Q.Find the Mode from the following distribution: Marks: 0-10, 10-20, 20-30, 30-40, 40-50; Number of students (f): 5, 8, 15, 12, 5.

Measures of central tendency are single values that summarise the centre of a data set. The three mathematical averages are the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM); the two positional averages are the median and mode.

How it works

The arithmetic mean adds and divides; the geometric mean multiplies and takes a root (ideal for ratios and growth rates); the harmonic mean handles rates like speed. The median is the middle value in order, and the mode is the most frequent value — both unaffected by extreme outliers.

For two positive numbers a neat identity ties the three mathematical means together, so any two of them determine the third.

Key properties

• $\text{AM}\ge\text{GM}\ge\text{HM}$ for any set of positive values, with equality only when all values are equal.
• The identity $\text{GM}^{2}=\text{AM}\times\text{HM}$ holds exactly for two observations.
• Median and mode ignore extreme values; the arithmetic mean does not.

Quick example

Given $\text{AM}=64$ and $\text{HM}=16$, find the GM.

1. Use $\text{GM}=\sqrt{\text{AM}\times\text{HM}}$. …