Glossary of statistical terms
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)
Want to learn more about this topic?
Statistics.com offers over 100 courses in statistics from introductory to advanced level. Most are 4 weeks long and take place online in series of weekly lessons and assignments, requiring about 15 hours/week. Participate at your convenience; there are no set times when you must to be online. Ask questions and exchange comments with the instructor and other students on a private discussion board throughout the course.
This course provides an easy introduction to statistics and statistical terminology through a series of practical applications. Once you've completed this course you'll be able to summarize data and interpret reports and newspaper accounts that use statistics and probability. You'll use simulation and resampling to fully grasp the difficult concept of "statistical significance."
This course, the first of a 3-course sequence, provides an introduction to statistics for those with little or no prior exposure to basic probability and statistics. It runs every eight weeks.
This course covers the analysis of data gathered in surveys.
Promoting better understanding of statistics throughout the worldTo celebrate the International Year of Statistics in 2013, we started a program to provide a statistical term every week, delivered directly to your inbox. The Word of the Week program proved to be quite popular, and continues. The Institute for Statistics Education offers an extensive glossary of statistical terms, available to all for reference and research. Make it your New Year's resolution to improve your own statistical knowledge! Sign up here. Rather not have more email? Simply bookmark our home page and check our “Stats Word of the Week” feature.
Want to be notified of future courses?Yes