### Statistics Phase 4 -: Dilemma of Variation and Coefficient of Variation

__MEASURE OF VARIABILITY__**VARIANCE**

Variance measures the dispersion/distance of a set
of data points around their mean.

It is the difference between the expected and the
actual results such as that between budget and actual expenditure.

First lets talk about the numerator. It’s the square of the sum of difference between
the observations and the mean. Closer the number to the mean, the lower the
result will be and vice versa.

The reasons why we do squaring are so that we do
not obtain a negative variance because distance is never negative and to
amplify the effect of large differences.

The only difference in the formula is because of
the denominator as we subtract 1 from the sample size in sample variance but
not in population variance.

Due to this, sample variance is always bigger than
the population variance justifying the fact that there is more uncertainty
about sample variance because we have randomly drawn a sample from a
population.

Let’s take an example of a stock market or other
investment returns. The stock market has return on average 7% per year. This
does not mean that every year you get a 7% return, some years you get more and
some years less. This variability (volatility in stock) is an example of
variance.

Let’s have one more example. You drive to work
every day and take the same route. There is both variation in the time it takes
you to get to work (traffic, stop light timing etc.) and variation in the
amount of gas you use(traffic may affect this). All of this variability can be
measured with variance.

Basically, anywhere you see movements away from
the centre of something it can be measured with variance.

**STANDARD DEVIATION**

It is the measure of dispersion of a set of data
from its mean. The higher the dispersion, the greater is the standard
deviation.

Standard deviation is the square root of variance.

In financial terms, standard deviation is used to
measure the risks involved in an investment instrument.

**COEFFICIENT OF VARIATION**

It is also known as relative standard deviation
meaning standard deviation relative to its mean.

We can use this while comparing the variability of
2 or more series. The series of data for which the coefficient of variation is
large indicates that the group is more variable and it is less stable or less
uniform.

For example: A researcher is comparing 2 multiple-choice
tests with different conditions. Trying to compare the 2 test results is
challenging. Comparing standard deviations doesn’t really work, because the
means might be different. So in this case calculating coefficient of variation
does make some sense here.

Also it does not have a unit of measurement cause
it is simply a comparison.

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**credits: Simran Agrawal**
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