Q.The marks scored (out of 50) by 12 first-year Commerce students in an Economics unit test are: 22, 26, 14, 30, 18, 25, 35, 40, 28, 33, 20, 45. Calculate Q1 and Q3.

For an individual series, Q1 and Q3 are located by first arranging the data in order and finding the item at position N/4 and 3N/4 respectively.

Step 1 — Arrange in ascending order (N = 12).

14, 18, 20, 22, 25, 26, 28, 30, 33, 35, 40, 45

Step 2 — Locate Q1. $k = N/4 = 12/4 = 3$, a whole number, so Q1 is the average of the 3rd and 4th items:

Step 3 — Locate Q3. $k = 3N/4 = 3(12)/4 = 9$, a whole number, so Q3 is the average of the 9th and 10th items:

Cross-check via Q2/Median. Using the identical rule with i=2, $k=2N/4=6$ (whole), so $Q_2 = (26+28)/2 = 27$ — the same value the Measures of Central Tendency method gives for the median of this data (average of the 6th and 7th items), confirming that the same positional rule correctly reproduces Q2 = Median.

Forgetting to arrange the data in ascending order FIRST is the most common error — the position formula (iN/4) is only meaningful once the series is sorted; also, when the position is a whole number, some students take only the item AT that position instead of averaging it with the next item.