Q.For a certain distribution, the arithmetic mean is 45 and the median is 48. Estimate the mode using the empirical relationship, and verify your answer is internally consistent.

Mode: The Most Frequent Value

Imagine you ask a class of 30 students how many siblings they have. The answers come in: some say 0, some say 1, some say 2, a few say 3, and one says 5. You write the numbers on the board. One number appears more times than any other — say, 1 appears 12 times. That number, 1, is the mode of the data.

The mode is simply the value that occurs most often. It's the most popular, the most common, the typical-in-the-sense-of-frequent. No calculation, no formula — just counting.

The Precise Definition

For a given set of data, the mode is the observation (or value) that has the highest frequency — that is, the one that appears the maximum number of times.

If no value repeats, the data has no mode. If two values tie for the highest frequency, the data is bimodal (two modes). If more than two tie, it is multimodal.

Examples

Example 1 (Unimodal):

Data: 2, 3, 5, 3, 7, 3, 8, 3, 9

Frequency: 3 appears 4 times, all others appear once.

Mode = 3

Example 2 (Bimodal):

Data: 4, 6, 4, 7, 6, 8, 4, 6

Frequency: 4 appears 3 times, 6 appears 3 times.

Modes = 4 and 6

Example 3 (No mode):

Data: 1, 2, 3, 4, 5

Each value appears exactly once.

No mode.

Why Mode Matters

Unlike the mean (average) or median (middle value), the mode works for any type of data — numbers, yes, but also categories. You can find the mode of favourite colours, car brands, or blood groups. The mean of "red, blue, red, green" is meaningless; the mode — red — tells you the most common choice. …