Confidence Interval Calculator

Calculate two-sided confidence intervals for population means and proportions with z-score and Student's t-distribution support, visual interval diagrams, and margin of error breakdowns.

CI Parameters

Live calculation
Example Scenarios:

When sample size n < 30 with unknown population σ, Student's t-distribution is automatically applied.

95% Confidence Interval
[74.08, 81.92]
Margin of Error (ME)±3.92
Point Estimate: 78Distribution: Standard Normal (z)

Interval Range Visualization

Lower: 74.08Center: 78Upper: 81.92
Standard Error2
Critical Value1.96
Sample Size (n)36
Confidence Level95%

Step-by-Step Calculation Solution

1. SE = σ ÷ √n = 12 ÷ √36 = 2

2. Critical Value (z) for 95% = 1.96

3. Margin of Error = 1.96 × 2 = 3.92

4. 95% CI = [74.08, 81.92]

What Is a Confidence Interval?

A confidence interval (CI)is a range of values derived from sample data that is likely to enclose the true value of an unknown population parameter. Rather than offering a single point estimate (like “average test score is 78”), a confidence interval quantifies sampling uncertainty by stating “we are 95% confident the true population average lies between 74.08 and 81.92.”

1. Confidence Interval for Population Mean (x̄)
CI=x̄ ± [ z* (or t*) ×
σ√n
]
2. Confidence Interval for Proportions (p̂)
CI=p̂ ± [ z* × √[ p̂(1 − p̂) / n ]]
Step-by-Step Calculation Breakdown (Example: x̄ = 78, n = 36, σ = 12, 95% Confidence)
Step 1: Compute Standard Error of the Mean (SE)
• SE = σ ÷ √n = 12 ÷ √36 = 12 ÷ 6 = 2.000
Step 2: Calculate Margin of Error (ME = z* × SE)
• For 95% confidence, z* = 1.960
• ME = 1.960 × 2.000 = 3.920
Step 3: Establish Interval Bounds [x̄ − ME, x̄ + ME]
• Lower Bound = 78 − 3.920 = 74.080
• Upper Bound = 78 + 3.920 = 81.920
• 95% CI = [74.08, 81.92]

Critical Value Reference

Confidence LevelCritical z*Significance (α)Each Tail Area
90%1.6450.105.0%
95%1.9600.052.5%
99%2.5760.010.5%

Frequently Asked Questions

What is a confidence interval in simple terms?
A confidence interval is a plausible range of values constructed from sample data likely to contain the unknown true population parameter. For example, a 95% confidence interval of [48, 52] means that if you repeated the sampling protocol 100 times, roughly 95 of the constructed intervals would capture the true population mean.
What is the difference between 90%, 95%, and 99% confidence levels?
The confidence level governs how broad your interval is. A 95% confidence level (the standard in academic research) uses a z-score of 1.96. A 99% level (z = 2.576) yields a wider margin of error with greater safety, while a 90% level (z = 1.645) gives a narrower interval with a higher 10% risk of missing the parameter.
When should I use a z-distribution vs. a t-distribution?
Use the z-distribution (standard normal) when the population standard deviation (σ) is known or sample size is large (n ≥ 30). Use Student's t-distribution when σ is unknown and estimated by sample s for smaller samples (n < 30), because t-distributions account for extra estimation variance via heavier tails.
How does sample size affect the confidence interval width?
The margin of error is inversely proportional to the square root of sample size: ME = z* × (σ / √n). Quadrupling your sample size (4×) cuts the width of the confidence interval in half (50%).
What is the margin of error and how is it calculated?
The margin of error (ME) is the plus/minus radius added to and subtracted from the point estimate. For means: ME = z* × (σ / √n). For proportions: ME = z* × √[p̂(1 − p̂) / n].
Can I calculate confidence intervals for proportions (percentages)?
Yes. In Proportion Mode, enter the number of successes (x) and total trials (n). The calculator calculates p̂ = x / n and determines the Wald confidence interval with standard error verification.

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