Central Tendency (Measures):
Any measure of central tendency provides a typical value of a set of values . Normally, it is a value around which values are grouped. The most widely used measures of central tendency are (arithmetic) mean , median , trimmed mean , mode . Measures of central tendency are defined for a population and for a sample .
For example, two samples – (8,9,10,11,12) and (18,19,20,21,22) have central locations differing by 10 units, and most measures of central location would give values 10 and 20 of the two samples, respectively.
Measures of central tendency normally meet the following requirements:
- If all values coincide – i.e. all are equal to the same value – then the measure is equal to , or formally
- The value is within the interval between the minimal and the maximal value of the set :
- Measure has a property of shift invariance :
where may be negative.
- Measure has a property of scale invariance :
Most measures of central tendency also have the following property: If the set of values is symmetrical with respect to a value (the center), then the value of the measure coincides with the center . “Symmetrical” here means that, for each value different from , there is another value deviating from the center by the same magnitude as , but in the opposite direction. (More generally, “Symmetrical” means that the right and left sides of a distribution look the same, in mirror image.)
Note that some measures are often classified as measures of central tendency (and have “mean” in their names) but do not meet the requirement of shift invariance. Such measures are usually defined mathematically only for non-negative values and, practically, are applicable to quantities that are non-negative in principle – e.g. price, time or space interval, weight, etc. Strictly speaking, such descriptive statistics measure “effective magnitude” or “average magnitude” rather than central tendency. Some examples of such measures are: the power mean , the harmonic mean , the geometric mean , root mean square .
See also Mean Values (Comparison)