How Sure, Give or Take
A confidence interval reports an estimate plus a margin of error, and the confidence level states how often intervals built this way capture the true value. · 12 min
You have a sample, and from it a single number — 52% of the people asked support the measure. That number is your best guess for the whole population, but it is almost certainly not exactly right. A different sample would have handed you a slightly different figure. So instead of reporting one number and pretending it is exact, statisticians report a range, together with a statement of how much to trust it. That range is what this folio builds.
Guess before you learn
Two surveys both estimate support at 50%. Survey A asked 100 people; Survey B asked 1,600 people. Survey B's margin of error is about how many times smaller than Survey A's?
Multiplying the sample by 16 does not shrink the margin by 16. The margin depends on the square root of the sample size, and the square root of 16 is 4 — so Survey B's margin is about 4 times smaller. Keep your pencil mark; the whole idea of a margin of error is coming.
Reporting a range has a name and a shape. It is called a confidence interval, and it always has two parts: an estimate in the middle, and a margin of error reaching out on either side. The confidence level — the percentage you see quoted, like 95% — describes how often this method lands on the truth.
9–12
3–5
You asked a few classmates and 6 out of 10 liked the new lunch. That is a clue about the whole school, not the exact answer. A fair report says 'about 6 in 10, give or take a bit,' and adds how sure you are that the truth is nearby.
6–8
A confidence interval reports an estimate plus or minus a margin of error. If a poll finds 52% support with a margin of 5%, the interval runs from 47% to 57%. The confidence level, usually 95%, tells you how dependable the method is: build intervals this way many times, and about 95% of them will contain the true value.
9–12
A confidence interval has the form estimate ± (margin of error), where the margin is a multiplier times the standard error of the estimate. For 95% confidence the multiplier is about 2, borrowed straight from the normal curve's 68–95–99.7 rule. The confidence level describes the procedure, not one interval: if you repeated the sampling many times and built an interval each time, about 95% of those intervals would capture the true parameter. The parameter itself is a fixed number; the interval is what changes from sample to sample.
K–2
You guess a jar holds about 40 jellybeans. You are not sure, so you say 'maybe 5 more or 5 less.' The 5 is your give-or-take. Saying a range is honest when you cannot be exact.
Undergrad
Treat the interval as random. For a parameter θ estimated by θ̂ with standard error SE, the interval θ̂ ± 1.96·SE is a random interval whose endpoints vary with the sample. Its coverage probability — the chance the random interval contains the fixed θ — is 0.95 by construction. A realized interval, once computed, either contains θ or does not; the 95% is a property of the method. That is exactly why 'there is a 95% probability θ lies in this interval' is the wrong reading.
Postgrad
A 1 − α confidence procedure is a map from data to sets C(X) satisfying P_θ(θ ∈ C(X)) ≥ 1 − α for every θ. Coverage is a frequentist guarantee over the sampling distribution, not a posterior probability — the Bayesian credible interval answers the probability question and generally does not coincide unless the prior is chosen to match. Confidence intervals are dual to hypothesis tests: C(X) is the set of null values not rejected at level α. The width scales with SE ∝ 1/√n, fixing the sample-size arithmetic behind any stated margin.
confidence interval
An estimate together with a margin of error: a range of plausible values for a population number, reported with a confidence level.
margin of error
How far the interval reaches on each side of the estimate. It equals a multiplier times the standard error.
Why is this true?
Why must a 99% interval be wider than a 95% interval from the same data?
To capture the true value a larger share of the time, the interval must reach further on each side. With the same data the center and the standard error are fixed, so more confidence can only come from a larger multiplier — and a larger multiplier lengthens the interval both ways.
Where does the margin come from? Two ingredients. The first is the standard error — the typical distance between a sample estimate and the truth, which you met when a statistic bounced from one sample to the next. The second is a multiplier set by the confidence level: about 2 for 95%, drawn from the normal curve. Multiply them, and you have the margin of error.
Build a 95% confidence interval: 52% support, n = 400 — the steps fade as you master them
√(0.2496 ÷ 400) = √0.000624
2 × 0.025 = 0.05
0.52 − 0.05 and 0.52 + 0.05
47% to 57%
So a confidence interval is an honest estimate: a center you computed, a margin that admits your uncertainty, and a confidence level that says how often the method works. Next you turn from one variable to two — and start asking whether they move together.
Practice — new ink and old, interleaved
1.You spread a dataset out so every value's distance from the mean doubles. What happens to the standard deviation?
2.A value has a z-score of exactly 0. What does that tell you?
3.The sample mean's distribution is approximately normal even when the population is skewed. This result is:
4.For an approximately normal sampling distribution, about what percent of sample means fall within 2 standard errors of the true mean? Enter a whole number.
5.An estimate of 30 has a standard error of 5. Give the upper end of the 95% interval, using a multiplier of 2.
6.A value of 74 comes from a distribution with mean 65 and standard deviation 6. What is its z-score?
7.Standard error measures which of these?
8.'95% confident' means the true value...
9.You take a larger sample. What happens to the sampling variability of the sample mean?