How do I interpret a confidence interval?

A 95% CI of (42, 48) for mean height means we are 95% confident the population mean is between 42 and 48. Narrower intervals indicate more precision. If comparing groups, non-overlapping CIs suggest a significant difference.

How does sample size affect confidence intervals?

Larger samples give narrower confidence intervals. Margin of error decreases with √n - quadrupling sample size halves the margin of error. This is why larger studies provide more precise estimates. Diminishing returns kick in at very large samples.

What confidence level should I use?

95% is standard in most fields. Use 99% when errors are costly (medical research, safety testing). Use 90% for preliminary analysis or when resources are limited. Higher confidence = wider interval. Choose before collecting data, not after.

How do I calculate sample size for a desired margin of error?

Use n = (z × σ / E)², where z is the critical value, σ is standard deviation, and E is desired margin of error. For 95% CI with SD=10 and margin of error ±2: n = (1.96 × 10 / 2)² = 96. Always round up.

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Confidence Interval Calculator

Find margin of error for surveys and research

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Last verified: February 2026Accuracy tested

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Interval Type

Distribution

Confidence Level

Sample Mean (x̄)

Std Dev (s)

Sample Size (n)

95% Confidence Interval for μ

(0.00, 0.00)

Visual Representation

Sample Mean (x̄)

Std Error

2.1909

Critical (t*)

2.060

Margin of Error

±4.513

Step-by-Step Calculation

Step 1: Standard Error

SE = s / √n = 12 / √30 = 2.1909

Step 2: Critical Value (t with df=29)

t* = 2.060

Step 3: Margin of Error

ME = t* × SE = 2.060 × 2.1909 = 4.5132

Step 4: Confidence Interval

CI = x̄ ± ME = 85 ± 4.5132 = (80.4868, 89.5132)

Interpretation

We are 95% confident that the true population mean lies between 80.49 and 89.51.

Critical Z-Values

80%

z* = 1.282

85%

z* = 1.44

90%

z* = 1.645

95%

z* = 1.96

99%

z* = 2.576

When to Use Each

t-interval

Use when population σ is unknown (most real-world cases). Requires sample standard deviation (s).

z-interval

Use when population σ is known or n ≥ 30 and you have σ. Also used for proportions.

Sources & Methodology

Formula: CI = x̄ ± z × (σ/√n)

Confidence interval for mean

Source: Statistical inference theory

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How to Use the Confidence Interval Calculator

The confidence interval calculator computes confidence intervals for means and proportions at various confidence levels. Visualize the interval with error bars and determine the sample size needed for your desired margin of error.

What is a Confidence Interval?

A confidence interval is a range of values that likely contains the true population parameter. A 95% confidence interval means: if we repeated the sampling process many times, about 95% of the intervals would contain the true value.

The Formula

Confidence Interval = Sample Statistic ± Margin of Error

Margin of Error = Critical Value × Standard Error

Components

Sample Mean (x̄): The center of your interval
Standard Deviation (s): Measure of spread
Sample Size (n): Number of observations
Confidence Level: Usually 90%, 95%, or 99%

Critical Values

Common z-values for different confidence levels:

90% confidence: z = 1.645
95% confidence: z = 1.96
99% confidence: z = 2.576

Use t-values instead of z-values for small samples (n < 30).

Interpreting Confidence Intervals

A 95% CI of (42.5, 47.5) means: "We are 95% confident that the true population mean is between 42.5 and 47.5."

Sample Size Determination

To achieve a specific margin of error:

n = (z × σ / E)²

where E is the desired margin of error.

Related Calculators

For p-values and hypothesis testing, use our p-value calculator. For standard deviation, try our standard deviation calculator.

Frequently Asked Questions

95% confidence means: if we repeated the sampling and calculated intervals many times, about 95% of those intervals would contain the true population parameter. It does NOT mean theres a 95% chance the true value is in this specific interval - the true value either is or isnt in there.