A set of values with mean 20 and its standard deviation is 6 what is the coefficient of variation

Coefficient of Standard Deviation

The standard deviation is the absolute measure of dispersion. Its relative measure is called the standard coefficient of dispersion or coefficient of standard deviation. It is defined as:

\[{\text{Coefficient}}\,{\text{of}}\,{\text{Standard}}\,{\text{Deviation}} = \frac{S}{{\overline X }}\]

Coefficient of Variation

The most important of all the relative measures of dispersion is the coefficient of variation. This word is variation not variance. There is no such thing as coefficient of variance. The coefficient of variation $$\left( {C.V} \right)$$ is defined as:
\[\left( {C.V} \right) = \frac{S}{{\overline X }} \times 100\]

Thus $$C.V$$ is the value of $$S$$ when $$\overline X $$ is assumed equal to 100. It is a pure number and the unit of observation is not mentioned with its value. It is written in percentage form like 20% or 25%. When its value is 20%, it means that when the mean of the observations is assumed equal to 100, their standard deviation will be 20. The $$C.V$$ is used to compare the dispersion in different sets of data particularly the data which differ in their means or differ in their units of measurement. The wages of workers may be in dollars and the consumption of meat in families may be in kilograms. The standard deviation of wages in dollars cannot be compared with the standard deviation of amount of meat in kilograms. Both the standard deviations need to be converted into a coefficient of variation for comparison. Suppose the value of $$C.V$$ for wages is 10% and the values of $$C.V$$ for kilograms of meat is 25%. This means that the wages of workers are consistent. Their wages are close to the overall average of their wages. But the families consume meat in quite different quantities. Some families consume very small quantities of meat and some others consume large quantities of meat. We say that there is greater variation in their consumption of meat. The observations about the quantity of meat are more dispersed or more variant.

Example:

Calculate the coefficient of standard deviation and coefficient of variation for the following sample data: 2, 4, 8, 6, 10, and 12.

Solution:

$$X$$	$${\left( {X – \overline X } \right)^2}$$
$$2$$	$${(2 – 7)^2} = 25$$
$$4$$	$${(4 – 7)^2} = 9$$
$$8$$	$${(8 – 7)^2} = 1$$
$$6$$	$${(6 – 7)^2} = 1$$
$$10$$	$${(10 – 7)^2} = 9$$
$$12$$	$${(12 – 7)^2} = 25$$
$$\sum X = 42$$	$$\sum {\left( {X – \overline X } \right)^2} = 70$$

$$\overline X = \frac{{\sum X}}{n} = \frac{{42}}{6} = 7$$ $$S = \sqrt {\frac{{\sum {{\left( {X – \overline X } \right)}^2}}}{n}} $$

$$S = \sqrt {\frac{{70}}{6}} = \sqrt {\frac{{35}}{3}} = 3.42$$

Coefficient of Standard Deviation $$ = \frac{S}{{\overline X }} = \frac{{3.42}}{7} = 0.49$$

Coefficient of Variation \[\left( {C.V} \right) = \frac{S}{{\overline X }} \times 100 = \frac{{3.42}}{7} \times 100 = 48.86\% \]

Example:

Calculate the coefficient of standard deviation and coefficient of variation from the following distribution of marks:

Marks	No. of Students
$$1 – 3$$	$$40$$
$$3 – 5$$	$$30$$
$$5 – 7$$	$$20$$
$$7 – 9$$	$$10$$

Solution:

Marks	$$f$$	$$X$$	$$fX$$	$${\left( {X – \overline X } \right)^2}$$	$$f{\left( {X – \overline X } \right)^2}$$
$$1 – 3$$	$$40$$	$$2$$	$$80$$	$$4$$	$$160$$
$$3 – 5$$	$$30$$	$$4$$	$$120$$	$$0$$	$$0$$
$$5 – 7$$	$$20$$	$$6$$	$$120$$	$$4$$	$$80$$
$$7 – 9$$	$$10$$	$$8$$	$$80$$	$$16$$	$$160$$
Total	$$100$$		$$400$$		$$400$$

$$\overline X = \frac{{\sum fX}}{{\sum f}} = \frac{{400}}{{100}} = 4$$
$$S = \sqrt {\frac{{\sum f{{\left( {X – \overline X } \right)}^2}}}{{\sum f}}} = \sqrt {\frac{{400}}{{100}}} = \sqrt 4 = 2$$ marks

Coefficient of Standard Deviation $$ = \frac{S}{{\overline X }} = \frac{2}{4} = 0.5$$

Coefficient of Variation \[\left( {C.V} \right) = \frac{S}{{\overline X }} \times 100 = \frac{2}{4} \times 100 = 50\% \]

Question

The mean of a distribution is 20 and the standard deviation is 5.
What is the value of the coefficient of variation?

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Statistics Xtramous 2 years 43 Answers 3265 views Contributor 0

The co-efficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. The co-efficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.

