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Amazing Mathematics Answer Key Mean Absolute Deviation Maze Answer Key

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Mean Deviation

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The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean. For a sample size N, the mean deviation is defined by

MD=1/Nsum_(i=1)^N|x_i-x^_|,

(1)

where x^_ is the mean of the distribution. The mean deviation of a list of numbers is implemented in the Wolfram Language as MeanDeviation[data].

The mean deviation for a discrete distribution P_i defined for i=1, 2, ..., N is given by

MD=sum_(i=1)^NP_i|x_i-x^_|.

(2)

Mean deviation is an important descriptive statistic that is not frequently encountered in mathematical statistics. This is essentially because while mean deviation has a natural intuitive definition as the "mean deviation from the mean," the introduction of the absolute value makes analytical calculations using this statistic much more complicated than the standard deviation

sigma=sqrt(1/Nsum_(i=1)^N(x_i-x^_)^2).

(3)

As a result, least squares fitting and other standard statistical techniques rely on minimizing the sum of square residuals instead of the sum of absolute residuals.

For example, consider the discrete uniform distribution consisting of n possible outcomes with P_i=1/N for i=1, 2, ..., N. The mean is given by

x^_=sum_(i=1)^NiP_i=1/Nsum_(i=1)^Ni=1/2(N+1).

(4)

The variance (and therefore its square root, namely the standard deviation) is also straightforward to obtain as

sigma^2=sum_(i=1)^N(i-x^_)^2P_i=1/(12)(N-1)(N+1).

(5)

On the other hand, the mean deviation is given by

MD=sum_(i=1)^N|i-x^_|P_i=1/Nsum_(i=1)^N|i-x^_|.

(6)

This can be obtained in closed form, but is much more unwieldy since it requires breaking up the summand into two pieces and treating the cases of n even and odd separately.

The following table summarizes the mean absolute deviations for some named continuous distributions, where B(z;a,b) is an incomplete beta function, B(a,b) is a beta function, Gamma(z) is a gamma function, gamma is the Euler-Mascheroni constant, G_(2,1)^(0,2)(a,b|m) is a Meijer G-function, Ei(z) is the exponential integral function, erf(z) is erf, and erfc(z) is erfc.

distribution M.D.
beta distribution (2alphaB(alpha/(alpha+beta);alpha,beta)-2(alpha+beta)B(alpha/(alpha+beta);alpha+1,beta))/((alpha+beta)B(alpha,beta))
chi-squared distribution ((2e)^(-nu/2)nu^(nu/2+1)(nu+2))/(Gamma(2+1/2nu))
exponential distribution 2/(elambda)
gamma distribution (2e^(-alpha)alpha^alphabeta)/(Gamma(alpha))
Gumbel distribution beta[G_(2,1)^(2,0)(e,gamma^(-1)|1,1; 0)-Ei(sinhgamma-coshgamma)]
half-normal distribution 2/theta[e^(-1/pi)+erf(1/(sqrt(pi)))]
Laplace distribution beta
logistic distribution 2betaln2
log normal distribution 2e^(mu+sigma^2/2)erf(sigma/(2sqrt(2)))
Maxwell distribution 4e^(-4/pi)sqrt(2/pi)sigma[1+e^(4/pi)erf(2/(sqrt(pi)))]
normal distribution sqrt(2/pi)sigma
Pareto distribution alpha/(sqrt(2pi))
Rayleigh distribution sqrt(2pi)erfc((sqrt(pi))/2)
Student's t-distribution (sqrt(nu))/((nu-1)B(1/2nu,1/2))
Student's t-distribution (2sqrt(nu)sigma-nu^(nu/2)sigma^nusqrt(mu^2+nusigma^2))/((nu-1)B(1/2nu,1/2))
triangular distribution 1/6(b-a)
triangular distribution {(2(b+c-2a)^3)/(81(a-b)(a-c)) for a+b<2c; (2(a+c-2b)^3)/(81(a-b)(b-c)) for a+b>2c
uniform distribution 1/4(b-a)

The following table summarizes the mean absolute deviations for some named discrete distributions, where mu=zeta(rho)/zeta(rho+1).

distribution M.D.
Bernoulli distribution 2p(1-p)
binomial distribution 2(1-p)^(N-|_Np_|)p^(|_Np_|+1)(|_Np_|+1)(N; |_Np_|+1)
discrete uniform distribution {1/4N for N even; ((N-1)(N+1))/(4N) for N odd
geometric distribution 2(1-p)^(|_1/p_|)|_1/p_|
Poisson distribution (2e^(-nu)nu^(|_nu_|+1))/(|_nu_|!)
Zipf distribution (2[zeta(rho+1)zeta(rho,|_mu_|+1)-zeta(rho)zeta(rho+1,|_mu_|+1)])/(zeta^2(rho+1))

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Amazing Mathematics Answer Key Mean Absolute Deviation Maze Answer Key

Source: https://mathworld.wolfram.com/MeanDeviation.html

Posted by: mcculloughbefor1969.blogspot.com

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