?It is useful to summarize data by methods that lead to numerical results, called
descriptive measures. We discuss two types of descriptive measures: measures of
central tendency and measures of dispersion. They may be computed from the data
contained in a sample or from the data of a finite population. A descriptive measure
computed from or used to describe a sample of data is called a statistic while a
descriptive measure computed from or used to describe a population of data is called a
Even when you draw a collection of data from a common source, individual
observations are not likely to have the same value. It is impractical to keep in mind all
the values that may be present in a set of data. What we need is some single value that
we may consider typical of the set of data as a whole. The need for such a single value
is usually met by Links Of London Charms one of the three measures of central
tendency: the arithmetic mean, the median, and the mode.
The Arithmetic Mean. The most familiar measure of central tendency is the arithmetic
mean. Popularly known as the average, it is sometimes called the arithmetic average,
or simply the mean. We find it by adding all the values in a set of data and dividing
the total by the number of values that were summed.
The properties of the arithmetic mean include the following: (1) For a given set of
data, there is one, and only one, arithmetic mean. (2) Its meaning is easily understood.
(3) Since every value goes into its computation, it is affected by the magnitude of
each value. Because of this property, the arithmetic mean may not be the best measure
of central tendency when one or two extreme values are present in a set of data. And
(4) the mean, unlike some descriptive measures whose values may be determined by
inspection, is a computed measure, and therefore it can be manipulated algebraically.
This property makes it an especially useful measure for statistical inference purposes.
The Weighted Mean. When the frequency of occurrence of the individual
measurements to be averaged varies, we may refer to the frequencies as weights and
to the resulting mean as a weighted mean. Sometimes the measurements to be
averaged vary in importance rather than frequency of occurrence. In such cases, a
weighted mean will provide an average that reflects the relative importance of the
The Median. The median is that value above which half the values lie and below
which the other half lie. If the number of items is odd, the median is the value of the
middle item of an ordered array, when the items are arranged in ascending (or
descending) order of magnitude. If the number of items is even, none of the items has
an equal number of values above and below it. In this event, the median is equal the
mean, or average, of the two middle values. The Mode. The mode for ungrouped
discrete data is the value that occurs most frequently. If all the values in a set of data
are different, there is no mode.
The Geometric Mean
There are some problems requiring the calculation of an average for which none of
the averages discussed so far is appropriate. For example, when we wish to obtain the
Links Of London Bracelets average value of a series of ratios, percentages, or rates of
change, the arithmetic mean proves to be an inadequate choice for the job. The
measure needed in these situations is the geometric mean. The geometric mean of a
series of n measurements is the nth root of the product of the n measurements.
Calculating the geometric mean according to its basic definition can be a laborious
task. By the use of logarithms, however, the measure may be computed with relative
The following are some characteristics of the geometric mean: (1) It is not unduly
influenced by extreme values. (2) It is always smaller than the arithmetic mean. (3) It
is a meaningful measure only when all of the measurements are positive. And (4) the
product of a series of measurements remains unchanged if the geometric mean of the
measurements is substituted for each measurement in the series. The Harmonic Mean
mother average that is preferred over other such measures in certain situations is the
harmonic mean. The harmonic mean of a series of measurements is the reciprocal of
the arithmetic mean) f the reciprocals of the individual measurements. The harmonic
mean is the average of choice when the average of time rates is required. It lass
decided advantages when the data to be averaged are certain types of price data,
^consequently, the harmonic mean finds frequent use in the field of economics.