How to Calculate the Standard Error of a Slope

How to Calculate the Standard Error of a Slope
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In statistics, the parameters of a linear mathematical model can be determined from experimental data using a method called linear regression. This method estimates the parameters of an equation of the form y = mx + b (the standard equation for a line) using experimental data. However, as with most statistical models, the model will not exactly match the data; therefore, some parameters, such as the slope, will have some error (or uncertainty) associated with them. The standard error is one way of measuring this uncertainty and can be accomplished in a few short steps.

    Find the sum of square residuals (SSR) for the model. This is the sum of the square of the difference between each individual data point and the data point that the model predicts. For example, if the data points were 2.7, 5.9 and 9.4 and the data points predicted from the model were 3, 6 and 9, then taking the square of the difference of each of the points gives 0.09 (found by subtracting 3 by 2.7 and squaring the resulting number), 0.01 and 0.16, respectively. Adding these numbers together gives 0.26.

    Divide the SSR of the model by the number of data point observations, minus two. In this example, there are three observations and subtracting two from this gives one. Therefore, dividing the SSR of 0.26 by one gives 0.26. Call this result A.

    Take the square root of result A. In the above example, taking the square root of 0.26 gives 0.51.

    Determine the explained sum of squares (ESS) of the independent variable. For example, if the data points were measured at intervals of 1, 2 and 3 seconds, then you will subtract each number by the mean of the numbers and square it, then sum the ensuing numbers. For example, the mean of the given numbers is 2, so subtracting each number by two and squaring gives 1, 0 and 1. Taking the sum of these numbers gives 2.

    Find the square root of the ESS. In the example here, taking the square root of 2 gives 1.41. Call this result B.

    Divide result B by result A. Concluding the example, dividing 0.51 by 1.41 gives 0.36. This is the standard error of the slope.


    • If you have a large set of data, you may want to consider automating the calculation, as there will be large numbers of individual calculations that need to be done.


About the Author

Thomas Bourdin began writing professionally in 2010. He writes for various websites, where his interests include science, computers and music. He holds a Bachelor of Science degree in physics with a minor in mathematics from the University of Saskatchewan and a Master of Science in physics from Ryerson University.