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Unit 4: Central Tendency: Mean, Median and Mode and their Properties

5. The median of the sum or difference or pairs of corresponding items into two series is not equal Notes
to the sum or difference of the medians of the two series.
Mode
The term ‘mode’ has come from French in which it means ‘to be in fashion’. As a statistical language,
mode is the value that occurs most frequently in a statistical distribution. Thus ‘Mode’ is the most
representative average and is a position of greatest concentration of values. It has great value
conceptually. It is what the doctor means when he describes that a desease of cold and fever usually
takes a week to get cured. Similarly, average size of shirt/shoes sold, average family income etc. also
cannot be most frequently occurring value.
According to Tate, “The mode may be defined as the item which occurs most frequently in a statistical
series.”
In the words of Garrett, “Mode is that single measure or score which occurs most frequently.”
Merits
The merits are as follows:
(1) Easy to understand,
(2) Simple to calculate and locate,
(3) Quantitative data in ranking is possible, mode is very useful,
(4) It is the actual value that is in the series,
(5) Mode remains unaffected by dispersion of series,
(6) Not affected by extreme items,
(7) Can be calculated even if extreme values are not known.
Demerits

The demerits are as follows:
(1) Mode cannot be subject to further Mathematical treatment, because it is not obtained from any
alzebraic calculations,
(2) It is quite likely that there is no mode for a series,
(3) Cannot be used if relative importance of items have to be considered,
(4) Choice of grouping has a considerable influence on the value of the mode.

Harmonic mean is a type of average which has limited application only that too in a
restricted field.

Properties of mode

Assuming definedness, and for simplicity uniqueness, the following are some of the most interesting
properties.
• All three measures have the following property: If the random variable (or each value from the
sample) is subjected to the linear or affine transformation which repalces X by aX + b, so are the
mean, median and mode.
• However, if there is an arbitrary monotonic transformation, only the median follows; for
example, if X is replaced by exp (X), the median changes from m to exp (m) but the mean and
mode won’t.
• Except for extremely samples, the mode is insentive to “outliers” (such as occasional, rare, false
experimental readings). The median is also very robust in the presence of outliers, while the
mean is rather sensitive.

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