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Central Tendency (Measures)

Central Tendency (Measures):

Any measure of central tendency provides a typical value of a set of Math image values Math image . Normally, it is a value around which values Math image 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 Math image of central tendency normally meet the following requirements:

  • If all values Math image coincide – i.e. all are equal to the same value Math image – then the measure Math image is equal to Math image , or formally
  • The value Math image is within the interval between the minimal and the maximal value of the set Math image :
  • Measure Math image has a property of shift invariance :

    where Math image may be negative.

  • Measure Math image has a property of scale invariance :

Most measures of central tendency also have the following property: If the set of values Math image is symmetrical with respect to a value Math image (the center), then the value of the measure Math image coincides with the center Math image . “Symmetrical” here means that, for each value Math image different from Math image , there is another value Math image deviating from the center Math image by the same magnitude as Math image , 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 Math image 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)

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