Confidence Interval Calculator Formula

Understand the math behind the confidence interval calculator. Each variable explained with a worked example.

Use the Confidence Interval Calculator

Formulas Used

Lower Bound

lower = x_bar - moe

Upper Bound

upper = x_bar + moe

Margin of Error

margin_of_error = moe

Standard Error

standard_error = se

Variables

Variable	Description	Default
`x_bar`	Sample Mean	50
`sigma`	Standard Deviation	10
`n`	Sample Size	36
`z`	Z-Value (e.g., 1.96 for 95%)	1.96
`se`	Derived value`= sigma / sqrt(n)`	calculated
`moe`	Derived value`= z * sigma / sqrt(n)`	calculated

How It Works

How to Calculate a Confidence Interval

Formula

CI = x_bar +/- z * (sigma / sqrt(n))

The confidence interval gives a range of plausible values for the population mean. The standard error (sigma/sqrt(n)) measures how much the sample mean varies across samples. The z-value determines the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.

Worked Example

Sample mean = 50, SD = 10, n = 36. Build a 95% confidence interval.

x_bar = 50sigma = 10n = 36z = 1.96

01Standard Error = 10 / sqrt(36) = 10 / 6 = 1.6667
02Margin of Error = 1.96 * 1.6667 = 3.2667
03Lower bound = 50 - 3.2667 = 46.7333
04Upper bound = 50 + 3.2667 = 53.2667
0595% CI: (46.7333, 53.2667)

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