Margin of error: formula, examples and how to calculate it

What is margin of error?

The margin of error is the "plus or minus" around a number estimated from a sample. It tells you how far the estimate could plausibly be from the true value because of random sampling alone. "Conversion rate 4.0% ± 0.8 points at 95% confidence" means that if you repeated the measurement many times, intervals built this way would contain the true rate about 95% of the time.

The margin of error is half the width of a confidence interval: estimate − margin to estimate + margin.

Margin of error formula

For a proportion such as a conversion rate, click-through rate or share of users:

MoE = z × √( p × (1 − p) ÷ n )

For a mean such as average order value:

MoE = z × s ÷ √n

• p is the observed proportion, s the sample standard deviation, n the sample size
• z depends on the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%

How to calculate it: a conversion-rate example

A landing page converted 100 of 2,500 visitors, p = 0.04 (sample data).

1. Standard error: √(0.04 × 0.96 ÷ 2,500) = 0.00392
2. Multiply by z for 95%: 1.96 × 0.00392 = 0.0077
3. Margin of error = ±0.77 percentage points. The 95% interval is 3.23% to 4.77%.

The same 4% rate with different sample sizes:

Quadrupling the sample halves the margin of error. Going from ±1.2 to ±0.2 points takes 40 times the traffic.

For a mean: 400 orders with an average of $62 and a standard deviation of $48 give a margin of 1.96 × 48 ÷ √400 = ±$4.70.

Margin of error for a difference

Most analytics questions compare two numbers: this week vs last week, variant A vs B, mobile vs desktop. The margin on a difference is larger than either margin alone:

MoE(p₁ − p₂) = z × √( p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂ )

Two pages with 5,000 visitors each convert at 4.0% and 4.6%. The difference is 0.6 points, but its margin is ±0.80 points. The interval runs from −0.2 to +1.4 points and includes zero, so the data cannot tell the pages apart yet. This is the same logic as a significance test; see statistical significance and p-value.

What changes the margin of error

• Sample size. The margin shrinks with √n.
• Confidence level. 99% confidence widens the margin by about 31% compared with 95% (2.576 vs 1.96).
• The proportion itself. p × (1 − p) is largest at 50%, so a 50% rate has the widest margin for a given n. Rates near 0% or 100% have narrower absolute margins but wider relative ones.
• Population size, sometimes. When the sample is a large share of a small population, multiply by the finite population correction √((N − n) ÷ (N − 1)). Surveying 500 of 3,000 customers at p = 0.5 gives ±4.0 points instead of ±4.4.

How many visitors do you need?

Rearranged for n, with a target margin E:

n = z² × p × (1 − p) ÷ E²

To measure a 4% conversion rate to within ±0.5 points at 95% confidence you need 1.96² × 0.04 × 0.96 ÷ 0.005² ≈ 5,901 visitors; within ±1 point, about 1,476. To plan a test that detects a specific lift, use minimum detectable effect and statistical power instead.

What the margin of error does not cover

The margin of error only measures random sampling error. It says nothing about:

• Biased samples: a survey answered only by happy customers, or analytics that miss visitors who block scripts. See sampling bias.
• Measurement error: bots counted as people, duplicate events, wrong time zone.
• Non-independent data: many events from the same user counted as separate observations. Count users, not events, when users are the unit.

Does analytics data even have a margin of error?

A dashboard that counts every tracked visit is not a survey, so in one sense there is no sampling. In practice the margin still matters whenever you treat this week's numbers as evidence about how the site behaves in general: comparing periods, segments or variants, or judging whether a small segment's rate is real. A segment with 300 visitors and a 4% rate carries a margin of about ±2.2 points, wide enough that the "best converting" segment is often just noise.

If your analytics tool samples data before reporting, the sample adds its own margin on top; see data sampling. TapCub reports on all collected events rather than a sample, so the margin you need to think about is the statistical one described here.

Sources

• Pew Research Center. 5 key things to know about the margin of error in election polls.
• OpenStax. Introductory Statistics, chapter 8: Confidence Intervals.
• NIST/SEMATECH. e-Handbook: confidence limits for the mean.