Mode — Computation, Merits and Limitations

Mode is the value that occurs most frequently in a data set — the observation with the highest concentration of items around it.

1. Individual series: the mode is simply the value that repeats the maximum number of times. If every value appears only once (or all appear equally often), the series has no mode (or is "amodal").

2. Discrete series: the mode is the $x$ value carrying the highest frequency. When the highest frequency does not clearly stand out (values clustering close to two neighbouring points), a grouping table is used to confirm the modal value by re-grouping frequencies in twos and threes — a standard technique for an "ill-defined" mode, taught alongside the direct-inspection method.

3. Continuous series: first identify the modal class — the class with the highest frequency — then apply:

where $L$ = lower limit of the modal class, $f_1$ = frequency of the modal class, $f_0$ = frequency of the class PRECEDING the modal class, $f_2$ = frequency of the class SUCCEEDING the modal class, and $h$ = class width.

Empirical relationship (approximation, for moderately skewed distributions): when a distribution's mode is hard to determine directly (ill-defined or bimodal data), mode may be ESTIMATED from the already-computed mean and median:

This is always an approximation, never an exact formula, and is used only when direct computation of the mode is genuinely unreliable.

Merits of mode:

• Not affected by extreme values, similar to the median.
• Can be located graphically (from a histogram).
• Easy to understand — it is simply "the most common value," useful in everyday business language (e.g. "the modal shoe size sold").
• Useful even for qualitative-leaning data with a clear most-frequent category. …