# Understanding Statistical Moments: Formulas, Properties, and Applications

Classified in Mathematics

Written at on English with a size of 4.61 KB.

## 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).

Order | Origin (Zero) | Central (Mean) |
---|---|---|

r=0 | M_{0}=1 | M_{0}=1 |

r=1 | M_{1}= x̅ | M_{1}= 0 |

r=2 | M_{2}= (Σ(X_{i}^{2}.n_{i})/N | M_{2}= (Σ(X_{i} – x̅)^{2}.n_{i})/N |

r=3 | M_{3}=Σ(X_{i}^{3}.n_{i})/N | M_{3}= (Σ(X_{i} – x̅)^{3}.n_{i})/N |

### Change of Origin of the Mean

x̅ + k

### Change of Origin of the Variance

M_{2}= (Σ((X_{i} +K)–(K+ x̅))^{2}.n_{i})/N= M_{2}= (Σ(X_{i} – x̅)^{2}.n_{i})/N - not affected

### Change of Scale

M_{2}= (Σ(X_{i}.k – x̅.k)=M_{2}= (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

r^{2}

### 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)=y_{1}.f_{xy1}+y_{2}.f_{xy2}+y_{2}.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}=B_{0}+B_{1}.T_{i}+B_{2}.Q_{1} (indicate the quarter)+B_{3}.Q_{2}+B_{4}.Q_{3}+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

w_{1}x_{1}+w_{2}x_{2}+w_{3}x_{3}/w_{1}+w_{2}+w_{3}

### Geometric Mean

GM=n√(x_{1}).(x_{2}).(x_{3}) (1.8) use to calculate average change in percentage

### Variation Rate

T=X_{t5}-x_{t4}/x_{t4}

### Index Number

I=X_{t}/x_{0} (base) and after the variation rate.

### Value Index

I=P_{t}.Q_{t}/P_{0}.Q_{0}=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