Business Mathematics and Basic Statistics · Ch 4 — Measures of Central Tendency

Median — Discrete Data

The median is the value that lies exactly in the middle of a data set once it has been arranged in ascending (or descending) order — half the observations lie at or below it, and half lie at or above it.

For individual (raw) observations:

Note

Median — Individual Observations

Arrange the $n$ observations in ascending order.

If $n$ is odd, the median is the value of the $\left(\dfrac{n+1}{2}\right)$ -th observation.
If $n$ is even, the median is the average of the $\left(\dfrac{n}{2}\right)$ -th and $\left(\dfrac{n}{2}+1\right)$ -th observations.

For a discrete frequency distribution, the same idea is applied using a cumulative frequency (c.f.) column, which, against each value of $x$ , records the running total of all frequencies up to and including that value.

Note

Median — Discrete Frequency Distribution (Cumulative Frequency Method)

Build the cumulative frequency column and let $N = \sum f$ .

If $N$ is odd, the median is the value of $x$ whose cumulative frequency first reaches or exceeds $\dfrac{N+1}{2}$ .
If $N$ is even, locate the values of $x$ corresponding to the $\left(\dfrac{N}{2}\right)$ -th and $\left(\dfrac{N}{2}+1\right)$ -th observations from the c.f. column, and take their average. …

Definition 1Cumulative Frequency

The running total of frequencies up to and including a given value of $x$ ; used to locate the position of the median in a f …

Definition 2Median

The middle value of a data set arranged in ascending order — the value such that half the observations lie at or below it and h …

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