Measures of Central
Tendency
A measure of central
tendency refers to the middle point or typical value around which the values in
a distribution tend to cluster. This tendency of the distribution is known as
central tendency and measures devised to consider this tendency is known as
measures of central tendency. An effective measure of central
tendency accurately represents the overall distribution of data. To serve its
purpose well, a good measure of central tendency should have the following
characteristics:
1. Clearly
Defined: The measure should have a precise and unambiguous
definition, yielding a single, consistent value for a given dataset.
2. Easy
to Understand and Compute: It should be simple to interpret
and calculate, making it accessible for general use.
3. Based
on All Observations: A reliable measure should consider every
value in the dataset, ensuring a complete representation of the distribution.
4. Capable
of Further Mathematical Treatment: It should be amenable
for further mathematical treatment.
5. Stable
Across Samples: It should be minimally affected by the
fluctuation of sampling
There are three most
commonly used measures of central tendency. These are: 1) Arithmetic Mean 2)
Median, and 3) Mode.
1)
Arithmetic Mean: This is calculated by
dividing the sum of the values in a data set by the number of values. It is
also a useful measure for further statistics and comparisons among different
data sets. However, a major limitation of arithmetic mean is that it cannot be
computed for open-ended class-intervals.
2)
Median: Median is the middle value in an
ordered data set. It divides the distribution into two equal parts so that
exactly one half of the observations is below and one half is above that point.
Because the median reflects the position rather than the magnitude of values,
it is also known as a positional average. If the number of observations
is odd, the median is the middle value; if even, it is the average of the two
middle values. The median is not affected by extreme values or outliers, making
it a robust measure of central tendency.
3)
Mode: Mode is the value in a distribution
that corresponds to the maximum concentration of frequencies. Unlike the mean
and median, the mode can be used with nominal data and may be useful in
identifying the most popular category or value.
Measures
of Dispersion
Knowing only the mean,
median, or mode of a dataset does not provide a complete understanding of how
the data is distributed. Average does not tell us about how the score or
measurements are arranged in relation to the center. It is possible that two
sets of data with equal mean or median may differ in terms of their
variability. To understand this aspect, we use measures of dispersion, which
describe the extent to which values differ from each other or from the average.
Measures of these variations are known as the ‘measures of dispersion’. The
most common measures of dispersion include range, average deviation,
quartile deviation, variance, and standard deviation.
i)
Range:
Range is one of the simplest measures of dispersion and is represented
by ‘R’. The range is defined as the difference between the largest score and
the smallest score in the distribution. A large value of range indicates higher
variability while a smaller range suggests lower variability. Range can be a
good measure if the distribution is not much skewed.
Range =Maximum Value−Minimum Value
ii)
Average deviation (Mean Deviation): Average
deviation is the arithmetic mean of the differences between each score and the
mean. This deviation is usually measured from mean or median.
Merits: It is less affected by extreme
values as compared to standard deviation. Useful for comparing the spread of
multiple datasets.
iii)
Standard deviation Standard deviation is
the most widely used and reliable measure of dispersion. In standard deviation,
instead of the actual values of the deviations we consider the squares of
deviations and the outcome is known as variance. Further, the square root of
this variance is known as standard deviation and represented as SD. Thus,
standard deviation is the square root of the mean of the squared deviations of
the individual observations from the mean.
Merits: It uses all data points in the calculation. It
is suitable for advanced statistical analysis and further mathematical
operations.