Standard error of coefficient

The standard deviation of the estimate of a regression coefficient. It measures how precisely your data can estimate the coefficient's unknown value. Its value is always positive, and smaller values indicate a more precise estimate.

The standard error of a coefficient helps determine whether the value of the coefficient is significantly different than zero – in other words, whether the predictor has a significant effect on the response. Dividing the coefficient by its standard error calculates a t-value; if the p-value associated with this t-statistic is less than your alpha-level, you conclude that the coefficient is significantly different from zero.

For example, you are studying the effect of fertilizers on plant height with a regression model that includes the quantity of nitrogen and phosphorus as two predictor variables. You produce the following linear regression output. The standard errors of the coefficients are in bold text.

The standard error of the nitrogen coefficient is smaller than that of phosphorus. Therefore, your data was able to estimate the coefficient for nitrogen with greater precision. In fact, the standard error of the phosphorus coefficient is very large relative to the value of the coefficient itself, so the t-value of -0.23 is too small to declare statistical significance. The resulting p-value is much greater than common alpha-levels, so you cannot conclude this coefficient differs from zero. You remove the phosphorus variable from your regression model and continue the analysis.