03 // Bidirectional STEM Calculators // STATISTICSMOE = z × (σ / √n)

Confidence Interval Margin of Error Calculator

The Margin of Error (MOE) defines the half-width ± radius of a two-sided confidence interval around a sample mean at critical score z: MOE = z × (σ / √n).

MOE = z × (σ / √n)

Pick which value you want to calculate (or type into any field):Solving for: MOE (±) — Margin of Error

Margin of ErrorCalculated

Half-width of the confidence interval (± units)

0.011,000.00

unit

Critical Z-Value

1.645 (90% CI), 1.960 (95% CI), or 2.576 (99% CI)

0.54

unit

Standard Deviation

Population or sample standard deviation

0.015,000.00

unit

Sample Size

Number of independent observations

2100,000.00

unit

Calculated Result · Margin of Error (MOE (±))

1.96

Step-by-Step Calculation

01Canonical Law: MOE = z × (σ / √n)

02Given Inputs (Base Units): z* = 1.96, σ = 10, n = 100

03Solve for MOE (±) (Margin of Error): MOE (±) = 1.96

The Margin of Error (MOE) defines the half-width ± radius of a two-sided confidence interval around a sample mean at critical score z: MOE = z × (σ / √n).

Real-World Reference Values

90% Confidence Levelz* = 1.6449

10% two-tailed α risk

95% Confidence Level (Standard)z* = 1.9600

5% two-tailed α risk

99% Confidence Levelz* = 2.5758

1% two-tailed α risk

20-Row Comparison Table

Margin of Error (MOE (±)) · Critical Z-Value (z*)

Compare across:

Confidence Interval Margin of Error Calculator — Critical Z-Value → Margin of Error
#	Critical Z-Value (z*)	Margin of Error (MOE (±))	Δ %
01	0.7252	0.7252	-63.00%
02	0.869621	0.869621	-55.63%
03	1.014	1.014	-48.26%
04	1.1585	1.1585	-40.89%
05	1.3029	1.3029	-33.53%
06	1.4473	1.4473	-26.16%
07	1.5917	1.5917	-18.79%
08	1.7361	1.7361	-11.42%
09	1.8806	1.8806	-4.05%
10	1.96 Current	1.96	0.00%
11	2.1694	2.1694	+10.68%
12	2.3138	2.3138	+18.05%
13	2.4583	2.4583	+25.42%
14	2.6027	2.6027	+32.79%
15	2.7471	2.7471	+40.16%
16	2.8915	2.8915	+47.53%
17	3.0359	3.0359	+54.89%
18	3.1804	3.1804	+62.26%
19	3.3248	3.3248	+69.63%
20	3.4692	3.4692	+77.00%

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Frequently Asked Questions

How do I find the sample size n needed for a specific Margin of Error?+

Click SOLVE FOR on n (Sample Size) and enter your target MOE, standard deviation σ, and confidence critical value z*.

Why does increasing confidence from 95% to 99% widen the Margin of Error?+

Being more certain that the interval captures the true parameter requires a wider net, raising z* from 1.960 to 2.576 (+31.4% wider).

How do I construct the full Confidence Interval from MOE?+

Add and subtract MOE from your sample mean: CI = [x̄ - MOE, x̄ + MOE].