Partition Values for a Discrete Frequency Distribution

When data are given as distinct values with frequencies (a discrete frequency distribution), the observations are not listed one by one, so we locate a partition value through the less-than cumulative frequency column.

Method.

1. Arrange the values $x$ in ascending order and write the frequencies $f$.
2. Build the less-than cumulative frequency ($cf$) column and find $N = \sum f$ (the total frequency).
3. Compute the required position, using $(N+1)$ exactly as for raw data:

4. Read down the $cf$ column: the partition value is the value of $x$ whose cumulative frequency first reaches (or exceeds) that position.

Worked illustration. For $x = 10, 20, 30, 40, 50$ with $f = 4, 7, 12, 9, 3$:

Here $N = 35$, so $N + 1 = 36$. The $Q_1$ position is $\tfrac{1(36)}{4} = 9$; the first $cf \ge 9$ is $11$, against $x = 20$, so $Q_1 = 20$. The full working for $Q_1, Q_2, Q_3$ is in the Worked Examples.