How To Calculate The Sum Of Squares?

How To Calculate The Sum Of Squares?

Sum Of Squares: Sum of squares is a demographic technique used in backsliding analysis to determine the distribution of data points. In a regression analysis, the goal is to verify how well a data series can be fitted to a gathering that might help to explain how the data series was generated. Sum of squares is used as a scientific way to find the function that honestest fits (varies least) from the data.

The sum of squares is a means of deviation from the mean. In statistics, the mean is the average of a set of numbers and is the most ordinarily used measure of central movement. The arithmetic mean is simply anticipated by summing up the values in the data set and partitioning by the number of values.

 How To Calculate The Sum Of Squares?

Sum Of Squares Formula

The sum of squares custom is used to calculate the sum of two or more spaces in an expression. To describe how well a model represents the data being modelled, the sum of spaces formula is used. Also, the sum of spaces is the measure of deviation from the data’s mean value. Hence, it is calculated as the total summation of the spaces minus the mean.

The series \sum\limits_{k=1}^n k^a = 1^a + 2^a + 3^a + \cdots + n^a gives the sum of the a^\text{th} powers of the first n positive numbers, where a and n are positive integers. Each of these series can be anticipated through a closed-form formula. The case a=1,n=100 is famously said to have been solved by Gauss as a young schoolboy: given the tedious task of adding the first 100 positive integers, Gauss quickly used a formula to calculate the sum of 5050.

Sum of spaces can be anticipated using two custom i.e. by algebra and by the mean. The formula to calculate the sum of the squares of two values are given below,

Sum of Squares Formulas
In Statistics Sum of Squares:

= Σ(xi + x̄)2

In Algebra Sum of Squares of Two Values:

= a2 + b2 = (a + b)2 − 2ab

For “n” Terms Sum of Squares Formula for “n” numbers

= 12 + 22 + 32 ……. n2 = [n(n + 1)(2n + 1)] / 6

  • ∑ = sum
  • xi = each value in the set
  • x̄ = mean
  • xi – x̄ = deviation
  • (xi – x̄)2 = square of the deviation
  • a, b = numbers
  • n = number of terms

Faulhaber’s formula, which is derived below, provides a generalized formula to compute these sums for any value of a.

Manipulations of these sums yield useful results in areas including string theory, quantum mechanics, and complex numbers.

Sum Of Squares Calculator

 Sum Of Squares Calculator

We can use the same trick here that we used with the sum of the natural numbers, using differences.

n 0 1 2 3 4 5 6
n2 0 1 4 9 16 25 36
Sn 0 1 5 14 30 55 91
Δ1 1 4 9 16 25 36
Δ2 3 5 7 9 11
Δ3 2 2 2 2

As usual, the first n in the table is zero, which isn’t a natural number.
Because Δis a constant, the sum is a cubic of the form
an3+bn2+cn+d, [1.0]

and we can find the coefficients using concurrent equations, which we can make as we wish, as we know how to add spaces to the table and to sum them, even if we don’t know the formula.

In the table below, we create three equations, noting that d=0 from the first one (revealing the reason for the non-natural number zero)

n 0 1 2 3
Sn 0 1 5 14
d=0 a+b+c=1 8a+4b+2c=5 27a+9b+3c=14

Rewriting our equations:
a+b+c=1 [1.1]
8a+4b+2c=5 [1.2]
27a+9b+3c=14 [1.3]

Using 1.1 with 1.2 and 1.3 we can make two new equations:
6a+2b=3 [1.4]
24a+6b=11 [1.5]

By subtracting 3x Equation 1.4 from 1.5, we get:
6a=2
So a=1/3 [Also noting, by the way, that Δ3/3!=1/3]

Substituting a=1/3 in 1.4 gives
2b=1
So b=1/2

Finally, substituting these values into 1.1 we get
1/3+1/2+c=1
So c=1/6

[Actually, with the sum of the powers, the sum of the coefficients in the formula is always 1]

So, we can substitute our values into 1.0, to get the sum of the squares of the first n natural numbers (or first n positive integers):
n3/3+n2/2+n/6
Or, in various forms:
sumSquaresNatNumbersFormulae.gif

Residual Sum Of Squares

Let’s say the closing prices of Microsoft (MSFT) in the last five days were 74.01, 74.77, 73.94, 73.61, and 73.40 in US dollars. The sum of the total prices is $369.73 and the mean or average price of the textbook would thus be $369.73 5 = $73.95.

