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The purpose of this paper is to achieve decomposition formulas of sums regarding deviation cubes, the sum of deviation raised to the power of four and codeviance, because they allow to evaluate the contribution of different components of the above three absolute measures regarding asymmetry, disnormality and concordance. We have obtained more significant formulas that are valid only for two groups, in addition to the formulas valid for
*r* groups, and we have prepared an example to emphasize how useful those formulas were.

In case, a collective unit can be split in groups, decomposition of deviance in two parts is quite known in statistics (Girone, 2009) [_{k} is the high number of a k-group and by (x_{ki}, y_{ki}) the i-observation of two types in a k-group.

The averages, deviances and deviation cubed sums of X-type groups times k = 1 , 2 , ⋯ , r are

x ¯ k = ∑ i = 1 N k x k i N k (1)

D e v ( X k ) = ∑ i = 1 N k ( x k i − x ¯ k ) 2 (2)

S c ( X k ) = ∑ i = 1 N k ( x k i − x ¯ k ) 3 (3)

General average, general deviance and general sum of deviation cubes are

x ¯ = ∑ k = 1 r ∑ i = 1 N k x k i N (4)

D e v ( X ) = ∑ k = 1 r ∑ i = 1 N k ( x k i − x ¯ ) 2 N (5)

S c ( X ) = ∑ k = 1 r ∑ i = 1 N k ( x k i − x ¯ ) 3 N (6)

Similarly, to what is done for deviance decomposition, we start from the formula of a general sum of deviation cubes: by subtracting and adding the average of k-group within brackets

S c ( X ) = ∑ k = 1 r ∑ i = 1 N k [ ( x k i − x ¯ k ) + ( x ¯ k − x ¯ ) ] 3 (7)

calculating the cube and simplifying it, the outcome is

S c ( X ) = ∑ k = 1 r ∑ i = 1 N k ( x k i − x ¯ k ) 3 + 3 ∑ k = 1 r ( x ¯ k − x ¯ ) D e v ( X k ) + ∑ k = 1 r ( x ¯ k − x ¯ ) 3 N k (8)

that is,

S C ( X ) = ∑ k = 1 r S c ( X k ) + 3 ∑ k = 1 r ( x ¯ k − x ¯ ) D e v ( X k ) + ∑ k = 1 r ( x ¯ k − x ¯ ) 3 N k (9)

which shows that a general sum of deviation cubes is primarily equal to the sum of deviation cube partial sums plus the sum of deviations between partial averages and general average, rated by partial deviances, and even more the sum of deviation cubes between partial averages and general average rated by an high number of groups. In other words, the sum of deviation cubes, apart from the inside, is completed by two components depending on differences between partial averages and general average, on deviances and an high number of groups.

It is hardly necessary to emphasize that if partial averages are all mutually equal, the last two parts cancel each other out. Therefore, the general sum of deviation cubes is equal to the sum of partial sums in deviation cubes.

Other special cases of similar partial deviances are equally interesting as well as high numbers of similar groups.

The above formula is simple only in the case of two groups:

S C ( X ) = ∑ k = 1 2 S C ( X k ) + 3 ( x ¯ 1 − x ¯ 2 ) ( σ 1 2 − σ 2 2 ) N 1 N 2 N 1 + N 2 + ( x ¯ 1 − x ¯ 2 ) 3 ( N 2 − N 1 ) N 1 N 2 ( N 1 + N 2 ) (10)

where σ 1 2 and σ 2 2 are variables of the two groups. The second addend is positive (negative) if averages and variances are concordant (discordant). Instead, third addend is positive (negative) if averages and high numbers of the two groups are discordant (concordant).

The sums of deviation raised to the power of four for X-type groups, that is, the partial sums of deviation fourth exponents as to k = 1 , 2 , ⋯ , r , are

S q ( X k ) = ∑ i = 1 N k ( x k i − x ¯ k ) 4 (11)

The general sum of the deviation raised to the power of four is

S q ( X k ) = ∑ k = 1 r ∑ i = 1 N k ( x k i − x ¯ k ) 4 (12)

Similarly, to what has been done in previous paragraph, by subtracting and adding the average of k-group within brackets, in the general sum of deviation raised to the power of four, the outcome is

S q ( X ) = ∑ k = 1 r ∑ i = 1 N k [ ( x k i − x ¯ k ) + ( x ¯ k − x ¯ ) ] 4 (13)

Calculating to the power of four and simplifying it, it comes out

S q ( X ) = ∑ k = 1 r ∑ i = 1 N k ( x k i − x ¯ k ) 4 + 4 ∑ k = 1 r ( x ¯ k − x ¯ ) S c ( X k ) + 6 ∑ k = 1 r ( x ¯ k − x ¯ ) 2 D e v ( X ) + ∑ k = 1 r ( x ¯ k − x ¯ ) 4 N k (14)

that is,

S q ( X ) = ∑ k = 1 r S q ( X k ) + 4 ∑ k = 1 r ( x ¯ k − x ¯ ) S c ( X k ) + 6 ∑ k = 1 r ( x ¯ k − x ¯ ) D e v ( X k ) + ∑ k = 1 r ( x ¯ k − x ¯ ) 4 N k (15)

which shows that general sum of deviation raised to the power of four is firstly equal to the sum of partial sums in deviation raised to the power of four plus the sum of deviation between partial averages and general average, rated by partial sums of deviation cubes, plus the sum of squares of deviations between partial averages and general average rated by deviances of groups, as well as even the sums of deviation raised to the power of four between partial averages and general average rated by high numbers of groups. In other words, the general sum of deviation raised to the power of four, apart from the inside, is completed by three components depending on differences between partial averages and general average, on partial sums of deviation cubes, on partial deviances and an high number of groups.

