#### 3.1.1 Measures of Central Tendency

Central tendency refers to the middle point of a distribution. Its aim is to
characterize the statistical data by a single number, representing the order of
magnitude of the whole set of observations:

##### Arithmetic mean

The most commonly used measure of the central tendency is the arithmetic
mean or simple average. The arithmetic mean of a variable *X* with N
observations *X*_{i} is the ratio of the sum of all observations divided
by the number of observations, *N*.

This measure is commonly used in statistical inference procedures. A major
disadvantage of this measure is its sensitivity to departure from symmetry
between the left and right extremes of the distribution. In such cases, the
preferable measure of centrality is the *Median.*

##### Median

The value of the variable *X* that splits the sample into two halves is
called the *Median*. Fifty percent of cases fall below the median and
fifty percent fall above the median. If there is an even number of cases, the
median is the average of the two middle cases when they are ordered in
ascending or descending order.

If the median value is very different from the mean, then the distribution
of the data is *skewed*.

The median has certain disadvantages. Certain statistical procedures that
use the median are more complex than those, which use the mean.

##### Mode

The mode is the value that occurs in the data set with the highest
frequency. If several values share the highest frequency, each of them is a
mode. Like the median, the mode is not unduly affected by extreme values.

If the data come from a normal distribution, the mean, median and mode are
all equal. If mean and median are very different, most likely there are
outliers or the distribution is skewed. If this is the case, then the median is
probably a better measure of location. The mean is very sensitive to extreme
values and can be seriously contaminate even by one observation!

##### Quantiles

Consider a variable, whose values *X*_{i} are set in ascending
order. The quantiles, denoted by *q*_{1}, *q*_{2},*
q*_{p–1} are the values of *X* that divide the objects
into *p* equal parts.

The *deciles*, denoted by *q*_{1}*, q*_{2}*,……,q*_{9}
are the values of *X* that divide the objects into ten equal parts.
Similarly, the *quartiles* denoted by *q*_{1}*, q*_{2}*,
q*_{3} are the values of *X* that divide the objects into four
equal parts.