# Uncertainties in gradients and best-fit lines

1. Oct 25, 2013

### Downwinds

Hi, I hope I'm asking this in the right place. I need to understand this in order to complete a project, but it's not exactly a 'homework question'.

I have some data which has a linear trend. The x values all have the same (random) uncertainty of ≈5cm and the y values all have the same random uncertainty of ≈0.5ms
I've got all my data plotted on graphs using a spreadsheet, which can put a best-fit line (of the form y=mx+c) through the points and tell me the equation of the line. I need to be able to propagate my errors through though, as I need to find the inverse of the gradient and the error associated with that in order to get the result I'm interested in. I also want to find the y value at which the best fit line crosses a certain x value (ie. the y-intercept) and its error.

I have a very basic understanding of statistics - standard deviation, averages, simple error propagation, and I understand that the best-fit lines are found using linear regression.

I've attached one of my graphs.

Background:
I'm in my second year of university, studying geophysics. This is my first fieldwork project which is a refraction seismology survey, and I'm interested in finding out the velocity at which the wave travels through the ground (velocity = 1000/gradient, in these graphs). In the graph I've posted, the source of the waves is at 48 m and the graph shows the wave travelling in forward and backward directions and refracting in response to subsurface changes.

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2. Oct 25, 2013

### Stephen Tashi

A simple regression program is liable to assume that there are no errors in the X data and that all random errors happen in the Y data. If you want to do a least squares fit in a case where both the X and Y data have random errors, you could consider "total least squares" regression. http://en.wikipedia.org/wiki/Total_least_squares

"Uncertainty" is not a precisely defined term in statistics, but often people in physics use it to mean "standard deviation". Where are your getting the numbers you gave for the uncertainty in the measurement? Are they from the specifications for the measuring equipment that was used? (We have to distinguish among "sample standard deviations" and "population standard deviations" and "estimators of population standard deviations".)