Unit 04.04: Width is the honest measure of what you know

An interval says what range of values the data is consistent with. Its width is the honest measure of how much you know.

The interval is the result

A confidence interval for a mean comes from t.test() and one for a proportion from prop.test(). Both report the estimate with a range around it, and the range is the part worth reading.

Width is driven by the sample size, through the same square root as the standard error. Quadrupling n roughly halves the width, which is the practical fact behind every sample-size argument you will ever have.

Reporting the interval instead of the point estimate changes the conversation. '42%' invites a decision; '42%, somewhere between 35% and 49%' invites the right one.

This block builds intervals for a mean and a proportion, then varies n.

set.seed(105)

# Mean
values <- rnorm(50, mean = 72, sd = 11)
tt <- t.test(values)
cat("Mean:", round(mean(values), 2), "\n")
cat("95% CI:", round(tt$conf.int[1], 2), "to", round(tt$conf.int[2], 2), "\n")
cat("Width :", round(diff(tt$conf.int), 2), "\n\n")

# Proportion
successes <- 84; n <- 200
pt <- prop.test(successes, n)
cat("Proportion:", successes / n, "\n")
cat("95% CI:", round(pt$conf.int[1], 4), "to", round(pt$conf.int[2], 4), "\n")
cat("Width :", round(diff(pt$conf.int), 4), "\n\n")

cat("Width against sample size, same proportion:\n")
for (nn in c(50, 200, 800, 3200)) {
  ci <- prop.test(round(0.42 * nn), nn)$conf.int
  cat(sprintf("  n = %4d  CI %.3f to %.3f  width %.3f\n",
              nn, ci[1], ci[2], diff(ci)))
}
cat("\nEach fourfold increase in n roughly halves the width.\n")

The mean of 71.36 carries an interval from 68.73 to 73.98, width 5.26. The proportion of 0.42 from 200 observations has an interval from 0.351 to 0.492, width 0.140 — fourteen percentage points, which is a great deal more uncertainty than '42%' suggests. The sample-size table shows the width falling from 0.282 at n = 50 to 0.034 at n = 3200: each fourfold increase roughly halves it.

The mistake this prevents

The mistake is reporting the point estimate alone. '42% of users abandoned' sounds precise; from 200 sessions the data is consistent with anything from 35% to 49%, which may well straddle the threshold the decision turns on.

Takeaway

Report an interval with every estimate and read its width before its centre. When someone asks for more precision, the square-root rule tells you what it will cost.

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