Ch 5Measures of Central Tendency — Class 11 Economics, concept-first.

A large mass of raw data — the marks of every student in a class, the rainfall recorded across an area, the output of a factory month after month, or the income of every family in a locality — is diff…

A measure of central tendency condenses a whole set of data into a single, representative value — the "typical" figure around which the observations cluster.

The way the arithmetic mean is worked out depends on how the data is presented. Broadly, the calculation falls into two categories:

For an ungrouped series (a plain list of individual values) the mean can be found by three equivalent methods.

For grouped data the observations carry frequencies. There are two cases.

The arithmetic mean has two properties worth remembering.

In an ordinary mean every item counts equally. Sometimes, though, items differ in importance, and it is right to give the more important ones greater influence.

The median is a positional average — the value that divides an ordered distribution into two equal halves.

For an individual (ungrouped) series the median is found by sorting the data from smallest to largest and reading off the middle value.

In a discrete series each value carries a frequency, and the median item — the item — is located using the cumulative frequency.

In a continuous series the data is grouped into class intervals, so the median is not a single listed value but lies inside a class.

Quartiles are positional measures that divide an ordered distribution into four equal parts, each holding the same number of observations. There are three quartiles:

Where quartiles cut a distribution into four parts, percentiles cut it into one hundred equal parts, giving 99 dividing positions denoted . The percentile, , coincides with the median.

Quartiles are located just like the median. For individual and discrete series the positions are given (with = number of observations) by:

The mode is the value that occurs most frequently in a series — the most "typical" value, around which the greatest concentration of items clusters.

For a given distribution the three averages usually stand in a fixed order relative to one another. Writing arithmetic mean , median and mode , their relative magnitudes are:

Measures of central tendency, or averages, summarise a data set by a single most-representative value.

- Central tendency is a single value that represents the centre of a data set. The three main measures are the mean, median, and mode. - Arithmetic mean () for ungrouped data: .

Solved examples, worked out step by step.