It is hardly necessary to emphasize that the last three parts cancel each other out if partial averages are all mutually similar. Therefore, the general sum of deviation raised to the power of four is equal to the sum of partial sums in deviation raised to the power of four.

There are other equally interesting special cases of mutually similar partial sums in deviation cubes, of mutually similar deviances as well as similar high numbers of groups.

The above formula, also in this case, is simple only in the case of two groups:

S q ( X ) = ∑ k = 1 2 S q ( X k ) + 4 ( x ¯ 1 − x ¯ 2 ) ( γ 1 − γ 2 ) N 1 N 2 N 1 + N 2 + 6 ( x ¯ 1 − x ¯ 2 ) 2 ( N 1 σ 1 2 + N 2 σ 2 2 ) N 1 N 2 N 1 + N 2 + ( x ¯ 1 − x ¯ 2 ) 4 ( N 1 2 − N 1 N 2 + N 2 2 ) N 1 N 2 ( N 1 + N 2 ) 3 (16)

where γ_{1} e γ_{2} are asymmetrical indexes of the two groups. The second addend is positive (negative) if averages and asymmetrical indexes are concordant (discordant). The third and fourth addends are always not negative.

Regarding partial averages and totals of Y-type, formulas in previous paragraph are to be taken into account by substituting all x with all y.

Codeviances between types X and Y of the groups, as to k = 1 , 2 , ⋯ , r , are

C o d e v ( X k , Y k ) = ∑ i = 1 N k ( x k i − x ¯ k ) ( y k i − y ¯ k ) . (17)

General codeviance is

C o d e v ( X , Y ) = ∑ k = 1 r ∑ i = 1 N k ( x k i − x ¯ k ) ( y k i − y ¯ ) . (18)

Similarly, to what has been done in previous paragraphs, by subtracting and adding the average of k-group within brackets, the outcome is

C o d e v ( X , Y ) = ∑ k = 1 r ∑ i = 1 N k [ ( x k i − x ¯ k ) + ( x ¯ k − x ¯ ) ] [ ( y ¯ k i − y ¯ k ) + ( y ¯ k − y ¯ ) ] . (19)

By calculating the product and eliminating two zero-value terms, we have

C o d e v ( X , Y ) = ∑ k = 1 r ∑ i = 1 N k ( x k i − x ¯ k ) ( y k i − y ¯ k ) + ∑ k = 1 r ( x ¯ k − x ¯ ) ( y ¯ k − y ¯ ) N k . (20)

that is

C o d e v ( X , Y ) = ∑ k = 1 r C o d e v ( X k , Y k ) + ∑ k = 1 r ( x ¯ k − x ¯ ) ( y ¯ k − y ¯ ) N k (21)

which shows that general codeviance is equal to the sum of partial codeviances increased by the sum of results from deviations of partial averages out of X-general average, as to corresponding Y-deviations rated by high numbers of groups. The latter sum can be also called codeviance of averages.

It is hardly necessary to emphasize that codeviance of averages is zero value in similar partial averages (regarding one or both types), so that general codeviance is equal to the sum of partial codeviances.

The above formula, also in this case, is simple only in the case of two groups:

C o d e v ( X ) = ∑ k = 1 2 C o d e v ( X k , Y k ) + ( x ¯ 1 − x ¯ 2 ) ( y ¯ 1 − y ¯ 2 ) N 1 N 2 N 1 + N 2 (22)

Second addend is positive (negative) depending on whether averages of the above two types are concordant (discordant).

As an application, we refer to a group of 278 students (144 males and 134 females) attending a first year course at the University of Bari whose height and body weight were detected. The averages, deviations, sums of deviation cubes and deviation raised to the power of four are as follows (Tables 1-4).

As to previous results, the following (Tables 5-8) decompositions are made.

