions are needed for confidence intervals?
A: For means: the sample should be random, and either the population should be normally distributed or the sample size should be large (Central Limit Theorem). For proportions: np ≥ 5 and n(1-p) ≥ 5 (at least 5 successes and 5 failures). For difference of means: independent samples and similar assumptions for each group.

Q: Can confidence intervals be used for hypothesis testing?
A: Yes! If a confidence interval for a difference or effect size does not include zero (or the null value), it's equivalent to rejecting the null hypothesis at the corresponding significance level. For example, a 95% CI that doesn't include zero corresponds to p < 0.05.

Q: What if my confidence interval includes impossible values?
A: This can happen with proportions near 0 or 1, or with small samples. For proportions, consider using the Wilson score interval or exact methods (Clopper-Pearson) instead of the normal approximation. Always check that your interval makes practical sense.

Q: How does sample size affect the confidence interval?
A: Larger sample sizes reduce the standard error, leading to narrower confidence intervals and more precise estimates. The relationship is proportional to 1/√n, so quadrupling the sample size halves the margin of error.

Q: What's the difference between confidence and probability?
A: Once calculated, a specific confidence interval either contains the true parameter or it doesn't - there's no probability involved. The confidence level refers to the long-run frequency: 95% of similarly constructed intervals will contain the true parameter. It's about the method's reliability, not the probability for a specific interval.