Mathematics

Mathematics

Problem

The marks obtained out of 100 by 80 students in a Mathematics test are grouped below:
Marks0–1010–2020–3030–4040–5050–6060–7070–8080–9090–100
Number of students26711151410753
(a) Prepare both the ‘less than’ and the ‘more than’ cumulative frequency tables for the above distribution. (b) Draw the ‘less than’ and ‘more than’ ogives on the same axes, choosing suitable scales. (c) From the point of intersection of the two curves, determine the median mark. (d) Verify the median obtained graphically by calculating it using the median formula for grouped data.
Intuitive explanation

Cumulative frequency tables organize raw data into running totals—either counting upward from the lowest marks ('less than') or downward from the highest ('more than'). When you plot these as ogives (smooth curves), they tell a visual story: the 'less than' curve climbs from bottom-left to top-right, while the 'more than' curve descends. Where they intersect, you've found the median—the mark that splits the class exactly in half. This graphical insight connects to the algebraic median formula, which pinpoints the same value by finding where cumulative frequency equals n/2n/2.

Intuitive explanation

Cumulative frequency tables organize raw data into running totals—either counting upward from the lowest marks ('less than') or downward from the highest ('more than'). When you plot these as ogives (smooth curves), they tell a visual story: the 'less than' curve climbs from bottom-left to top-right, while the 'more than' curve descends. Where they intersect, you've found the median—the mark that splits the class exactly in half. This graphical insight connects to the algebraic median formula, which pinpoints the same value by finding where cumulative frequency equals n/2n/2.

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