Box-and-Whisker Plots and Skewness

Lesson 13 of STA301 explores visual data screening and the quantitative measurement of distributional asymmetry through Box-and-Whisker Plots and Karl Pearson's Skewness coefficients. Developed by John Tukey, the Box-and-Whisker Plot provides a graphical representation of the five-number summary. A central rectangular box spans the Interquartile Range (IQR = Q3 - Q1), encompassing the central fifty percent of the data, with a vertical line marking the median. 'Whiskers' extend outward from the box edges to the smallest and largest observations situated within designated boundary thresholds known as fences. Students learn to compute Inner Fences (Q1 - 1.5*IQR and Q3 + 1.5*IQR) and Outer Fences (Q1 - 3.0*IQR and Q3 + 3.0*IQR). Any observation falling between the inner and outer fences is flagged as a mild outlier, whereas observations falling beyond the outer fences are classified as extreme outliers. This diagnostic protocol enables researchers to identify data anomalies and examine distribution symmetry without distributional assumptions. The lecture then investigates Skewness—the absence of symmetry in a frequency distribution. A symmetrical distribution exhibits equal frequencies at equal distances from the central mean, with mean, median, and mode coinciding. When the distribution possesses a long, extended tail toward higher values, it is positively skewed (right-skewed), with Mean > Median > Mode. Conversely, when the tail extends toward lower values, it is negatively skewed (left-skewed), with Mean < Median < Mode. To quantify asymmetry, Karl Pearson's Coefficients of Skewness are derived: Pearson's First Coefficient of Skewness: Sk1 = (Mean - Mode) / s, and Pearson's Second Coefficient of Skewness: Sk2 = 3*(Mean - Median) / s. Both coefficients yield zero for symmetrical distributions, positive values for right-skewed data, and negative values for left-skewed data. In Virtual University midterm examinations, calculating inner fences, classifying outliers, and interpreting Pearson's skewness coefficients are frequent objective and subjective exam questions. Studying sta301 handouts pdf and past papers ensures thorough exam preparation.