square

(noun)

The second power of a number, value, term or expression.

Related Terms

Examples of square in the following topics:

  • Chi-Square Probability Table

    • Areas in the chi-square table always refer to the right tail.

  • Chi Square Distribution

    • Define the Chi Square distribution in terms of squared normal deviates
    • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
    • Therefore, Chi Square with one degree of freedom, written as χ2(1), is simply the distribution of a single normal deviate squared.
    • A Chi Square calculator can be used to find that the probability of a Chi Square (with 2 df) being six or higher is 0.050.
    • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.

  • Factoring Perfect Square Trinomials

    • When a trinomial is a perfect square, it can be factored into two equal binomials.
    • Note that if a binomial of the form $a+b$ is squared, the result has the following form: $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.$ So both the first and last term are squares, and the middle term has factors of $2, $ $a$, and $b,$ where the latter are the square roots of the first and last term respectively.
    • For example, if the expression $2x+3$ were squared, we would obtain $(2x+3)(2x+3)=4x^2+12x+9.$ Note that the first term $4x^2$ is the square of $2x$ while the last term $9$ is the square of $3$, while the middle term is twice $2x\cdot3$.
    • Suppose you were trying to factor $x^2+8x+16.$ One can see that the first term is the square of $x$ while the last term is the square of $4$.
    • Since the first term is $3x$ squared, the last term is one squared, and the middle term is twice $3x\cdot 1$, this is a perfect square, and we can write:

  • Student Learning Outcomes

  • Completing the Square

    • The method of completing the square allows for the conversion to the form:
    • Once completing the square has been performed, the quadratic is easy to solve; because there is only one place where the variable $x$ is squared, the $(x-h)^2$ term can be isolated on one side of the equation, and then the square root of both sides can be taken.
    • This quadratic is not a perfect square.  
    • The closest perfect square is the square of $5$, which was determined by dividing the $b$ term (in this case $10$) by two and producing the square of the result.
    • Solve for the zeros of a quadratic function by completing the square

  • Root-Mean-Square Speed

    • The root-mean-square speed measures the average speed of particles in a gas, defined as $v_{rms}=\sqrt{\frac{3RT}{M}}$ .
    • By squaring the velocities and taking the square root, we overcome the "directional" component of velocity and simultaneously acquire the particles' average velocity.
    • The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas.
    • What is the root-mean-square speed for a sample of oxygen gas at 298 K?
    • Recall the mathematical formulation of the root-mean-square velocity for a gas.

  • Notation

    • where df = degrees of freedom depend on how chi-square is being used.
    • (If you want to practice calculating chi-square probabilities then use df = n−1.
    • The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.

  • The Root-Mean-Square

    • Its name comes from its definition as the square root of the mean of the squares of the values.
    • However, using the RMS method, we would square every number (making them all positive) and take the square root of the average.
    • The root-mean-square is always greater than or equal to the average of the unsigned values.
    • Physical scientists often use the term "root-mean-square" as a synonym for standard deviation when referring to the square root of the mean squared deviation of a signal from a given baseline or fit.
    • $G$ is the geometric mean, $H$ is the harmonic mean, $Q$ is the quadratic mean (also known as root-mean-square).

  • Structure of the Chi-Squared Test

    • Chi-square statistics use nominal (categorical) or ordinal level data.
    • Data used in a chi-square analysis has to satisfy the following conditions:
    • If a chi squared test is conducted on a sample with a smaller size, then the chi squared test will yield an inaccurate inference.
    • First, we calculate a chi-square test statistic.
    • Second, we use the chi-square distribution.

  • Randomization for two-way tables and chi-square

    • In short, we create a randomized contingency table, then compute a chi-square test statistic.
    • When the minimum threshold is met, the simulated null distribution will very closely resemble the chi-square distribution.