# Understanding Statistical Moments: Formulas, Properties, and Applications

Classified in Mathematics

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## 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)

Mr = (Σ(Xi – O)r.ni)/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).
OrderOrigin (Zero)Central (Mean)
r=0M0=1M0=1
r=1M1= x̅M1= 0
r=2M2= (Σ(Xi2.ni)/NM2= (Σ(Xi – x̅)2.ni)/N
r=3M3=Σ(Xi3.ni)/NM3= (Σ(Xi – x̅)3.ni)/N

x̅ + k

### Change of Origin of the Variance

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

### Change of Scale

M2= (Σ(Xi.k – x̅.k)=M2= (Σ(K(Xi – x̅)

### Covariance

dxy=Σxi.yi.fi-(x.y)

r=dxy/dx.dy

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

hi=ni/ri; Rank: ri=Li-Li+1

### Conditional Mean

E(Y|X=0)=y1.fxy1+y2.fxy2+y2.fxy3.

### Coefficient of Variability

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

### Standardization

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

### Trend and Seasonality

Yi=B0+B1.Ti+B2.Q1 (indicate the quarter)+B3.Q2+B4.Q3i

MD=Σ|X-X|/ n

### Median (Me)

Middle point value of ni

### Mode (Mo)

Most repeated value

Σ(xi.fi)

### Weighted Mean

w1x1+w2x2+w3x3/w1+w2+w3

### Geometric Mean

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

T=Xt5-xt4/xt4

### Index Number

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

I=Pt.Qt/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