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Intro Dispersion
Summarizing univariate data (II):
Measures of dispersion
Facultad de Comercio, Turismo y Ciencias Sociales Jovellanos
Measures of dispersion
Intro Dispersion
Introduction
Measures of dispersion Variance Standard Deviation Coefficient of variation Interquartile range Measures of inequality
Measures of dispersion
Dispersion^ Intro
Why do we need measures of dispersion?
- Example 1. Discuss the example about reliability of suppliers provided in page 34.
- Example 2. Angela, Mary, John and Emma have been asked about how many hours a day they spent in home study? Their answers are provided in the following table.
Angela Mary John Emma October 3 0 0 0 November 3 2 0 0 December 3 3 6 3 January 4 8 7 10
Who is the most regular student, in your opinion? And the least regular one?
Dispersion^ Intro
Variance Std.Dev.Coef.Var. Int.RangeIneq.
Measures of dispersion
I (^) The arithmetic mean, the median, and the mode provide measures of central location for the data. I (^) Percentiles are most often used for determining rank positions of individuals in a sample. I (^) We also need information about how much the data are spread out. We can define different measures of dispersion (or variability) based on: I (^) the difference between maximum and minimum values in the sample I (^) the distances of scores of all individuals in the sample wrt the arithmetic mean I (^) the difference between some specific percentiles (for instance the 75th percentile and the 25th percentile) I (^) etc
Intro Dispersion Coef.Var.Int.Range Ineq.
Measures of dispersion
I (^) Variance I (^) Standard deviation I (^) Coefficient of variation I (^) Interquartile range I (^) Measures of inequality or concentration I (^) Lorenz curve and Gini index
Measures of dispersion
Intro Dispersion Coef.Var.Int.Range Ineq.
Variance
I (^) Idea: Average of squared distances. x xi
I (^) Mathematical formula: Var(X ) =
∑k i=1(xi^ −x)^2 ·^ ni n. I (^) Example:
10 20 30 0 20 40 20
low variance high variance (^) null variance
I (^) Since the deviation scores are squared, the variance is in square units. Measures of dispersion
Dispersion^ Intro
Variance Std.Dev.Coef.Var. Int.RangeIneq.
Properties of the variance
I (^) Var(cX + d) = c^2 Var(X ), ∀ c, d I (^) Var(X ) = 0 if and only if k = 1 (Variance is zero when all observations are the same, i.e., all individuals in the sample take the same value.) I (^) Computational formula: Var(X ) = x^2 − x^2 =
∑k i=1 x i^2 ni n −^ x
Example. Let us compute the variance of the figures: 2, 3, 4, 2, 8. I (^) First, we calculate the arithmetic mean:
x =
2 + 3 + 4 + 2 + 8 5
=
19 5
= 3. 8 I (^) Second, we calculate arithmetic mean of squares:
x^2 =
22 + 3^2 + 4^2 + 2^2 + 8^2 5
=
97 5
= 19. 4 I (^) Finally, we calculate the variance Var (X ) = 19. 4 − 3. 82 = 4. 96
Dispersion^ Intro
Variance Std.Dev.Coef.Var. Int.RangeIneq.
Standard deviation
I (^) Mathematical formula: SD(X ) =
Var(X ) =
√ (^) ∑k i=1(xi^ −x)^2 ·^ ni n. I (^) Unlike variance, it is expressed in the same units as the data. I (^) Example: Let us compute the standard deviation of the figures: 2, 3, 4, 2, 8. I (^) First, we calculate the variance, i.e: I (^) First, we calculate the arithmetic mean:
x = 2 + 3 + 4 + 2 + 8 5 =^195 = 3. 8 I (^) Second, we calculate arithmetic mean of squares:
x^2 =^2
(^2) + 3 (^2) + 4 (^2) + 2 (^2) + 8 2 5 =
97 5 = 19.^4 I (^) Third, we calculate the variance Var (X ) = 19. 4 − 3. 82 = 4. 96 I (^) Finally, we compute the squared root: SD(X ) =
√ 4 .96 = 2. 23.
Intro Dispersion Coef.Var.Int.Range Ineq.
Example: Gini coefficient of national income around the
world
Measures of dispersion
Intro Dispersion Coef.Var.Int.Range Ineq.
Example: Gini coefficient since WW-II
Measures of dispersion
Dispersion^ Intro
Variance Std.Dev.Coef.Var. Int.RangeIneq.
Example: the variance is not the best dispersion measure
to quantify “inequality”
I (^) Income in the first sample (in thousand Euro): xi ni 10 1 20 1 30 1 40 1 50 1 I (^) Income in the second sample (in thousand Euro): yi n′ i 10 3 40 1 50 1 I (^) Var(X ) <Var(Y ), but income is better distributed in the second case.
Dispersion^ Intro
Variance Std.Dev.Coef.Var. Int.RangeIneq.
Introduction to calculation of Gini coefficient (I)
The following frequency table indicates the distribution of income (in Euro) on a sample:
xi ni Ni Pi xi ni cum. income Qi 10 5 5 50% 50 50 25% 20 1 6 60% 20 70 35% 30 3 9 90% 90 160 80% 40 1 10 100% 40 200 100%
I (^) Cumulative income of 50% poorest people (5 persons)= 50 Eur. = 25% of total income. I (^) Cumulative income of 60% poorest people (6 persons)= 70 Eur. = 35% of total income. I (^) Cumulative income of 90% poorest people (9 persons)= 160 Eur. = 80% of total income.
Intro Dispersion Coef.Var.Int.Range Ineq.
Introduction to calculation of Gini coefficient (II): the
Lorenz curve
Lorenz curve
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90 100 Cumulative percent
Cumulative income
Measures of dispersion
Intro Dispersion Coef.Var.Int.Range Ineq.
Mathematical formula of Gini coefficient
IG =
∑k− 1 i=1 ∑ (Pi^ −^ Qi^ ) k− 1 i=1 Pi
I (^) Pk = Qk = 100% I (^) Pi ≥ Qi , for all i. Therefore IG ≥ 0. I (^) Qi ≥ 0 , for all i. Therefore
∑k− 1 i=1 (Pi^ −^ Qi^ )^ ≤^
∑k− 1 i=1 Pi^ ,^ and so IG ≤ 1. I (^) IG = 0 corresponds to total equality. (A society in which everyone earns the same) I (^) IG = 1 corresponds to total inequality. (One person has all the income)
Measures of dispersion
Dispersion^ Intro
Variance Std.Dev.Coef.Var. Int.RangeIneq.
Example: Brazil vs Hungary
Dispersion^ Intro
Variance Std.Dev.Coef.Var. Int.RangeIneq.
Gini coefficient: graphical view
IG =
area(red) area(red) + area(blue)
(0 ≤ IG ≤ 1)
