If $\rm \sum x_i = 20, \sum x_i^2=200$ and n = 10 for an observed variable x, then what is the coefficient of variation?

Question

If $\rm \sum x_i = 20, \sum x_i^2=200$  and n = 10 for an observed variable x, then what is the coefficient of variation?

Prepp Team · Accepted Answer

Understanding the Coefficient of Variation

The coefficient of variation (CV) is a statistical measure of the relative variability of data. It is the ratio of the standard deviation to the mean, often expressed as a percentage. It helps compare the degree of variation between data sets, even if their means are drastically different.

Coefficient of Variation Formula

The formula for the coefficient of variation (CV) is:

$ \text{CV} = \frac{\sigma}{\bar{x}} \times 100\% $

Where:

$ \sigma $ is the standard deviation
$ \bar{x} $ is the mean

To find the coefficient of variation, we first need to calculate the mean ($ \bar{x} $) and the standard deviation ($ \sigma $) from the given data.

Calculating the Mean ($ \bar{x} $)

The mean is the average of the observations. The formula for the mean is:

$ \bar{x} = \frac{\sum x_i}{n} $

Given:

$ \sum x_i = 20 $
$ n = 10 $

Substitute the values into the formula:

$ \bar{x} = \frac{20}{10} $

$ \bar{x} = 2 $

So, the mean of the observed variable x is 2.

Calculating the Standard Deviation ($ \sigma $)

The standard deviation is the square root of the variance. The variance ($ \sigma^2 $) can be calculated using the formula:

$ \sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2 $

Given:

$ \sum x_i^2 = 200 $
$ n = 10 $
$ \bar{x} = 2 $ (calculated above)

Substitute the values into the variance formula:

$ \sigma^2 = \frac{200}{10} - (2)^2 $

$ \sigma^2 = 20 - 4 $

$ \sigma^2 = 16 $

Now, calculate the standard deviation ($ \sigma $) by taking the square root of the variance:

$ \sigma = \sqrt{\sigma^2} = \sqrt{16} $

$ \sigma = 4 $

The standard deviation is 4.

Calculating the Coefficient of Variation

Now that we have the standard deviation ($ \sigma = 4 $) and the mean ($ \bar{x} = 2 $), we can calculate the coefficient of variation:

$ \text{CV} = \frac{\sigma}{\bar{x}} \times 100 $

$ \text{CV} = \frac{4}{2} \times 100 $

$ \text{CV} = 2 \times 100 $

$ \text{CV} = 200 $

The coefficient of variation is 200. The options are given as numerical values without the percentage symbol, so the answer is 200.

Statistic	Formula	Calculation	Result
Mean ($\bar{x}$)	$\frac{\sum x_i}{n}$	$\frac{20}{10}$	2
Variance ($\sigma^2$)	$\frac{\sum x_i^2}{n} - (\bar{x})^2$	$\frac{200}{10} - (2)^2 = 20 - 4$	16
Standard Deviation ($\sigma$)	$\sqrt{\sigma^2}$	$\sqrt{16}$	4
Coefficient of Variation (CV)	$\frac{\sigma}{\bar{x}} \times 100$	$\frac{4}{2} \times 100$	200

Final Coefficient of Variation Result

Based on the calculations, the coefficient of variation is 200.

Revision Table: Key Statistics Concepts

Concept	Definition	Formula
Mean	The average value of a dataset.	$\bar{x} = \frac{\sum x_i}{n}$
Variance	A measure of how spread out the data points are from the mean (average of squared differences).	$\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ or $\frac{\sum x_i^2}{n} - (\bar{x})^2$
Standard Deviation	The square root of the variance; measures the typical distance of data points from the mean.	$\sigma = \sqrt{\sigma^2}$
Coefficient of Variation	A standardized measure of dispersion of a probability distribution or frequency distribution.	$\text{CV} = \frac{\sigma}{\bar{x}} \times 100\%$

Additional Information: Using Coefficient of Variation

The coefficient of variation is a useful tool when comparing datasets with different scales or units. For example, it can be used to compare the variability in heights (measured in meters) and weights (measured in kilograms) within a population. A higher CV indicates greater variability relative to the mean, while a lower CV indicates less relative variability.

It is particularly valuable in fields like finance to compare the risk (measured by standard deviation) of investments with different expected returns (measured by the mean). A lower CV might indicate a more favorable risk-return trade-off.

However, the coefficient of variation should be used cautiously, especially when the mean ($ \bar{x} $) is close to zero, as small changes in the mean can lead to large fluctuations in the CV.

Prepp Team · Answer

80

Prepp Team · Answer

100

Prepp Team · Answer

150

Statistic	Formula	Calculation	Result
Mean (\(\bar{x}\))	\(\frac{\sum x_i}{n}\)	\(\frac{20}{10}\)	2
Variance (\(\sigma^2\))	\(\frac{\sum x_i^2}{n} - (\bar{x})^2\)	\(\frac{200}{10} - (2)^2 = 20 - 4\)	16
Standard Deviation (\(\sigma\))	\(\sqrt{\sigma^2}\)	\(\sqrt{16}\)	4
Coefficient of Variation (CV)	\(\frac{\sigma}{\bar{x}} \times 100\)	\(\frac{4}{2} \times 100\)	200

	Expenditure (in Rs.)
Items	Family A	Family B
Food	3,500	2,700
Clothing	500	800
Rent	1,500	1,000
Education	2,000	1,800
Miscellaneous	2,500	1,800

Age	Frequency
15 – 25	2
25 – 35	4
35 – 45	6
45 – 55	5
55 - 65	3

Understanding the Coefficient of Variation

Coefficient of Variation Formula

Calculating the Mean (\( \bar{x} \))

Calculating the Standard Deviation (\( \sigma \))

Calculating the Coefficient of Variation

Final Coefficient of Variation Result

Revision Table: Key Statistics Concepts

Additional Information: Using Coefficient of Variation

Similar Questions

Important Questions from Variance and Standard Deviation

NDA 2026 Mock Test Series (Latest Pattern)

NDA Mock Test Series for Maths & GAT

Start Your Preparation with Prepp Mobile App

Concept	Definition	Formula
Mean	The average value of a dataset.	\(\bar{x} = \frac{\sum x_i}{n}\)
Variance	A measure of how spread out the data points are from the mean (average of squared differences).	\(\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}\) or \(\frac{\sum x_i^2}{n} - (\bar{x})^2\)
Standard Deviation	The square root of the variance; measures the typical distance of data points from the mean.	\(\sigma = \sqrt{\sigma^2}\)
Coefficient of Variation	A standardized measure of dispersion of a probability distribution or frequency distribution.	\(\text{CV} = \frac{\sigma}{\bar{x}} \times 100\%\)