Page 52 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes • In continuous unimodal distributions the median lies, as a rule of thumb, between the mean
and the mode, about one third of the way going from mean to mode. In a formula, median ≈ (2
× mean + mode)/3. This rule, due to Karl Pearson, often applies to slightly non-symmetric
distributions that resemble a normal distribution, but it is not always true and in general the
three statistics can appear in any order.

• For unimodal distributions, the mode is within 3 standard deviations of the mean, and the
root mean square deviation about the mode is between the standard deviation and twice the
standard deviation.
Self-Assessment
1. Fill in the blanks:
(i) The consecutive addition of frequencies is called ............
(ii) Below 10, more than 40 are the examples of ............ class-intervals.
(iii) Sum of the deviations of the items from the ............ is always zero (taking + ve and –ve signs).
(iv) n root or ‘n’ items of a series is termed as ............
th
(v) ............ of a series is the reciprocal of the arithmetic avrage of the reciprocals of the values of
its various items.
4.3 Summary

• Measures of central tendency or averages reduce the large number of observations to one figure.
Actually the measures of central tendency describe the tendency of items of group around the
middle in a frequency distributions of numerical values.
• For the average to be good, it is essential that it is capable of further alzebraic treatment, otherwise
its use will become very limited. For example, in the absence of this quality, the combined
average of two or more series from their individual averages will not be calculated. This would
hinder the possibility to study the average relationship of various parts of a variable, if it is
expressed as the sum of two or more variables.
• An average should be capable of being calculated with reasonable ease and within reasonable
time. If the time taken is long or the calculations are tedious and complicated, the average shall
have only limited use.
• Arithmetic mean is the most widely used method of calculated averages, so much so that when
only ‘mean’ is indicated it is assumed to be arithmetic mean universally. It is obtained by adding
up all the observations and dividing it by number of observations.
• Weighted arithmetic mean is the method of calculating a more representative central value and
takes into consideration the relative importance of the various figures in the series. Whereas in
simple arithmetic mean, equal weight or importance is given to each item. If the central value
has to more representative and the data is such that few items are more important than other,
the method of weighted arithmetic mean is used.
• An average rate like kilometer per hour, per day items manufactured etc. are required to be
found, harmonic mean is calculated. The harmonic mean of a series of values is the reciprocal
of the arithmetic mean of the reciprocals of the individual values. Reciprocal tables are used
with ease for this. The Harmonic Mean is less than the geometric mean of the same observations.
• The sum of the squared deviations of the items from arithmetic mean is minimum, that is less
than the sum of the squared deviations of the items from any other value.
• Median may be defined as ‘the middlemost or central value of the series when the values are
arranged in ascending or descending order of magnitude’.
• If there is an odd number of an item, then the median is found out by taking the middle most
items of the series only after arranging the data in order of magnitude.

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