University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

83

Pages:

3
Author:

Niklas

One of the first measures of asymmetry, attributed to A.L. Bowley and G. U. Yule, is based on the quartiles. B = (q3 - 2m + q1) / (q3 - q1) -1 ≤ B ≤ 1 Left asymmetry: q3 - m < m - q1, B > 0 Symmetry: q3 - m = m - q1, B = 0 Right asymmetry: q3 - m > m - q1, B < 0 Example: Data: 22 24 18 30 18 30 20 19 28 19 N = 10 Mean = 22.8 Median = 20 q1 = 19, q2 = 21, q3 = 28 B = (28 - 2 * 22.8 + 19) / (28 - 19) = 0.5 (right symmetry) IQR = q3 - q1 = 9 Lw = q1 - 1.5IQR = 5.5 Uw = q3 + 1.5IQR = 41.5 The most used index for the asymmetry is the Pearson index: standardized asymmetry index, introduced by K. Pearson and defined as: P = (1/N) * Σ(i=1 to N) * ((xi - μ)^3 / σ^3) Left asymmetry, symmetry, right asymmetry: Visual representations with box plots and dot plots Examples: Cinema: Bowley = 0.33, Pearson = 2.2 Books: Bowley = 0, Pearson = 5.1 High school marks: Bowley = 0, Pearson = 0.35 Distribution shape: Kurtosis Platykurtic: Light tails, first to reach zero Leptokurtic: Heavy tails, highest peak An observation potentially anomalous (outlier) is defined as an observation that appears "extreme" with respect to the rest of the distribution. In a boxplot, it goes over the whiskers (stache). Example: Box plot with labeled quartiles and whiskers, indicating outliers. Heading: Transformations To transform a variable Y with observations y1, ..., yN according to a function g(y), means to create a new variable T = g(Y) of which modalities are transformed according to g(y) of the original modalities of Y. New variable has observations: t1 = g(y1), ..., tn = g(yn). We encounter transformations in cases such as: Changing units of measure Applying a function to improve the readability of a distribution Understanding what happens to the positions and variability measures of T in relation to the corresponding measures of Y. T = constant * Y + g with g = constant Example: Original observations: y1 = 1, y2 = 4, y3 = 9 Transformation: g(y) = √y Transformed observations: t1 = 1, t2 = 2, t3 = 3