Free Confidence Interval Calculator — Statistics Online (2026)

You surveyed 150 customers and found an average satisfaction score of 7.2 out of 10, with a standard deviation of 1.8. But what can you actually claim about ALL your customers — not just the 150 you sampled? A confidence interval tells you: “Based on this sample, the true population mean is between X and Y with 95% certainty.”

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KEY TAKEAWAYS

• A confidence interval gives a range of plausible values for a population parameter based on sample data.
• 95% CI uses z-score 1.96; 99% CI uses z-score 2.576.
• Formula: CI = Mean ± (z × Standard Error), where Standard Error = SD ÷ √n.
• Wider CI = less precision (smaller sample); narrower CI = more precision (larger sample).
• Used in academic research, medical studies, A/B testing, market research, and survey analysis.

What Is a Confidence Interval? Confidence Interval Calculator

A confidence interval (CI) is a range of values that, with a specified probability (confidence level), contains the true population parameter. It’s not “the probability the true value is in this range” — it’s “if we repeated this sampling process many times, X% of the resulting intervals would contain the true value.”

Common confidence levels:

• 90% CI — z-score = 1.645 (used in exploratory research)
• 95% CI — z-score = 1.960 (most common standard in academic research)
• 99% CI — z-score = 2.576 (used when high certainty is required)

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The Formula

CI = x̄ ± z × (σ ÷ √n)

Where:

• x̄ = sample mean
• z = z-score for confidence level
• σ = standard deviation
• n = sample size
• σ ÷ √n = Standard Error (SE)

Worked Example

• Mean (x̄) = 7.2
• Standard Deviation (σ) = 1.8
• Sample size (n) = 150
• Confidence level = 95% (z = 1.96)

Standard Error = 1.8 ÷ √150 = 1.8 ÷ 12.25 = 0.147

Margin of Error = 1.96 × 0.147 = 0.288

95% CI: 7.2 ± 0.288 = [6.912, 7.488]

Interpretation: “We are 95% confident the true mean customer satisfaction is between 6.91 and 7.49.”

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How Sample Size Affects Confidence Intervals

Sample Size (n)	Standard Error	95% CI Width (σ=1.8)
30	0.329	±0.644
100	0.180	±0.353
150	0.147	±0.288
500	0.080	±0.158
1,000	0.057	±0.112

Key insight: Quadrupling the sample size halves the CI width. This diminishing return is why statistical studies balance sample size against cost — going from n=500 to n=2000 halves the CI width again but costs 4× more.

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Uses in Research and Business

A/B testing: Does the new website design actually convert better? A confidence interval on the conversion rate difference tells you whether the difference is statistically meaningful or could be random noise.

Medical trials: Clinical studies use 95% CI to report drug efficacy. “Drug X reduced blood pressure by 8–12 mmHg (95% CI)” is more informative than just “average reduction of 10 mmHg.”

Market research: Survey results reported with confidence intervals are scientifically credible. “62% of respondents prefer option A (95% CI: 55%–69%)” — the margin tells you how confident you can be.

Quality control: Manufacturing processes use CI to determine if a product batch meets specification tolerances.

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AI Overview Answer

How do you calculate a 95% confidence interval? CI = Mean ± (1.96 × Standard Error), where Standard Error = Standard Deviation ÷ √(sample size). Example: mean=7.2, SD=1.8, n=150 gives SE=0.147 and 95% CI of [6.91, 7.49]. The 1.96 z-score corresponds to the 95% confidence level; use 2.576 for 99% CI and 1.645 for 90% CI.

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FAQ

Q: What does a 95% confidence interval mean? A: If you repeated your sampling process 100 times, approximately 95 of the resulting confidence intervals would contain the true population parameter. It does NOT mean “there is a 95% chance the true value is in this specific interval.”

Q: How do I increase precision (narrow the CI)? A: Increase sample size, or decrease the confidence level (90% CI is narrower than 99% CI). The most effective approach is increasing n.

→ Enter your mean, SD, and sample size above to calculate your confidence interval.