how to find coefficient of variation

In mathematics, a coefficient is defined as an integer that is multiplied with the variable of a single element or the terms of a polynomial. It is usually a number, but sometimes may be followed by a letter in an expression. For example : \(ax^2+bx+c\)

Here x denotes the variable and 'a' and 'b' are the coefficients of the equation.

What is the coefficient of variation? In the statistics and probability approach, the coefficient of variation or CV is a measure of scattering/dispersion of given information details around the mean value. The coefficient of variation expresses the ratio of the standard deviation to the mean, and it is a helpful statistic for comparing the degree of variation from one data set to another, even if the means are drastically distinct from one another.

With this article on the coefficient of variation, we will aim to learn about cv definition, coefficient of variance formula with examples.

Coefficient of Variance

Formula

The coefficient of variation is also recognised as relative standard deviation. The coeff. of variation is represented as the ratio of standard deviation (denoted by σ) to the mean (denoted by μ). Sometimes it is stated in percentage. The coefficient of variation formula is especially practised in those cases where we require correlating results from two different studies having different values. The formula to calculate the coefficient of variation is as follows:

\( \text{Coefficient of Variation}=\frac{{\text{Standard Deviation}}}{\text{Mean}}\times100\%\)

\( \text{Coefficient of Variation}=\frac{\sigma}{\mu}\times100\%\)

Some terms related to the coefficient of variance formula

A mean in simple words can be understood as the mathematical average of a set of two or more numbers. The mean for a provided set of numbers can be estimated in more than one way, comprising the arithmetic mean method, which is practised to compute the sum of the numbers in the series, and the geometric mean approach, which is the average of a set of products. Though, all of the primary means of computing a simple average produce the same estimated result is obtained most of the time.

\( X=\frac{\sum_{i=1}^nX_i}{N}\)

Here N= Total number of observations.

There are majorly 3 distinct types of mean value that you will find in statistics.

Arithmetic Mean
Geometric Mean
Harmonic Mean

Check out this article on the Binomial Theorem.

The extent of statistical data is estimated by the standard deviation. Standard Deviation is the square root of variance. It is a measure of the extent to which data deviates from the mean. It is indicated by the symbol, 'σ'. The variance of the given data set is the average square distance between the mean value and specific data value.

The Population variance is given by the formula:

\( σ^{2}=\frac{1}{N}\sum_{i=1}^N(X_i−μ)^{2}\)

Where:

\( σ^2=\text{Population variance}\)

N = Number of observations in the population.

\( X_i=\text{ith observation in the population}\)

\( \mu=\text{Population mean or Assumed mean.}\)

The Population Standard Deviation is given by the formula:

\( σ=\sqrt{\frac{1}{N}\sum_{i=1}^N(X_i−μ)^2}\)

Where:

\( \sigma \) = Population standard deviation

The sample variance formula is as follows.

\( s^2=\frac{1}{n−1}\sum_{i=1}^n(x_i−\overline{x})^2\)

Where:

\( s^2=\text{Sample Variance}\)

n= Number of observations in the sample.

\( x_{i}=\text{ith observation in the population}.\)

\( \overline{x}=\text{Sample mean or Arithmetic mean}\)

Also, learn about Permutations and Combinations here.

The sample standard deviation formula is as follows.

\( s=\sqrt{\frac{1}{n−1}\sum_{i=1}^n(x_i−\overline{x})^2}\)

Where:

s = Sample standard deviation

How to Find Coefficient of Variation?

The Coefficient of variation formula or the cv formula, also identified as relative standard deviation (RSD), is a conventional measure of the dispersion of a probability distribution or frequency allocation. If the value of the coefficient of variation is lower it states that the data is less variability and has high stability. Let us understand the coefficient of variation with an example.

Question: Calculate the arithmetic mean if the coefficient of variation of distribution is 60 and the standard deviation is 25.

Solution:

\(CV=60\)
\(σ=25\)
\(CV=\left(\frac{\sigma}{\mu}\right)\times100\%\)
\(60=\frac{25}{\mu}\times100\%\)
\(\mu=41.66\%\)

Question: Find the coefficient of variation if the Mean is 50.1 and the standard deviation is 11.2.

Solution:

\(CV=?\)
\(σ=11.2\)
\(\mu=50.1\)
\(CV=\left(\frac{\sigma}{\mu}\right)\times100\%\)
\(CV=\left(\frac{11.2}{50.1}\right)\times100\%\)
\(CV=22.355\%\)

Applications of Coefficient of Variation

The coefficient of variation or cv is a statistical method to define the relative dispersion of data points in a data set throughout the mean.
It has major importance in finance as it enables investors to determine the uncertainty in comparison to the expected amount of revenue.
The coefficient of variation is applied to know the consistency of the data. that is the uniformity in the values of the information/distribution from the arithmetic mean of the data/number. A distribution with a smaller CV is taken as more consistent than the one with a larger CV.
C.V is also very beneficial in comparing two or more groups of data that are measured in distinct units of measurement.
The complex operative relationship may be converted into a manageable form by applying the concept of coefficient of variation.
By defining the simplified relationship between the coefficient of variation and statistical parameters, one can examine the characteristics of statistical distributions by means of the coefficient of variation.
By means of a mathematical example, it is conferred that the method of the coefficient of variation can yield an identical result to that of the Taylor expansion method.

Learn more about Sequences and Series here.

Coefficient of Variance: Key Takeaways

A coefficient is always connected to a variable. Also, a variable without a number has one as its coefficient.
Standard deviation is defined as the positive square root of variance.
Standard deviation is the indicator that displays the dispersion of the input points about the mean.
The coefficient of variation is given by the formula:

\(\text{C V}=\frac{\sigma}{\mu}\times100\%\)

The coefficient of variation or CV is a statistical method of the relative dispersion of information provided in a data set throughout the mean.
In banking and finance, the coefficient of variation enables investors to determine how much volatility, or uncertainty, is expected in comparison to the amount of return expected from investments.
The lower is the ratio of the standard deviation to mean return, the more reliable the risk-return trade-off.
The coefficient of variation formula can be practised to determine the deviation between the traditional mean price and the current price performance of a property, stock, or bond, related to other assets.

We hope that the above article on Coefficient of Variation is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

Coefficient of Variation FAQs

Q.1 What is the coefficient of variation in statistics?

Ans.1 The coefficient of variation also known as CV is the ratio of the standard deviation to the mean. The higher the coefficient of variation, the larger the level of dispersion around the mean.

Q.2 Why do we calculate the coefficient of variation?

Ans.2 The coefficient of variation shows the extent of variability of data in a sample about the mean of the population. In banking and finance, the coefficient of variation enables investors to determine how much volatility, or uncertainty, is expected in comparison to the amount of return expected from investments.

Q.3 What is the formula of coefficient of variation?

Ans.3 \( \text{C V}=\frac{\sigma}{\mu}\times100\%\)

Q.4 What does the coefficient of variation tell you?

Ans.4 The coefficient of variation is applied to know the consistency of the data. that is the uniformity in the values of the information/distribution from the arithmetic mean of the data/number. A distribution with a smaller CV is taken as more consistent than the one with a larger CV.

Q.5 What is the use of the coefficient of variation?

Ans.5 C.V is also very beneficial in comparing two or more groups of data that are measured in distinct units of measurement. The complicated operative relationship may be converted into a simple form by applying the concept of coefficient of variation. The coefficient of variation formula or calculation can be practised to determine the deviation between the traditional mean price and the current price performance of a property, stock, or bond, related to other assets.

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