Previous results allow the following considerations to be made. Let’s start with the values:

- the two groups are quite similarly large (Males are slightly prevailing);

- average values, as to both types, are bigger among Males;

- standard deviations, for both types, indicates a bigger variability among Males;

Males | Females | Total | |
---|---|---|---|

Number of cases | 144 | 134 | 278 |

Males | Females | Total | |
---|---|---|---|

Averages | 177.42 | 165.74 | 171.79 |

Deviances | 5883.16 | 4317.86 | 19,677.90 |

Standard deviations | 6.39 | 5.68 | 8.41 |

Sums of cubed deviations | −2020.46 | −81.22 | 14,918.91 |

Asymmetrical indexes | −0.053 | −0.003 | 0.090 |

Sums of deviations raised to four | 657,677.65 | 67,978.08 | 3,475,668.57 |

Disnormality indexes | −0.264 | 0.364 | 0.504 |

Males | Females | Total | |
---|---|---|---|

Averages | 71.36 | 56.52 | 64.21 |

Deviances | 16,919.23 | 7283.43.00 | 39,485.90 |

Standard deviations | 10.84 | 7.37 | 11.92 |

Sums of cubed deviations | 159,004.06 | 36,036.56 | 381,979.28 |

Asymmetrical indexes | 0.867 | 0.671 | 0.812 |

Sums of deviations raised to four | 8,273,289.29 | 1,273,444.32 | 9,546,733.61 |

Disnormality indexes | 1.162 | 0.217 | −1.298 |

Males | Females | Total | |
---|---|---|---|

Codeviances | 6329.97 | 2183.28 | 20,548.10 |

Correlation coefficients | +0.634 | +0.389 | +0.737 |

Decomposition of deviance | Heights | Weights | ||
---|---|---|---|---|

Values | % | Values | % | |

Deviances (Males) | 5883.16 | 29.9 | 16,919.23 | 42.9 |

Deviances (Females) | 4317.86 | 21.9 | 7283.43 | 18.4 |

Internal Deviances | 10,201.02 | 51.8 | 24,202.66 | 61.3 |

External Deviances (between) | 9476.88 | 48.2 | 15,283.24 | 38.7 |

General Deviances | 19,677.90 | 100.0 | 39,485.90 | 100.0 |

Decomposition of the sum of deviation cubes | Heights | Weights | ||
---|---|---|---|---|

Values | % | Values | % | |

Sums of cubes (Males) | 2020.46 | −13.5 | 159,004.06 | 41.6 |

Sums of cubes (Females) | −81.22 | −0.6 | 36,036.56 | 9.5 |

Internal Sums of cubes | −2101.68 | −14.1 | 195,040.62 | 51.1 |

Sums of average deviations by deviances | 21,003.88 | 140.8 | 195,096.35 | 51.1 |

Sums of average deviation cubes by high numbers | −3983.29 | −26.7 | −8157.69 | −2.2 |

Total sums of cubes | 14,918.91 | 100.0 | 381,979.28 | 100.0 |

Decomposition of the sum of deviation raised to the power of four | Heights powers | Weights | ||
---|---|---|---|---|

Values | % | Values | % | |

Sums of fourth powers (Males) | 657,677.65 | 18.9 | 8,273,289.29 | 38.3 |

Sums of fourth powers (Females) | 467,978.08 | 13.5 | 1,273,444.32 | 5.9 |

Sums of internal fourth powers | 1,125,655.73 | 32.4 | 9,546,733.61 | 44.2 |

Sums of average deviations by cubed deviations | −43,552.68 | −1.3 | 3,441,148.82 | 15.9 |

Sums of squares average deviations bydeviations | 2,068,829.12 | 59.5 | 7,775,075.23 | 36.0 |

Sums of fourth powers in average deviations by an high number | 324,736.40 | 9.4 | 844,561.34 | 3.9 |

Total sums of fourth powers | 3,475,668.57 | 100.0 | 21,607,519.00 | 100.0 |

Decomposition of codeviance | Values | % |
---|---|---|

Codeviance (Males) | 6329.97 | 30.8 |

Codeviance (Females) | 2183.28 | 10.6 |

Internal codeviance | 8513.25 | 41.4 |

Codeviance of averages | 12,034.85 | 58.6 |

General codeviance | 20,548.10 | 100.0 |

- height asymmetrical indexes in both sexes take on slight negative values, those for Weights, instead, are surely asymmetrical positive;

- height disnormality indexes take on mild and contrasting values (negative for Males and positive for Females), the Weight ones, for both sexes indeed, are similarly positive, even though only the Male one takes on an high value;

- both correlation coefficients are positive, the Male one is the highest.

Let us now turn to decompositions.

The decomposition of deviance highlights a relevant incidence of deviance between the two sexes, due to a marked difference of averages.

It should be added that such incidence affects more distinctly the height, whose variability depends almost exclusively on genetic factors, than weight whose variability also depends on exogenous factors.

Regarding decomposition of the sum of deviation cubes, it is necessary to make a difference as to both types.

Regarding height (

As it can be seen in

It is necessary to make a difference between height and weight regarding the sum of deviation fourth powers.

As to height (

As to weight (

To end up, in

Similarly, to what is done for deviance decomposition (Scheffè, 1999) [

The authors declare no conflicts of interest regarding the publication of this paper.

Manca, F. and Marin, C. (2020) Decomposition of the Sum of Cubes, the Sum Raised to the Power of Four and Codeviance. Applied Mathematics, 11, 1013-1020. https://doi.org/10.4236/am.2020.1110067