But knowing the mean of a measurement set is not always enough. Sometimes, it is helpful to know how much variation there is in a set of measurements. How far apart the individual values are from the mean may give some insight into how fit the observations or values are to the regression model that is created.

For example, if an analyst wanted to know whether the share price of MSFT moves in tandem with the price of Apple (AAPL), he can list out the set of observations for the process of both stocks for a certain period, say 1, 2, or 10 years and create a linear model with each of the observations or measurements recorded. If the relationship between both variables (i.e., the price of AAPL and price of MSFT) is not a straight line, then there are variations in the data set that need to be scrutinized.

In statistics speak, if the line in the linear model created does not pass through all the measurements of value, then some of the variability that has been observed in the share prices is unexplained. The sum of squares is used to calculate whether a linear relationship exists between two variables, and any unexplained variability is referred to as the residual sum of squares.

The sum of squares is the sum of the square of variation, where variation is defined as the spread between each individual value and the mean. To determine the sum of squares, the distance between each data point and the line of best fit is squared and then summed up. The line of best fit will minimize this value.

Total Sum Of Squares

  • For partitioning of variance, see Partition of sums of squares
  • For the “sum of squared deviations”, see Least squares
  • For the “sum of squared differences”, see Mean squared error
  • For the “sum of squared error”, see Residual sum of squares
  • For the “sum of squares due to lack of fit”, see Lack-of-fit sum of squares
  • For sums of squares relating to model predictions, see Explained sum of squares
  • For sums of squares relating to observations, see Total sum of squares
  • For sums of squared deviations, see Squared deviations
  • For modelling involving sums of squares, see Analysis of variance
  • For modelling involving the multivariate generalisation of sums of squares, see Multivariate analysis of variance
  • For the sum of squares of consecutive integers, see Square pyramidal number
  • For representing an integer as a sum of squares of 4 integers, see Lagrange’s four-square theorem
  • Legendre’s three-square theorem states which numbers can be expressed as the sum of three squares
  • Jacobi’s four-square theorem gives the number of ways that a number can be represented as the sum of four squares.
  • For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.
  • Fermat’s theorem on sums of two squares says which primes are sums of two squares.
    • A separate article discusses proofs of Fermat’s theorem on sums of two squares
    • The sum of two squares theorem generalizes Fermat’s theorem to specify which composite numbers are the sums of two squares.
  • Pythagorean triples are sets of three integers such that the sum of the squares of the first two equals the square of the third.
  • A Pythagorean prime is a prime that is the sum of two squares; Fermat’s theorem on sums of two squares states which primes are Pythagorean primes.
  • Pythagorean triangles with integer altitude from the hypotenuse have the sum of squares of inverses of the integer legs equal to the square of the inverse of the integer altitude from the hypotenuse.
  • Pythagorean quadruples are sets of four integers such that the sum of the squares of the first three equals the square of the fourth.
  • The Basel problem, solved by Euler in terms of {\displaystyle \pi }, asked for an exact expression for the sum of the squares of the reciprocals of all positive integers.
  • Rational trigonometry’s triple-quad rule and triple-spread rule contain sums of squares, similar to Heron’s formula.

What is sum of squares formula?

The mean of the sum of squares (SS) is the variance of a set of scores, and the square root of the variance is its standard deviation. This simple calculator uses the computational formula SS = ΣX2 – ((ΣX)2 / N) – to calculate the sum of squares for a single set of scores.

Is there a sum of squares?

Sum of two squares theorem. In number theory, the sum of two squares theorem says when an integer n > 1 can be written as a sum of two squares, that is, when n = a 2 + b 2 for some integers a, b. raised to an odd power.

What does sum of squares tell you?

The sum of squares represents a measure of variation or deviation from the mean. It is calculated as a summation of the squares of the differences from the mean. The calculation of the total sum of squares considers both the sum of squares from the factors and from randomness or error.