sum

(noun)

The result of adding two quantities.

Related Terms

  • Dijfe
  • identity element
  • Distributive
  • difference
  • quotient
  • product
  • associative
  • commutative
  • rational number

Examples of sum in the following topics:

  • Summation Notation

    • Use summation notation to express the sum of a subset of numbers
    • Many statistical formulas involve summing numbers.
    • because the expression on the left means to sum up all the values of X and then square the sum (19² = 361), whereas the expression on the right means to square the numbers and then sum the squares (90.54, as shown).
    • Some formulas involve the sum of cross products.
    • The sum of the cross products is 3 + 4 + 21 = 28.
  • Conjugation of "Sum"

  • Summary of Formulas

    • The symbol Σ means to add or to find the sum.
    • ∑ x ∗ f = The sum of values multiplied by their respective frequencies
    • $\bar{x} = \frac{\sum{x}}{n} or x = \frac{\sum{f} \cdot x}{n}$$\bar{x} = \frac{\sum{x}}{n} or x = \frac{\sum{f} \cdot x}{n}$
    • $s = \sqrt{\frac{\sum(x \bar{x})^2}{n 1}}or s = \sqrt{\frac{\sum{f} \cdot ( x \bar{x})^2}{n-1}}$
    • $s = \sqrt{\frac{\sum(x \bar{x})^2}{N}}or s = \sqrt{\frac{\sum{f} \cdot ( x \bar{x})^2}{N}}$
  • Fitting a Curve

    • $\sum_{i=1}^{n} y_{1}$
    • $\sum_{i=1}^{n} x_{i}$
    • $\sum_{i=1}^{n}x_{i}y_{i}-\frac{1}{n}\sum_{i=1}^{n}x_{i}\sum_{j=1}^{n}y_{j}$
    • Calculate the denominator:  The sum of the squares of the $x$-coordinates minus one-eighth the sum of the $x$-coordinates squared.
    • $\sum_{i=1}^{n}(x_{i}^{2})-\frac{1}{n}(\sum_{i=1}^{n}x_{i})^{2}$
  • Summing an Infinite Series

    • A series is the sum of the terms of a sequence.
    • The sequence of partial sums ${S_k}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence ${a_n}$ from $a_0$ to $a_k$:
    • $\displaystyle{S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k}$
    • A series is said to converge when the sequence of partial sums has a finite limit.
    • By definition the series $\sum_{n=0}^\infty a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums  converges to $L$.
  • The Sum of Draws

    • The sum of draws can be illustrated by the following process.
    • In this case your sum of draws would be $4+4=8$.
    • Your sum of draws is, therefore, subject to a force known as chance variation.
    • The sum of these 25 draws is 89.
    • Obviously this sum would have been different had the draws been different.
  • Series Solutions

    • $\displaystyle{f= \sum_{k=0}^\infty A_kz^k \ f'= \sum_{k=0}^\infty kA_kz^{k-1} \ f''= \sum_{k=0}^\infty k(k-1)A_kz^{k-2}}$
    • $\begin{aligned} & {} \sum_{k=0}^\infty k(k-1)A_kz^{k-2}-2z \sum_{k=0}^\infty kA_kz^{k-1}+ \sum_{k=0}^\infty A_kz^k=0 \\ & = \sum_{k=0}^\infty k(k-1)A_kz^{k-2}- \sum_{k=0}^\infty 2kA_kz^k+ \sum_{k=0}^\infty A_kz^k \end{aligned}$
    • $\begin{aligned} & = \sum_{k+2=0}^\infty (k+2)((k+2)-1)A_{k+2}z^{(k+2)-2}- \sum_{k=0}^\infty 2kA_kz^k+ \sum_{k=0}^\infty A_kz^k \\ & = \sum_{k=0}^\infty (k+2)(k+1)A_{k+2}z^k- \sum_{k=0}^\infty 2kA_kz^k+ \sum_{k=0}^\infty A_kz^k \\ & = \sum_{k=0}^\infty \left((k+2)(k+1)A_{k+2}+(-2k+1)A_k \right)z^k \end{aligned}$
    • The exponential function (in blue), and the sum of the first $n+1$ terms of its Maclaurin power series (in red).
  • Computing Pearson's r

    • Before proceeding with the calculations, let's consider why the sum of the xy column reveals the relationship between X and Y.
    • This would make negative values of xy as likely as positive values and the sum would be small.
    • To achieve this property, Pearson's correlation is computed by dividing the sum of the xy column ($\sum xy$) by the square root of the product of the sum of the x2 column ($\sum x^2$) and the sum of the y2 column ($\sum y^2$).
    • $r=\frac{\sum xy-\frac{\sum x \sum y}{N}}{\sqrt{\left( \sum x^2-\frac{\left( \sum x \right) ^2}{N} \right)} \sqrt{\left( \sum y^2-\frac{\left( \sum y \right) ^2}{N} \right)}}$
  • Series

    • A series is the sum of the terms of a sequence.
    • A series is, informally speaking, the sum of the terms of a sequence.
    • The sequence of partial sums $\{S_k\}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence $\{a_n\}$ from $a_0$ to $a_k$:
    • $\displaystyle{S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k}$
    • By definition, the series $\sum_{n=0}^{\infty} a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums $\{S_k\}$ converges to $L$.
  • Partitioning the Sums of Squares

    • Partition sum of squares Y into sum of squares predicted and sum of squares error
    • Define r2 in terms of sum of squares explained and sum of squares Y
    • The sum of squares error is the sum of the squared errors of prediction.
    • This can be summed up as:
    • First, notice that the sum of y and the sum of y' are both zero.
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