Business Mathematics and Basic Statistics · Ch 7 — Measures of Dispersion

Mean Deviation about the Median — Continuous (Grouped) Data

Extending mean deviation about the median to grouped/continuous data first requires the median of a continuous distribution itself — a construction the measures-of-central-tendency chapter deliberately left out, since that chapter's own scope was limited to discrete data. It is introduced here because this topic's syllabus explicitly requires mean deviation about the median for continuous data as well.

For continuous data, the median cannot simply be read off a cumulative frequency column the way it can for discrete data, because the individual observations inside a class are not separately known — only how many fall inside each interval. Instead, the median is estimated by assuming the observations in the median class (the class interval containing the middle of the distribution) are spread evenly across that class, and interpolating within it:

Note

Median of Continuous/Grouped Data (Interpolation Formula)

$M = L + \left(\dfrac{\frac{N}{2} - cf}{f}\right)\times h$

where $L$ = lower boundary of the median class, $N=\sum f$ , $cf$ = cumulative frequency of the class before the median class, $f$ = frequency of the median class itself, and $h$ = the median class's width. The median class is identified as the first class whose cumulative frequency reaches or exceeds $N/2$ .

Once $M$ is found this way, the mean deviation about the median for grouped data follows exactly the same pattern as every other grouped-data dispersion measure — using class marks, not class limits:

Note

Mean Deviation about the Median — Continuous/Grouped Data

$\text{MD}(M) = \dfrac{\sum f_i\,|x_i - M|}{N}$

where $x_i$ is the class mark of the $i$ -th class. …

Definition 1Median Class

The class interval in a continuous/grouped frequency distribution whose cumulative frequency first reaches or exceeds $N/2$ — the interval within w …

Definition 2Interpolation (for a continuous median)

Estimating the median's exact position inside the median class by assuming its observations are spread evenly across the class width, rather than reading off a single discr …

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