Mathematics

Mathematics

Problem

For a certain set of observations grouped in equal class intervals of width 10, the incomplete table below is obtained.
Class interval0–1010–2020–3030–4040–50
Frequency12a25b18
The following facts are known: 1. The total number of observations is 100. 2. The median of the distribution is 28. a. (i) Form the necessary equations and find the values of a and b. b. (ii) After obtaining a and b, write the cumulative frequencies for all classes.
Intuitive explanation

The key is using two constraints to find the missing frequencies aa and bb. First, all frequencies must sum to 100, which gives you one equation. Second, the median of 28 falls in the 20–30 class, so use the median formula for grouped data—the cumulative frequency up to the median class must equal half the total. These two conditions together uniquely determine both unknowns.

Intuitive explanation

The key is using two constraints to find the missing frequencies aa and bb. First, all frequencies must sum to 100, which gives you one equation. Second, the median of 28 falls in the 20–30 class, so use the median formula for grouped data—the cumulative frequency up to the median class must equal half the total. These two conditions together uniquely determine both unknowns.

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