Statistics vocabulary is replete with intervals, ranges, errors, and estimates. Even at the introductory level, there are standard errors, margins of error, confidence intervals, point estimates, interval estimates, min-to-max ranges, and interquartile ranges (IQR). For students, the rich vocabulary is a source of confusion. And do we really need it?
Consider the distinctions between the standard error, the margin of error, and the confidence interval. For the standard topics of intro stats, the margin of error is (roughly) twice the standard error; the confidence interval spans a range from the point estimate up and down by one margin of error. They are all about the same thing: the precision to which a statistic is known.
Perhaps if we could start over, we could simplify things: the standard precision, the margin of precision, the precision interval.1 There is no sign that this will ever happen.
Still, there is an easy path forward. The standard error and the margin of error are intermediates in calculations whose end result is the confidence interval. Let’s simply focus student attention on the confidence interval. That’s what matters in the end. Going further, there is little or no loss in dropping entirely the intermediates in the calculation. We don’t actually use the standard error or margin of error for anything except constructing a confidence interval. And there are important statistics, for instance the risk ratio, which are conveniently absent from the curriculum in part because the confidence interval doesn’t involve the familiar symmetry of ±.2
The confidence interval needs to be contrasted with the description of variability itself. It’s a common fallacy among students that 95% of the individual cases in the data will lie inside the 95% confidence interval. In fact, for large n, hardly any of the individual cases will fall into the confidence interval. To avoid this fallacy, we need a vocabulary for describing variation that is clearly distinct from the vocabulary of inference. What we routinely use presents a Goldilocks problem: The Papa Bear of variation, the range of the data – from the minimum to the maximum – is too large for informed practical use (and not robust to outliers). The Mama Bear of variation, the interquartile range, is robust to outliers, but is too small. The standard deviation might be the Baby Bear here, but (beyond the imposing name) it is in the wrong format: it doesn’t go from high to low. Just as confidence intervals are the right format for describing precision, ranges are the right format for describing individual-to-individual variation.
The 95% range is in the right format and covers a sensible fraction of the cases. It’s easily generalized: a 90% range, a 95% range, a 50% range (which, incidentally has the length given by the interquartile range).3
Or we could have the standard error, the margin of error, and the error interval. But the “confidence” of the confidence interval sounds so much more re-assuring than the “error” of the “error interval.”↩
Many statistics instructors aren’t aware that a proper confidence inteval on the sample proportion also doesn’t usually have ± symmetry, even though the textbook formulas have been selected to imply otherwise.↩
Another possibility is the “95% coverage interval,” since the interval covers 95% of the individual cases. But “coverage” and “confidence” are too easily confused, and “coverage” itself has another meaning in inferential statistics. So “ran ge”, which accords with the everyday sense of the term and with common use such as a doctor’s, “Your blood glucose level is a little high, but falls in the normal range.”↩