04 // Interactive Formula Compendium // STATISTICSCI = x̄ ± z_(α/2) · (s / √n), SE = s / √n
Standard Error & Confidence Interval of the Mean Formula
The margin of error ME = z_(α/2)·(s/√n) scales inversely with the square root of sample size √n, requiring 4× the sample size to halve interval width.
Formula Reference
Main Formula
CI = x̄ ± z_(α/2) · (s / √n), SE = s / √n
Rearranged Forms
n_req = (z_(α/2) · s / E)²
ME = z_(α/2) · SE
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| ME | Margin of Error (±) | units |
| z | Critical Value z_(α/2) | dimensionless |
| s | Sample Std Deviation s | units |
| n | Sample Size n | observations |
Inputs & Parameters
[dimensionless]
[units]
[observations]
Calculated Result (Margin of Error (±))
ME =2.94units
Formula Notes & Background
- For i.i.d. observations with variance σ², Var(x̄) = Var(∑X_i / n) = (1/n²)·nσ² = σ²/n ⇒ SD(x̄) = σ/√n.
- For small n (< 30) with unknown population σ, replace normal quantile z_(α/2) with Student's t_(α/2, n−1).
Step-by-Step Examples
#1: 95% confidence margin of error (z = 1.96) for latency measurements with s = 12.0 ms across n = 64 requests.
z = 1.96s = 12.0 msn = 64
SE = 12.0 / √64 = 1.50 ms; ME = 1.96 × 1.50
→ ME = ±2.94 ms
#2: Sample size n needed to narrow the 95% margin of error to E = ±1.0 ms when s = 12.0 ms.
z = 1.96s = 12.0E = 1.0
n = (1.96 × 12.0 / 1.0)² = 23.52²
→ n = 554 observations (rounded up from 553.2)
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Frequently Asked Questions
What does '95% confidence' rigorously mean?+
Across repeated independent samples, 95% of intervals constructed by x̄ ± 1.96·SE will cover the fixed true population parameter μ.
Does population size N affect the margin of error?+
Almost never unless you sample > 5% of the entire finite population (in which case multiply SE by √((N−n)/(N−1))).
What are the standard z critical values for 90%, 95%, and 99% confidence?+
90% → z = 1.6449; 95% → z = 1.9600; 99% → z = 2.5758.