Statistical Measures: Central Tendency, Dispersion, and Form

Measures of Central Tendency:

Arithmetic Mean

Used for:

Intervals and pooled data (xi = class mark)

Data are not grouped (tables).

Data are grouped (tables) but no intervals.

Median

The middle value in a sorted dataset.

For an odd number of observations, it's the middle number.

Sort data from lowest to highest before finding the median.

Mode

The value that appears most frequently in a dataset.

Mid-Range

RM = (Maximum Value + Minimum Value) / 2

Geometric Mean

G = ⁿ√(x₁ * x₂ * ... * xn)

Harmonic Mean

H = n / ( (1/x₁) + (1/x₂) + ... + (1/xn) )

Quadratic Mean (Root Mean Square)

Q = √[ (x₁² + x₂² + ... + xn²) / n ]

Percentile

A measure indicating the value below which a given percentage of observations in a group falls.

Kth Percentile

Measures of Dispersion:

Range

Re = Maximum Value - Minimum Value

Variance

S² = Σ(xi - μ)² / N (for population) or Σ(xi - x̄)² / (n-1) (for sample)

Measures the spread of data points around the mean.

Theoretical calculations.

V(x) ≥ 0

Source: Shift does not vary.

Change scale: V(kX) = k² * V(X)

V(k) = 0

Standard Deviation

The positive square root of the variance.

Interquartile Range (IQR)

IQR = Q₃ - Q₁ (Third quartile minus first quartile)

Rate of Opening (Range Ratio)

Coefficient of Variation (%)

CV = (Standard Deviation / Arithmetic Mean) * 100

As +> + scattered data

As +

Not always mean + as can be --

Moments:

• Moments with respect to the origin

• Moments with respect to the mean

Second moment (about the mean)

Measures of Form: Skewness and Kurtosis:

Measures of Asymmetry (Skewness)

Pearson Asymmetry Index

Ap = (Mean - Mode) / Standard Deviation or Ap = 3 * (Mean - Median) / Standard Deviation

Asymmetry increases with the value of Ap.

AP > 0: Distribution is asymmetric to the right (positive asymmetry).

AP < 0: Distribution is asymmetric to the left (negative asymmetry).

AP = 0: Distribution is symmetric (Mean = Median = Mode).

If Ap > 1, asymmetry is significant.

Third Moment (m₃)

Measures the asymmetry and maintains the sign and type of asymmetry.

m₃ > 0: Distribution is asymmetric to the right (+).

m₃ < 0: Distribution is asymmetric to the left (-).

m₃ = 0: Distribution is symmetrical.

Fisher Index (g₁)

g₁ = m₃ / (m₂)^(3/2)

This index normalizes the third-order moment.

Measures of Kurtosis

Kurtosis Rate (g₂)

g₂ = (m₄ / (m₂)² ) - 3

g₂ > 0: Leptokurtic distribution (high peak, heavy tails).

g₂ < 0: Platykurtic distribution (flat peak, light tails).

g₂ = 0: Mesokurtic distribution (normal kurtosis).

Measures of Concentration:

• Maximum concentration: Unequal distribution (e.g., X₁ = X₂ = ... = Xn-₁ = 0 and Xn is the total).

• Minimum concentration: Equitable distribution (X₁ = X₂ = ... = Xn).

Measures of concentration include the Gini index and the Lorenz curve.

Lorenz Curve

Visualizes income distribution within a population.

Data:

Initial products (Xini)

Rentiers and

Individual income (xi)

Absolute frequency (fi)

Cumulative absolute frequency (Ni)

Cumulative totals (Ui = Ri)

Ui -> Total income of all rentiers = A (%) ->

Cumulative relative wage (qi) (%)

Cumulative relative frequency (pi) (%)

pi - qi = 0: Lowest concentration.

Representation: The closer the curve is to the diagonal line (line of absolute equity), the less concentrated the distribution is, indicating more homogeneity.

Intervals of

Gini Index

Measures wealth concentration; it is twice the area between the line of absolute equity and the Lorenz curve.

[0,1] -> 0 = perfect equity, 1 = maximum concentration.