What does a standardised regression coefficient (beta) report that a raw coefficient does not?

## The definition A raw OLS slope `b_j` is in outcome units per predictor unit, which makes coefficients on differently-measured predictors incomparable in size. The standardised coefficient removes both sets of units: ``` beta_j = b_j * (s_xj / s_y) ``` where `s_xj` is the sample standard deviation of the predictor and `s_y` is the sample standard deviation of the outcome. The result is dimensionless and reads as: **a one-standard-deviation increase in this predictor is associated with `beta_j` standard deviations of change in the outcome, holding the other predictors fixed.** The equivalent procedure is to standardise every variable first - subtract each mean and divide by each standard deviation - and fit the model on the transformed variables. The coefficients that come out are the standardised betas, and the intercept is zero because all the centred variables have mean zero. A half-standardised version is sometimes more useful: multiply by `s_xj` only, leaving the outcome in its natural units, so the coefficient reads as "outcome units per standard deviation of the predictor". Stakeholders often find that easier than a fully dimensionless number, because the outcome stays in dollars or minutes. ## What problem it solves Put age in years, tenure in months and income in dollars into one model and the raw coefficients differ in magnitude mostly because the units differ in magnitude. Standardising asks the same question of every predictor - what does a typical-sized move in this variable buy? - so the numbers land on one scale and can at least be looked at side by side. That is a genuine gain in readability. It is the standard answer to "which of these matters more?" in fields where the predictors have no natural common unit. ## What it does not change Standardising is a linear rescaling of each variable, so it changes nothing about the fit. Fitted values, residuals and R-squared are identical. Every t-statistic and p-value is identical, because the coefficient and its standard error scale by exactly the same factor. If someone reports that standardising strengthened a result, something else changed - the model, the sample or the arithmetic. The sign and the ordering by magnitude within a single fitted model are also unchanged relative to the raw coefficients multiplied by their standard deviations; standardising just makes that comparison explicit rather than accidental. ## The limits worth knowing **Sample-dependence.** `s_xj` describes the spread of the predictor in *this* dataset. Restrict the sample to a narrow slice of a predictor and its beta shrinks, even though nothing about the underlying relationship changed. This makes betas awkward to compare across studies or across time periods with different sampling: the raw coefficient is the more portable quantity, and the beta is the more comparable-within-one-model quantity. **Binary predictors.** The standard deviation of a 0/1 indicator is `sqrt(p*(1-p))`, so it depends entirely on how common the category is, and it is at most 0.5. "A one-standard-deviation increase" in a variable that only takes two values is not a change anything can undergo. For indicators, report the effect in the model's own units instead of standardising them. **Correlated predictors.** Standardising does nothing about overlap. When two predictors carry much of the same information, the fit divides their shared association between them in a way that depends on the specification, so neither beta is a stable property of that variable alone. **Still not causal.** A standardised beta is a description of a comparison inside a fitted model. It carries no more claim about what happens if you intervene than the raw coefficient did. **Nonlinear terms.** The clean "one standard deviation in, beta standard deviations out" reading assumes the predictor enters the model in a single linear term. If it appears in more than one term, no single standardised number summarises moving it. ## Choosing between beta and a natural step In practice, decide by audience. For a technical comparison across predictors with no common unit, the standardised beta is the right summary and should be reported with a confidence interval like any other estimate. For a business audience, the effect of a concrete, decision-relevant change in the original units is almost always more useful, because a standard deviation is itself an abstraction most readers cannot picture. Reporting both is cheap and forestalls the argument. ## The interview answer Give the formula, give the reading in words, then volunteer the limitation before you are asked: it is comparable within this model and this sample, it does not change any p-value, and it is not a licence to declare one predictor causally more important than another. Volunteering the caveat is what separates someone who has used standardised betas from someone who has read about them.