Confidence Interval Calculator

Calculate confidence intervals for statistical analysis

Perfect for population mean and proportion estimation with customizable confidence levels

Last updated: December 2, 2025

CreatorFrank Zhao

Calculate confidence interval for...

What data do you have?

Sample mean (x̄)

Std deviation (s)

Sample size (n)

I know the desired value of:

C-level

Understanding Confidence Intervals

What is a Confidence Interval?

A confidence interval represents a range within which we expect a population parameter to fall, based on sample data. Rather than providing a single point estimate, it gives us both a lower and upper boundary, along with a confidence level that expresses our degree of certainty.

Think of it this way: A coffee shop owner samples 50 customers and finds their average spending is $4.20 per visit. The 95% confidence interval might be $3.80 to $4.60. This means: if we repeatedly sampled different groups of 50 customers and calculated the interval each time, approximately 95% of those intervals would capture the true average spending of all customers.

The confidence level (commonly 90%, 95%, or 99%) doesn't mean there's a 95% chance the true value is in this specific interval. Instead, it means our method produces intervals that contain the true value 95% of the time when repeated.

How to Use the Confidence Interval Calculator

Our calculator simplifies the entire process. Follow these straightforward steps:

Select calculation type: Choose between population mean (for numerical data like heights, prices) or population proportion (for categorical data like yes/no responses, pass/fail rates)
For Mean calculations: Specify whether you have the standard deviation or standard error, then enter your sample mean, standard deviation/error, and sample size
For Proportion calculations: Enter the sample proportion (as a decimal between 0 and 1) and your sample size
Set confidence level: Choose a standard level (90%, 95%, 99%) or enter a custom Z-score or p-value if you have specific requirements
View results: The calculator instantly displays your margin of error, lower bound, and upper bound

Confidence Interval Formulas

For Population Mean:

CI = x̄ ± (Z × SE)

Where:

• x̄ = Sample mean (average of your data points)
• Z = Critical Z-score (determined by confidence level)
• SE = Standard error = σ / √n
• σ = Standard deviation of the sample
• n = Sample size (number of observations)

For Population Proportion:

CI = p ± (Z × √[p(1-p)/n])

Where:

• p = Sample proportion (successes / total sample)
• Z = Critical Z-score
• n = Sample size

Margin of Error:

The margin of error (ME) is the amount added and subtracted from the point estimate. For means: ME = Z × SE. For proportions: ME = Z × √[p(1-p)/n]

Common Z-Scores for Confidence Levels

90% Confidence

Z = 1.645

One-tailed α = 0.05

95% Confidence

Z = 1.96

One-tailed α = 0.025

99% Confidence

Z = 2.576

One-tailed α = 0.005

Note: Higher confidence levels produce wider intervals. A 99% confidence interval will be wider than a 95% interval for the same data, reflecting greater certainty.

Calculation Example: Step by Step

Scenario: Quality Control in Manufacturing

A factory measures the weight of 40 product samples. The average weight is 250g with a standard deviation of 15g. Calculate the 95% confidence interval for the true average weight.

Step 1: Identify values — Sample mean (x̄) = 250g, Standard deviation (σ) = 15g, Sample size (n) = 40
Step 2: Find Z-score for 95% confidence — Z = 1.96
Step 3: Calculate standard error — SE = σ/√n = 15/√40 = 15/6.32 = 2.37g
Step 4: Calculate margin of error — ME = Z × SE = 1.96 × 2.37 = 4.65g
Step 5: Find interval bounds — Lower: 250 - 4.65 = 245.35g, Upper: 250 + 4.65 = 254.65g
Result: We are 95% confident the true average product weight is between 245.35g and 254.65g.

Real-World Applications

🏥

Healthcare & Medicine

Estimating the effectiveness of new treatments, average patient recovery times, or the proportion of patients responding to therapy

📊

Market Research

Determining consumer preferences, average purchase amounts, customer satisfaction scores, or brand awareness percentages

🔬

Scientific Research

Reporting experimental results with uncertainty ranges, comparing treatment groups, or validating measurement accuracy

🏭

Quality Control

Monitoring production processes, ensuring product specifications are met, or detecting manufacturing variations

📈

Finance & Economics

Forecasting stock returns, estimating risk metrics, analyzing economic indicators, or projecting sales figures

🗳️

Political Polling

Estimating election outcomes, measuring public opinion on policies, or gauging approval ratings with known margins of error

How to Interpret Your Results

✓ Correct Interpretation:

"If we repeated this sampling process many times, about 95% of the calculated intervals would contain the true population parameter."

✗ Common Misconception:

"There is a 95% probability that the true value is in this specific interval." (The true value either is or isn't in the interval — it's our method that's reliable 95% of the time.)

⚠ Important Considerations:

• Larger samples generally produce narrower, more precise intervals
• Higher confidence levels create wider intervals (more certainty requires more range)
• The interval's width reflects both sample variability and sample size
• Always ensure your sample is representative of the population

Frequently Asked Questions

What makes a confidence interval wider or narrower?

Three main factors affect interval width: (1) Confidence level — higher confidence = wider interval, (2) Sample size — larger sample = narrower interval, and (3) Variability — more variable data = wider interval.

When should I use 95% vs 99% confidence?

Use 95% for most general purposes — it balances precision and confidence well. Choose 99% when errors are costly (medical decisions, safety-critical engineering) and you need greater certainty, accepting a wider interval as the trade-off.

What's the difference between standard deviation and standard error?

Standard deviation measures the spread of individual data points in your sample. Standard error measures the precision of the sample mean as an estimate of the population mean. SE = SD / √n, so it decreases as sample size increases.

Can I use this for small samples?

This calculator uses Z-scores, which assume either: (1) the population is normally distributed, or (2) your sample size is large enough (usually n ≥ 30) for the Central Limit Theorem to apply. For small samples from non-normal populations, consider using a t-distribution instead.

What does the margin of error represent?

The margin of error is the ± amount around your point estimate. It shows the maximum expected difference between your sample statistic and the true population parameter, at your chosen confidence level. Smaller margins indicate more precise estimates.