Smoothing of data. Linear Regression:

In certain engineering problems we must perform some calculations and predictions based on experimental measurements. The experimental data collected is usually redundant and rough. For example if we measure the force required for different elongations of a given steel specimen and plot the results, they may look like this:

Here each small dot represents a single measurement. We can see a trend of increasing elongation for increased force but we have no clear mathematical relationship between the two. Linear regression provides a linear combination of functions that best represents the data in some sense.

Read the section of your book that deals with Linear Regression and also read the

NOTES ON THE METHOD OF LEAST SQUARES

The method described for determining the slope and y-intercept of a straight line that best fits the data can be worked on a table if the data points are few. This is illustrated below. Read the example and then try another one on your own.

	x	y	x-xbar	y-ybar	(x-xbar)(y-ybar)	(x-xbar)2
	12	0.48	-5.67	-0.12	0.68	32.11
	15	0.61	-2.67	0.01	-0.027	7.11
	18	0.60	0.33	0	0	0.11
	19	0.59	1.33	-0.01	-0.133	1.78
	20	0.62	2.33	0.02	0.0467	5.44
	22	0.70	4.33	0.1	0.433	18.78
sum	106	3.6	0	0	1.12	65.33
mean	17.67	0.6
m=	0.0171
b=	0.2971