# Measure of Central Tendency

The Measure of central tendency is a statistic that summarizes the entire quantitative set of data in a single value (a representative value of the data set) having a tendency to concentrate somewhere in the center of the data.

The tendency of the observations to cluster in the central part of the data is called the central tendency and the summary values as a measure of central tendency, also known as the measure of location or position, it is also known as averages too.

Note that

- The Measure of central tendency should be somewhere within the range of the data set.
- It should remain unchanged by a rearrangement of the observations in a different order.

**Criteria of a Satisfactory Measure of Location or Averages**

**Criteria of a Satisfactory Measure of Location or Averages**

There are several types of averages available to measure the representative value of a set of data or distribution. So it is desirable that an average should satisfy or possess all or most of the following conditions.

- It should be well defined i.e rigorously defined. There should be no confusion in its definition. Such as Sum of values divided by their total number is the well-defined definition of Arithmetic Mean.
- It should be based on all the observation made.
- Should be simple to understand and easy to interpret.
- Can be calculated quickly and easily.
- Should be amenable/manageable o mathematical treatment.
- Should be relatively stable in repeating sampling experiments.
- Should not be unduly influenced by abnormally large or small observations (i.e. extreme observations)

The mean, median, and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. There are several different kinds of calculations for central tendency where the kind of calculation depends on the type of the data i.e. level of measurement on which data is measured.

The following are the measure of central tendencies may be for univariate or multivariate data.

**The arithmetic mean:**The sum of all measurements divided by the number of observations in the data set**Median:**The middlemost value for sorted data. Median separates the higher half from the lower half of the data set i.e partition the data set into parts.**Mode:**The most frequent or repeated value in the data set.**Geometric mean:**The nth root of the product of the data values.**Harmonic mean:**The reciprocal of the arithmetic mean of the reciprocals of the data values**Weighted mean:**An arithmetic mean incorporating the weights to elements of certain data.**Distance-weighted estimator:**The measure uses weighting coefficients for $x_i$ that are computed as the inverse mean distance between $x_i$ and the other data points.**Truncated mean:**The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.**Midrange:**The arithmetic mean of the maximum and minimum values of a data set.**Midhinge:**The arithmetic mean of the two quartiles.**Trimean:**The weighted arithmetic mean of the median and two quartiles.**Winsorized mean:**An arithmetic mean in which extreme values are replaced by values closer to the median.

**Reference:**

1) Dodge, Y. (2003) *The Oxford Dictionary of Statistical Terms*, OUP. ISBN 0-19-920613-9

2) http://en.wikipedia.org/wiki/Central_tendency

3) Dodge, Y. (2005) The Concise Encyclopedia of Statistics. Springer,