Gauss showed that the arithmetic mean is indeed the best estimate of the location parameter by changing both the probability density and the method of estimation. He then turned the problem around by asking what form the density should have and what method of estimation should be used to get the arithmetic mean as estimate of the location parameter. The process of fitting the best-fit line is called linear regression. The idea behind finding the best-fit line is based on the assumption that the data are scattered about a straight line.

The line of best fit provides the analyst with coefficients explaining the level of dependence. An extension of this approach is elastic net regularization. It is necessary to make assumptions about the nature of the experimental errors to test the results statistically. A common assumption is that the errors belong to a normal distribution. The central limit theorem supports the idea that this is a good approximation in many cases. The following discussion is mostly presented in terms of linear functions but the use of least squares is valid and practical for more general families of functions.

## The formula

The goal we had of finding a line of best fit is the same as making the sum of these squared distances as small as possible. The process of differentiation in calculus makes it possible to minimize the sum of the squared distances from a given line. This explains the phrase “least squares” in our name for this line. As we look at the points in our graph and wish to draw a line through these points, a question arises.

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In any case, for a reasonable number of

noisy data points, the difference between vertical and perpendicular fits is quite

small. The linear least squares fitting technique is the simplest and most commonly applied form of linear regression and provides

a solution to the problem of finding the best fitting straight line through

a set of points. For this reason, standard forms for exponential,

logarithmic, and power

laws are often explicitly computed. The formulas for linear least squares fitting

were independently derived by Gauss and Legendre. Moreover there are formulas for its slope and \(y\)-intercept.

## Least Squares Criteria for Best Fit

Although it may be easy to apply and understand, it only relies on two variables so it doesn’t account for any outliers. That’s why it’s best used in conjunction with other analytical tools to get more reliable results. This method, the method of least squares, finds values of the intercept and slope coefficient that minimize the sum of the squared errors. Where the true error variance σ2 is replaced by an estimate, the reduced chi-squared statistic, based on the minimized value of the residual sum of squares (objective function), S. The denominator, n − m, is the statistical degrees of freedom; see effective degrees of freedom for generalizations.[13] C is the covariance matrix.

All the math we were talking about earlier (getting the average of X and Y, calculating b, and calculating a) should now be turned into code. We will also display the a and b values so we see them changing as we add values. At the start, it should be empty since we haven’t added any data to it just yet. We add some rules so we have our inputs and table to the left and our graph to the right.

## Least Squares Calculator

Linear models can be used to approximate the relationship between two variables. The truth is almost always much more complex than our simple line. For example, we do not know how the data outside of our limited window will behave. We mentioned earlier that a computer is usually used to compute the least squares line.

Some of the pros and cons of using this method are listed below. A spring should obey Hooke’s law which states that the extension of a spring y is proportional to the force, F, applied to it. The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth’s oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on land sightings for navigation.

## Alternative formulations

The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is, made as small as possible. Any other line you might choose would have a higher SSE than the best fit line. This best fit line is called the least-squares regression line . It is an invalid use of the regression equation that can lead to errors, hence should be avoided. Here the equation is set up to predict gift aid based on a student’s family income, which would be useful to students considering Elmhurst.

- The slope of the line, b, describes how changes in the variables are related.
- In actual practice computation of the regression line is done using a statistical computation package.
- If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
- In any case, for a reasonable number of
noisy data points, the difference between vertical and perpendicular fits is quite

small.

We loop through the values to get sums, averages, and all the other values we need to obtain the coefficient (a) and the slope (b). Each point of data is of the the form (x, y) and each point of the line of best fit using least-squares linear regression has the form (x, ŷ). The estimated intercept is the value of the response variable for the first category (i.e. the category corresponding to an indicator value of 0). The estimated slope is the average change in the response variable between the two categories.

## Line of Best Fit

Since the least squares line minimizes the squared distances between the line and our points, we can think of this line as the one that best fits our data. This is why the least a fitted least squares regression line squares line is also known as the line of best fit. Of all of the possible lines that could be drawn, the least squares line is closest to the set of data as a whole.

That event will grab the current values and update our table visually. Let’s assume that our objective is to figure out how many topics are covered by a student per hour of learning. For example, say we have a list of how many topics future engineers here at freeCodeCamp can solve if they invest 1, 2, or 3 hours continuously. Then we can predict how many topics will be covered after 4 hours of continuous study even without that data being available to us.

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