The mean of a distribution is 21 and the standard deviation is 7. What is the value of the coefficient variation?
33.33%
Calculating the Coefficient of Variation
The question asks us to find the value of the coefficient of variation for a distribution, given its mean and standard deviation. The coefficient of variation (CV) is a statistical measure that expresses the standard deviation as a percentage of the mean. It is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often used to compare the degree of variation between data sets, even if their means are drastically different.
Understanding Coefficient of Variation
The formula for the coefficient of variation (CV) is:
$$ \text{CV} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100\% $$
In this problem, we are given:
- Mean ($\bar{x}$) = 21
- Standard Deviation ($\sigma$) = 7
Step-by-Step Calculation
Let's substitute the given values into the formula for the coefficient of variation:
$$ \text{CV} = \left( \frac{7}{21} \right) \times 100\% $$
First, simplify the fraction $\frac{7}{21}$:
$$ \frac{7}{21} = \frac{1}{3} $$
Now, multiply the simplified fraction by 100%:
$$ \text{CV} = \frac{1}{3} \times 100\% $$
$$ \text{CV} \approx 0.3333 \times 100\% $$
$$ \text{CV} \approx 33.33\% $$
So, the coefficient of variation is approximately 33.33%.
Analyzing the Coefficient of Variation Result
A coefficient of variation of 33.33% tells us that the standard deviation is about one-third of the mean. This provides a relative measure of the dispersion of the data points around the mean. A lower coefficient of variation indicates less variability relative to the mean, while a higher one indicates more variability.
| Measure | Value |
|---|---|
| Mean ($\bar{x}$) | 21 |
| Standard Deviation ($\sigma$) | 7 |
| Coefficient of Variation (CV) | 33.33% (approx.) |
Comparing with Options
Let's compare our calculated value with the given options:
- 16.66%
- 66.66%
- 33.33%
- 100%
Our calculated value, 33.33%, matches option 3.
Revision Table: Key Statistical Concepts
| Concept | Definition | Formula | Purpose |
|---|---|---|---|
| Mean | The average of a dataset. | Sum of values / Number of values | Measure of central tendency. |
| Standard Deviation | A measure of the amount of variation or dispersion of a set of values. | $$ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} $$ (for sample) | Measure of absolute dispersion. |
| Coefficient of Variation | A measure of relative variability, standard deviation divided by the mean, often expressed as a percentage. | $$ \text{CV} = \left( \frac{\sigma}{\bar{x}} \right) \times 100\% $$ | Measure of relative dispersion, useful for comparing different datasets. |
Additional Information: Uses of Coefficient of Variation
The coefficient of variation is particularly useful in the following situations:
- Comparing Variability: When comparing datasets with different units or vastly different means, CV provides a standardized way to assess which dataset has more relative variability. For example, comparing the variability in heights of adults versus infants.
- Risk Assessment: In finance, CV can be used to compare the risk (measured by standard deviation) and return (measured by mean) of different investments. A lower CV suggests a better risk-return trade-off.
- Quality Control: It can be used to monitor the consistency of a production process. A low CV indicates high consistency.
It's important to note that the coefficient of variation is only meaningful when the mean is a non-zero value. If the mean is zero or very close to zero, the coefficient of variation can be misleading or undefined.
Important Questions from Variance and Standard Deviation
The sum of deviations of n numbers from 10 and 20 are p and q respectively. If (p - q)2 = 10000, then what is the value of n?
The mean and the variance of 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively.
If the data are moderately non-symmetrical, then which one of the following empirical relationships is correct?
If the total number of observations is 20, ∑ x i= 1000 and \(\sum {\rm{x}}_{\rm{i}}^2 = 84000\) , then what is the variance of the distribution?
Among these options, which one is NOT an example of relative measure of dispersion?
