Median, Quantiles and Empirical Relationship

Lesson 08 of STA301 examines positional measures of location, focusing on the grouped Median, quantile partition values, and the empirical relationship linking measures of central tendency. The median represents the middle value dividing an ordered frequency distribution into two equal halves, ensuring fifty percent of observations lie below and fifty percent above it. For continuous grouped data, the median class is located at cumulative frequency n / 2, and the exact value is calculated using the formula: Median = L + (h / f) * (n / 2 - C), where L is the lower class boundary of the median class, h is class width, f is class frequency, and C is the cumulative frequency of the preceding class. Unlike the arithmetic mean, the median is not distorted by extreme outliers and can be accurately computed for distributions with open-ended class intervals. For unimodal frequency distributions of moderate skewness, the lecture presents Karl Pearson's Empirical Relationship: Mean - Mode = 3 * (Mean - Median), which rearranges to Mode = 3*Median - 2*Mean. This relationship enables students to estimate one measure when the other two are known. In a perfectly symmetrical distribution, mean, median, and mode coincide at the central peak. In a positively skewed distribution, the mean is pulled farthest toward the long right tail (Mean > Median > Mode), whereas in a negatively skewed distribution, the order reverses (Mean < Median < Mode). The lecture also explores Quantiles—partition values dividing an ordered dataset into equal segments. Quartiles divide the distribution into four equal quarters (Q1, Q2, Q3); Deciles divide it into ten equal segments (D1 to D9); and Percentiles divide it into one hundred equal segments (P1 to P99). All quantiles follow the general interpolation formula Q_k = L + (h / f) * (k*n / 4 - C). In Virtual University midterm evaluations, quantile computations, empirical formula substitutions, and skewness comparisons are frequently tested. Studying sta301 past papers and reviewing official handouts solidifies conceptual and computational proficiency.