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The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial x 4 − 1 {\displaystyle x^{4}-1} can be factored as follows:
The squared Euclidean distance between two points, equal to the sum of squares of the differences between their coordinates; Heron's formula for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares) The British flag theorem for rectangles equates two sums of two squares
The result of fitting a set of data points with a quadratic function Conic fitting a set of points using least-squares approximation. In regression analysis, least squares is a parameter estimation method based on minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each ...
Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.
The sum of squares due to lack of fit is the weighted sum of squares of differences between each average of y-values corresponding to the same x-value and the corresponding fitted y-value, the weight in each case being simply the number of observed y-values for that x-value.
The prime decomposition of the number 2450 is given by 2450 = 2 · 5 2 · 7 2.Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4.
The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, y i = a + b 1 x 1i + b 2 x 2i + ... + ε i, where y i is the i th observation of the response variable, x ji is the i th observation of the j th ...
In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepancy between the data and an estimation model, such as a linear ...