Understanding Statistical Moments: Formulas, Properties, and Applications

Statistical Moments

A moment is a specific quantitative measure that characterizes a distribution. Two distributions are equal if their moments are equal.

Types of Moments:

• Related to the origin (0 as a reference)
• Related to the mean (μ as a reference)

M_r = (Σ(X_i – O)^r.n_i)/N

Where:

• X: individual observations
• r: Order of the moment (Order zero: r=0, First order: r=1, Second order: r=2, Third order: r=3)
• O: Origin or reference point
• n: frequency of each observation
• N: total number of observations

Properties:

• All moments of r=0 are equal to 1.
• Moments related to the mean are frequently called central moments.
• Moments with reference point 0 are frequently called ordinary moments.
• The arithmetic mean corresponds to the ordinary moment of the first order (r=1).
• The variance corresponds to the central moment of the second order (r=2).
• The skewness corresponds to the central moment of the third order (r=3).
• The kurtosis corresponds to the central moment of the fourth order (r=4).

Change of Origin of the Mean

x̅ + k

Change of Origin of the Variance

M₂= (Σ((X_i +K)–(K+ x̅))².n_i)/N= M₂= (Σ(X_i – x̅)².n_i)/N - not affected

Change of Scale

M₂= (Σ(X_i.k – x̅.k)=M₂= (Σ(K(X_i – x̅)

Covariance

d_xy=Σx_i.y_i.f_i-(x.y)

Coefficient of Correlation

r=d_xy/d_x.d_y

Coefficient of Determination

r²

Conditional Probability

Probability of an event (A) given that another event has occurred (B):

P(A|B)=P(A∩B)/P(B) (joint probability/marginal probability)

Probability of A or B

P(A or B)=P(A)+P(B)-P(A and B)

Dependent Event

P(A and B)=P(A).P(A|B)

Conditional Probability (Example)

Ask Alex knowing that it was incorrect: P(A|X)=P(A∩X)/P(X) (sum of incorrect)

Permutation Formula (Order Matters)

nPr=n!/(n-r)! (n - number of all possible objects, r - number of objects selected)

Combination Formula

nCr=nPr/r!=n!/r!(n-r)!

Density

h_i=n_i/r_i; Rank: r_i=L_i-L_i+1

Conditional Mean

E(Y|X=0)=y₁.f_xy1+y₂.f_xy2+y₂.f_xy3.

Coefficient of Variability

CV=d (standard deviation)/x (mean)

Standardization

Z=X_i-x/d (use to compare groups (each one with its mean and d), the higher the value the thought to get).

Trend and Seasonality

Y_i=B₀+B₁.T_i+B₂.Q₁ (indicate the quarter)+B₃.Q₂+B₄.Q₃+Σ_i

Mean Deviation (When Comparing Collections with the Same Mean)

MD=Σ|X-X|/ n

Median (Me)

Middle point value of n_i

Mode (Mo)

Most repeated value

Mean

Σ(x_i.f_i)

Weighted Mean

w₁x₁+w₂x₂+w₃x₃/w₁+w₂+w₃

Geometric Mean

GM=n√(x₁).(x₂).(x₃) (1.8) use to calculate average change in percentage

Variation Rate

T=X_t5-x_t4/x_t4

Index Number

I=X_t/x₀ (base) and after the variation rate.

Value Index

I=P_t.Q_t/P₀.Q₀=VI

Index Chain

I(18,15)=I(15,16).I(16,17).I(17,18)

Evolution on Average Salary

12Avgwage=salary2012/employ2012

13Avgwage=salary2013/employ2013