Q.Using the same daily-wages data as the previous example (0–10: 5, 10–20: 8, 20–30: 15, 30–40: 16, 40–50: 6 workers), find the modal wage, and verify your answer using the empirical relationship among mean, median and mode.

The modal class is the class with the highest frequency; the formula then interpolates within it to locate the exact mode. …

Step 1 — Locate the modal class. The highest frequency is 16, in the class 30–40 ($l=30$, $f_1=16$, $f_0=15$ [the 20–30 class before it], $f_2=6$ [the 40–50 class after it], $h=10$).

Step 2 — Apply the mode formula. …

Cross-check using the empirical relationship. From the previous example, Median = ₹28. The mean of this same distribution is $\bar X = \Sigma fm/N = 1350/50 = ₹27$ (mid-values 5,15,25,35,45 with frequencies 5,8,15,16,6). Karl Pearson's empirical formula estimates: $Mode \approx 3\,Median - 2\,Mean = 3(28)-2(27) = 84-54 = 30$. This estimate (₹30) is close to, though not identical with, the exact formula-based mode of ₹30.91 — exactly as expected, since the empirical relationship is on …

Treating the empirical-relationship estimate (₹30) and the exact grouping-formula value (₹30.91) as if one of them must be 'wrong' because they don't match exactly — the empirical relati …