In regression output, “s” is the standard error of the regression. It tells you the average distance that your actual data points fall from the regression line, measured in the same units as your dependent variable. If you’re predicting home prices and s equals $15,000, your model’s predictions are off by about $15,000 on average. Smaller values of s mean your model fits the data more tightly.
You’ll also see this called the “standard error of the estimate” depending on the software. It means the same thing.
What S Actually Measures
Think of the regression line as your best guess for the true relationship between your variables. The data points don’t land perfectly on that line. They scatter above and below it. S quantifies how much scatter there is.
More precisely, s measures the overall amount of “noise” in your data after accounting for the signal your model captures. The regression line represents your best estimate of the true signal, and s is your best estimate of the standard deviation of the true noise. Every residual (the gap between an observed value and the predicted value) is an estimate of the real error at that data point, and s summarizes the typical size of those errors.
Because s is expressed in the units of your dependent variable, it gives you an intuitive sense of prediction accuracy. If you’re modeling job satisfaction on a 10-point scale and s is 1.59, you can say: on average, the predicted satisfaction score is about 1.59 points away from the actual score. That kind of concrete statement is something you can’t get from R-squared alone.
How S Differs From R-Squared
R-squared and s both describe how well your model fits, but they do it in fundamentally different ways. R-squared is a relative measure. It tells you what percentage of the variation in your outcome variable the model explains. S is an absolute measure. It tells you how far off your predictions typically are, in real units.
This distinction matters. A model can have a high R-squared but still produce predictions that are too imprecise for your purposes. Suppose you’re predicting blood pressure and your R-squared is 0.85, which sounds great. But if s is 12 mmHg, your predictions are off by roughly 12 points on average, which may be too much to be clinically useful. R-squared alone wouldn’t reveal that problem. S forces you to ask: “Is this amount of error acceptable for what I’m trying to do?”
How S Is Calculated
The formula starts with the residuals, which are the differences between each observed value and the value your regression line predicts. You square all those residuals and add them up to get the sum of squared errors. Then you divide by the degrees of freedom, and take the square root of the result.
For a simple regression with one predictor, the degrees of freedom in the denominator is n minus 2, where n is the number of data points. You lose two degrees of freedom because the model estimated two parameters: a slope and an intercept. In multiple regression with several predictors, the denominator becomes n minus k minus 1, where k is the number of predictors. Each additional parameter you estimate costs you one more degree of freedom.
This adjustment matters because without it, s would systematically underestimate the true error. Dividing by a slightly smaller number corrects for the fact that the model was fitted to the same data you’re evaluating it on.
Using S to Build Prediction Intervals
One of the most practical uses of s is constructing prediction intervals, which tell you the range where a new observation is likely to fall. If the errors are roughly normally distributed, approximately 95% of data points will land within about two times s of the regression line.
For example, if your model predicts a skin cancer mortality rate of 150 per 100,000 at a particular latitude, and s is 20, then about 95% of actual mortality rates at that latitude should fall between 110 and 190 (that’s 150 plus or minus 40). The formal calculation uses a t-multiplier instead of exactly 2, and it also accounts for how far the predictor value is from its mean, but the core logic stays the same: s is the building block for every prediction interval your software generates.
This is where s becomes directly actionable. If someone asks you “how confident are you in this prediction?” you can give a concrete range rather than a vague answer.
How to Tell If S Is Good or Bad
There’s no universal threshold for what counts as a “good” s value. It depends entirely on your context and what level of prediction error you can tolerate. An s of 5 pounds might be perfectly acceptable if you’re predicting shipping weights for large packages but unacceptable if you’re formulating medication doses for infants.
The most useful approach is to compare s against the range or mean of your dependent variable. If your outcome variable ranges from 0 to 100 and s is 3, your model is predicting quite tightly. If s is 30, a third of the entire range, the model isn’t giving you much precision. You can also compare s across competing models. When you add a predictor and s drops meaningfully, that variable is contributing real predictive value. If s barely changes, the new variable isn’t helping.
S vs. Standard Error of Coefficients
One source of confusion: regression output contains multiple standard errors. The value labeled “s” (or “S” in some software) at the top of the output is the standard error of the regression, which describes overall model accuracy. Below that, in the coefficient table, each predictor has its own standard error, which describes how precisely the model has estimated that particular coefficient.
These two types of standard errors are related. The standard error of any individual coefficient equals s multiplied by a factor that depends on how much variability exists in that predictor. A predictor with a wider spread of values produces a more precisely estimated coefficient, which is why collecting data across a broad range of your predictor improves your regression results. But when someone asks “what is s in my regression output,” they’re almost always asking about the single summary value for the whole model, not the individual coefficient errors.