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Very often a
single number indicator does not give you much insight into the
characteristics of the data set. For example, recall the
scenario where an exceedingly wealthy person moves into your middle-class
neighbor hood. If you only use a measure of central tendency
then you will have no information that describes the
wide range of household incomes. In this scenario an additional
measure that could describe the income disparity would be useful.
This is where measures of variation come
in handy. The simplest measure of variation of a set of data
is the range. This number tells you the difference
between the highest and lowest data value so you have a feel for
how spread out the data are.
Another common
measure of variation is the standard deviation.
This number gives you a sense of how much 'on average' the
individual data values vary from the mean. For example, a
low standard deviation suggests that scores are clustered around
the mean, whereas a high standard deviation suggests that scores
are spread out around the mean.
Often we
are interested in how a particular score compares to the mean.
A Z-score is a standardized measure that
tells you how many standard deviations a particular score is from
the mean. A Z-score of 0 would mean that the score is 0 standard
deviations away from the mean and hence is equal to the mean.
A negative z-score would mean that the score is below the mean.
Percentiles
indicate what percentage of the data set fall at or below the score.
The 50th percentile equals the median. Percentiles
are often used in reporting scores on standardized exams (such as
the ACT, SAT, GRE, MCAT, etc.). For example, if you score
in the 89th percentile on the ACT then 89% of those who took the
test had scores at or below your score (i.e. you did well).
The coefficient
of variation is a measure which is used to determine what
percentage the standard deviation is of the mean.
This means that a high coefficient of variation tells you that the
standard deviation is a large percentage of the mean and so, relative
to the mean, the standard deviation is large. Hence, there
is a greater percentage of fluctuation in your data. Coefficient
of variation is used frequently in comparing two stocks. If
you monitor the stock prices for a specific time period and compute
the coefficient of variation of the data you collect, then, the
stock with the higher coefficient of variation has a higher percentage
of fluctuation (meaning it is a more volatile stock).
Pending on your investment strategy you may or may not prefer to
invest in the more volatile stock.
Recommended
Link: A Note
on Standard Deviation
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