Uncertainty is the gap between an estimate and the true value it stands in for, and statisticians measure that gap several ways depending on the question being asked.
This article maps four of those ways:
- Standard error, the unit uncertainty is measured in.
- Margin of error, the ± attached to a single estimate.
- Confidence interval, the range margin of error defines.
- Type I and Type II error, the risk of a wrong decision.
Standard Error: What It Measures
Standard error (SE) measures how much a sample statistic, such as a sample mean, is expected to vary from the true population value purely because of which sample happened to get drawn.
It is calculated as:
SE = σ / √n
where:
- σ: the population standard deviation (or s, the sample standard deviation, when σ is unknown)
- n: the sample size
The √n in the denominator is the important part. Doubling the sample size does not cut standard error in half. Cutting it in half requires quadrupling n.
That relationship sets the price of every other number in this piece. A smaller margin of error, a narrower confidence interval, and a more reliable hypothesis test all cost the same thing: more data, at a steep and non-linear rate.
Margin of Error: The Number Behind ±
Margin of error is the amount added to and subtracted from a point estimate to build a range the true value is likely to fall in.
It is calculated as:
Margin of Error = critical value × SE
where:
- critical value: a z-score or t-score tied to the desired confidence level (commonly 1.96 for 95% confidence)
- SE: the standard error of the estimate
A researcher reporting an estimate states the margin of error alongside it, in a fixed structure:
The estimated [parameter] was [point estimate] ± [margin of error] at the [X]% confidence level.
A wide margin of error is not a mistake in the analysis. It is the honest cost of a small sample, a variable population, or a higher confidence level demanded of the estimate.
Confidence Interval: The Range You’re Actually Claiming
A confidence interval is the finished range built from a point estimate and its margin of error, rather than the plus-or-minus figure by itself.
It is calculated as:
x̄ ± c(s/√n)
where:
- x̄: the sample mean, the point estimate
- c: the critical value
- s/√n: the standard error
A researcher states the finished interval in a fixed structure:
We are [X]% confident that the true [parameter] lies between [lower bound] and [upper bound].
Does “95% confident” mean there is a 95% chance the true value falls inside this particular interval? Not quite. The 95% describes the long-run procedure: if the same sampling process were repeated many times, about 95% of the intervals it produced would contain the true value. This one interval either does or it does not.
Type I and Type II Error: The Risk of Being Wrong
A Type I error is rejecting a null hypothesis that is actually true — a false positive. Its probability is denoted α, conventionally set at α = 0.05.
A Type II error is failing to reject a null hypothesis that is actually false — a false negative. Its probability is denoted β.
The two trade against each other. Tightening α to catch fewer false positives raises the chance of a false negative, for a fixed sample size, and loosening α does the reverse.
The way out of that tradeoff is the same lever named above. A larger sample lowers standard error, which lowers both risks at once instead of trading one for the other.
Statistical power is the probability of correctly detecting a real effect. It is calculated as:
Power = 1 − β
Which error costs more depends on what is being tested. A false positive in a medical screening sends a healthy patient for unnecessary follow-up. A false negative lets a real problem go undetected. Neither risk disappears on its own; choosing a test means choosing which one to weight more heavily.
A researcher reports a hypothesis test’s outcome in a fixed structure:
At α = [significance level], we [rejected / failed to reject] the null hypothesis that [null hypothesis statement].
Additional Resources
The Z-Table
t-Distribution Table
How to Report Regression Results
