Your OLS table shows a large insignificant coefficient beside a tiny significant one — what explains that?

## Why the comparison is invalid as posed An output table invites you to scan down the estimate column and rank predictors by size. That reading is wrong for a structural reason: each coefficient is denominated in *outcome units per one unit of its own predictor*, and the predictors have different units. A coefficient on tenure-in-years and a coefficient on revenue-in-dollars are not on a common scale, so `12000` versus `0.004` says nothing about which predictor matters more. Significance, meanwhile, is a comparison of a coefficient to **its own** standard error: `t = estimate / SE`. That ratio is unitless, so it survives rescaling while the raw magnitude does not. ## The rescaling argument Measure a spend predictor in dollars, then refit with the identical data measured in thousands of dollars. The coefficient is divided by 1000, and its standard error is divided by 1000 as well, because both carry the same units. Their ratio — the t-statistic — is unchanged, and so is the p-value. The fit is identical; only the printed magnitudes moved. This single fact settles the question. If simply changing a unit of measurement can make a coefficient a thousand times larger without changing its significance, then magnitude and significance cannot be measuring the same thing, and ranking predictors by raw coefficient size is meaningless. ## The precision argument The second half of the puzzle is real statistical content rather than bookkeeping. A big estimate becomes insignificant when its standard error is big, and the standard error is `s / (s_x * sqrt(n - 1))` in the simple case: large when the outcome is noisy, when that predictor barely varied in the observed data, or when the sample is small. So a predictor that genuinely swings the outcome hard, but was only ever observed across a narrow band, can produce a headline-grabbing estimate with an interval so wide it means nothing. Conversely, a tiny coefficient can clear the significance bar effortlessly. With tens of thousands of rows, standard errors get small enough that a per-unit effect too small to care about becomes overwhelmingly significant. Significance says the estimate is far from zero relative to its noise; it does not say the effect is worth acting on. ## What to do instead 1. **Put every predictor on a comparable change.** Report the outcome change per one standard deviation of the predictor, or per an interquartile shift, or per a decision-relevant change the business recognises (a 10k increase in spend, one extra year of tenure). Now the numbers are all in outcome units and can be compared. 2. **Always carry the interval.** A point estimate with no interval hides exactly the information that resolves this puzzle. 3. **Distinguish two kinds of insignificance.** If the large coefficient's interval spans, say, `-30,000` to `54,000`, the data are simply uninformative about it — you have not shown a null effect, you have shown you cannot tell. If instead a coefficient's interval is narrow and tightly hugs zero, you have positive evidence that any effect is small. These read identically in the significance column and mean opposite things. 4. **Separate statistical from practical significance.** For the tiny-but-significant coefficient, multiply it out over a realistic change in its predictor and ask whether the resulting outcome movement would change any decision. Very often it would not. ## Communicating it Stakeholders reliably read the biggest number in the estimate column as the most important driver. The fix is not to explain t-statistics to them; it is to never ship a table whose columns invite the comparison. Present effects per a named realistic change, with intervals, sorted by that quantity. When a large estimate is too imprecise to use, say so in those words — *we cannot yet size this one* — rather than reporting it as non-significant, which stakeholders hear as *no effect*. ## What an interviewer is checking That you do not confuse effect size with evidence, that you can produce the rescaling argument on demand, that you know why an insignificant large coefficient is a call for more data or more variation rather than a finding, and that you have a concrete plan for making coefficients comparable.