### 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.