The co-efficient of variation (CV) is a statistical measure of the relative dispersion of data points in a data series around the mean.
It represents the ratio of the standard deviation to the mean.
The CV is useful for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.
In finance, the co-efficient of variation allows investors to determine how much volatility, or risk, is assumed in comparison to the amount of return expected from investments.
The lower the ratio of the standard deviation to mean return, the better risk-return tradeoff.

The co-efficient of variation shows the extent of variability of data in a sample in relation to the mean of the population.

In finance, the co-efficient of variation allows investors to determine how much volatility, or risk, is assumed in comparison to the amount of return expected from investments. Ideally, if the co-efficient of variation formula should result in a lower ratio of the standard deviation to mean return, then the better the risk-return tradeoff.

They’re most often used to analyze dispersion around the mean, but quartile, quintile, or decile CVs can also be used to understand variation around the median or 10th percentile, for example.

The co-efficient of variation formula or calculation can be used to determine the deviation between the historical mean price and the current price performance of a stock, commodity, or bond, relative to other assets.

Below is the formula for how to calculate the co-efficient of variation:

$\begin{aligned} &\text{CV} = \frac { \sigma }{ \mu } \\ &\textbf{where:} \\ &\sigma = \text{standard deviation} \\ &\mu = \text{mean} \\ \end{aligned}$

To calculate the CV for a sample, the formula is:

$C V = s / x * 100$
where:
s = sample
x̄ = mean for the population

Multiplying the co-efficient by 100 is an optional step to get a percentage rather than a decimal.

The co-efficient of variation formula can be performed in Excel by first using the standard deviation function for a data set. Next, calculate the mean by using the Excel function provided. Since the co-efficient of variation is the standard deviation divided by the mean, divide the cell containing the standard deviation by the cell containing the mean.

The standard deviation is a statistic that measures the dispersion of a data set relative to its mean. It is used to determine the spread of values in a single data set rather than to compare different units.

When we want to compare two or more data sets, the co-efficient of variation is used. The CV is the ratio of the standard deviation to the mean. And because it’s independent of the unit in which the measurement was taken, it can be used to compare data sets with different units or widely different means.

In short, the standard deviation measures how far the average value lies from the mean, whereas the co-efficient of variation measures the ratio of the standard deviation to the mean.

The co-efficient of variation can be useful when comparing data sets with different units or widely different means.

That includes when the risk/reward ratio is used to select investments. For example, an investor who is risk-averse may want to consider assets with a historically low degree of volatility relative to the return, in relation to the overall market or its industry. Conversely, risk-seeking investors may look to invest in assets with a historically high degree of volatility.

When the mean value is close to zero, the CV becomes very sensitive to small changes in the mean. Using the example above, a notable flaw would be if the expected return in the denominator is negative or zero. In this case, the co-efficient of variation could be misleading.

If the expected return in the denominator of the co-efficient of variation formula is negative or zero, then the result could be misleading.

The co-efficient of variation is used in many different fields, including chemistry, engineering, physics, economics, and neuroscience.

Other than helping when using the risk/reward ratio to select investments, it is used by economists to measure economic inequality. Outside of finance, it is commonly applied to audit the precision of a particular process and arrive at a perfect balance.

For example, consider a risk-averse investor who wishes to invest in an exchange-traded fund (ETF), which is a basket of securities that tracks a broad market index. The investor selects the SPDR S&P 500 ETF, the Invesco QQQ ETF, and the iShares Russell 2000 ETF. Then, they analyze the ETFs’ returns and volatility over the past 15 years and assumes that the ETFs could have similar returns to their long-term averages.

For illustrative purposes, the following 15-year historical information is used for the investor’s decision:

If the SPDR S&P 500 ETF has an average annual return of 5.47% and a standard deviation of 14.68%, the SPDR S&P 500 ETF’s co-efficient of variation is 2.68.
If the Invesco QQQ ETF has an average annual return of 6.88% and a standard deviation of 21.31%, the QQQ’s co-efficient of variation is 3.10.
If the iShares Russell 2000 ETF has an average annual return of 7.16% and a standard deviation of 19.46%, the IWM’s co-efficient of variation is 2.72.

Based on the approximate figures, the investor could invest in either the SPDR S&P 500 ETF or the iShares Russell 2000 ETF, since the risk/reward ratios are approximately the same and indicate a better risk-return tradeoff than the Invesco QQQ ETF.

The co-efficient of variation (CV) indicates the size of a standard deviation in relation to its mean. The higher the co-efficient of variation, the greater the dispersion level around the mean.

That depends on what you’re looking at and comparing. No set value can be considered universally “good.” However, generally speaking, it is often the case that a lower co-efficient of variation is more desirable, as that would suggest a lower spread of data values relative to the mean.

To calculate the co-efficient of variation, first find the mean, then the sum of squares, and then work out the standard deviation. With that information at hand, it is possible to calculate the co-efficient of variation by dividing the standard deviation by the mean.

The co-efficient of variation is a simple way to compare the degree of variation from one data series to another. It can be applied to pretty much anything, including the process of picking suitable investments.

Generally speaking, a high CV indicates that the group is more variable, whereas a low value would suggest the opposite.