Confidence Level

Usually you pick a confidence level and solve for the margin of error. This calculator runs it backwards: given your sample size and the precision you want to report, what confidence level can you actually claim?

How the calculation works

It starts from the standard margin-of-error formula for a mean, E = z × σ / √n, and solves for z: z = E × √n / σ. The confidence level is the area under the standard normal curve between −z and +z, CL = 2Φ(z) − 1, where Φ is the normal cumulative distribution function. The significance level is α = 1 − CL, the probability left in the two tails.

Inputs: the margin of error E — the half-width of the interval (“± 2 points”), in the same units as your data; the standard deviation σ of the population, or a reliable estimate; and the sample size n. Outputs: the confidence level in percent, the critical value z and α in percent.

The calculation is two-sided and uses the normal (z) approximation. It is appropriate when σ is known or the sample is reasonably large (several dozen observations or more) and drawn at random.

Worked example

With the defaults E = 2, σ = 10 and n = 100: z = 2 × √100 / 10 = 2, so the confidence level is 2Φ(2) − 1 ≈ 95.45% and α ≈ 4.55%. Relax the margin to ±3 with the same σ and n and z becomes 3, giving 99.73%.

Things to keep in mind

• For a proportion (polls, conversion rates) use σ = √(p(1 − p)) and express E as a fraction: p = 0.5, E = 0.03 and n = 1,000 give z ≈ 1.897 and about 94.2% confidence.
• With a small sample and σ estimated from the same data, the Student t distribution is the correct reference; the true confidence will be lower than the z-based figure.
• Handy anchors: z = 1.645 → 90%, z = 1.96 → 95%, z = 2.576 → 99%.
• Confidence describes the method, not one interval: if you repeated the sampling many times, about that share of the intervals would contain the true value.
• The formula covers random sampling error only. Biased samples, non-response and measurement error are not fixed by any confidence level.