Confidence Interval Calculator - Margin of Error (95%, 90%, 99%)

Calculate statistical confidence intervals (90%, 95%, 99%) for sample means, margin of error (MOE), standard error of the mean (SEM), and critical Z/t-values.

⚙️ Configuration & Inputs

Sample Mean (x̄) Calculated average of sample data

Sample Standard Deviation (s) Sample spread or population sigma

Sample Size (n) Number of observations in sample

Confidence Level

📊 Results & Analysis

95% Confidence Interval [73.07, 77.93] 75.50 ± 2.43

Margin of Error (MOE) ± 2.430 ± 3.22% of mean

Standard Error (SE = s / √n) 1.240 Uncertainty of the mean

Critical Value (Z*) 1.960 Normal distribution multiplier

Lower Confidence Bound 73.07 Minimum estimated mean

Upper Confidence Bound 77.93 Maximum estimated mean

How Confidence Intervals & Margins of Error Work

In inferential statistics, we rarely know the exact true mean of an entire population. Instead, we draw a sample and construct a confidence interval to estimate where the true population parameter lies.

1. Margin of Error Formula

Margin of Error (MOE) = Z* × (s / √n)

Z* is the critical value corresponding to your confidence level.
s / √n is the Standard Error of the Mean (SEM).

2. Common Critical Z-Scores

90% Confidence: Z* = 1.645
95% Confidence: Z* = 1.960 (Gold standard in scientific research)
99% Confidence: Z* = 2.576 (Stricter threshold, wider interval)

Frequently Asked Questions

It means that if you repeated the exact same sampling process 100 times, approximately 95 of those constructed intervals would contain the true, unknown population mean.

Because sample size n is under a square root (√n) in the denominator, you must quadruple your sample size (4x) to cut your margin of error in half.

To be more confident (99% certain vs 95% certain) that your interval captures the true value, you must cast a wider net, increasing the margin of error.